diff --git a/problems/PA-05/Atcoder/abc184_f/editorial.md b/problems/PA-05/Atcoder/abc184_f/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..54e92791b3571fc78d44b1dcb5342072b452724c --- /dev/null +++ b/problems/PA-05/Atcoder/abc184_f/editorial.md @@ -0,0 +1,21 @@ +# F - Programming Contest (editorial) + +Source: https://atcoder.jp/contests/abc184/editorial/376 by evima + +We can see that this problem is NP-hard, since it is a difficult version of the subset sum problem. + +Let us focus on the fairly small \(N\) (\(N \leq 40\)). + +If you search all the combinations of tasks to solve, it takes \(O(2^N)\) and it will TLE. Instead, we can use a technique called “meet-in-the-middle” searching. + +1. Split a sequence \(A\) into two so that the number of elements are halved. Let us call them \(B\) and \(C\). +2. Enumerate all the possible sums of combinations from \(B\) and sort them, which we will call \(D\). It takes \(O(N 2^{N/2})\) time for this. +3. Brute force all the possible combinations from \(C\) and obtain the maximum value, with \(B\) combined. Here, the optimal choice of \(B\) can be found by performing binary searching on \(D\). The time complexity for these operations are \(O(N 2^{N/2})\). + +Now, the problem can be solved in a total of \(O(N 2^{N/2})\) time. + +We can improve each step so that the problem can be solved in a total of \(O(2^{N/2})\) time. + +Let \(D_i\) be a sorted sequence, each of whose element is a possible sum of choices from \(B_1, B_2, \ldots, B_i\). Initially, \(D_0 = [0]\). \(D_x\) can be obtained by merging \(D_{x-1}\) and “\(D_{x-1}\), each of whose element is added by \(B_x\)”. Since \(|D_0| + |D_1| + |D_2| + \cdots + |D_x| \leq 1 + 2 + 4 + \cdots + 2^x = O(2^x)\), we can obtain \(D_{N/2}\) in a \(O(2^{N/2})\) time. + +Enumerate all the possible sums of choices from \(B\), enumerate all the possible sums of choices from \(C\) too, and combine them with sliding windows, so that the problem can be solved in a total of \(O(2^{N/2})\) time. diff --git a/problems/PA-05/Atcoder/abc184_f/judge.json b/problems/PA-05/Atcoder/abc184_f/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..723601f7c4c1f753ade339b1139769b44c2172d5 --- /dev/null +++ b/problems/PA-05/Atcoder/abc184_f/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 35, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc184_f/meta.json b/problems/PA-05/Atcoder/abc184_f/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..76011982f82528e7c57c8c005ad6de1d7a7b22a6 --- /dev/null +++ b/problems/PA-05/Atcoder/abc184_f/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc184_f", + "contest": "abc184", + "type": "ABC", + "letter": "F", + "title": "Programming Contest", + "url": "https://atcoder.jp/contests/abc184/tasks/abc184_f", + "difficulty": 1423, + "score": 600, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 35, + "editorial_url": "https://atcoder.jp/contests/abc184/editorial/376", + "code_source": "model", + "pid": "A1482" +} diff --git a/problems/PA-05/Atcoder/abc184_f/solution.cpp b/problems/PA-05/Atcoder/abc184_f/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..15735a82de6952bd39b83f16fe08a202efbef5e8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc184_f/solution.cpp @@ -0,0 +1,52 @@ +// abc184_f: F - Programming Contest +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc184/editorial/376 +// Judged AC on all 35 tests, max 0.04 s of the 2 s limit (submission 1). + +#include +using namespace std; + +using ll = long long; + +vector subset_sums(const vector& a, int l, int r) { + vector sums{0}; + for (int k = l; k < r; ++k) { + size_t n = sums.size(); + vector next; + next.reserve(2 * n); + size_t i = 0, j = 0; + while (i < n || j < n) { + if (j == n || (i < n && sums[i] <= sums[j] + a[k])) { + next.push_back(sums[i++]); + } else { + next.push_back(sums[j++] + a[k]); + } + } + sums.swap(next); + } + return sums; +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int N; + ll T; + cin >> N >> T; + vector A(N); + for (ll& x : A) cin >> x; + + vector left = subset_sums(A, 0, N / 2); + vector right = subset_sums(A, N / 2, N); + + ll answer = 0; + int j = static_cast(right.size()) - 1; + for (ll x : left) { + while (j >= 0 && x + right[j] > T) --j; + if (j < 0) break; + answer = max(answer, x + right[j]); + if (answer == T) break; + } + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/abc184_f/statement.md b/problems/PA-05/Atcoder/abc184_f/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..b65bafaf89e4b5b2ac2a3c72f12f7b8058af95a1 --- /dev/null +++ b/problems/PA-05/Atcoder/abc184_f/statement.md @@ -0,0 +1,79 @@ +# F - Programming Contest + +**Score:** 600 points + +## Problem Statement +Takahashi will participate in a programming contest, which lasts for \(T\) minutes and presents \(N\) problems.
+With his extrasensory perception, he already knows that it will take \(A_i\) minutes to solve the \(i\)-th problem.
+He will choose zero or more problems to solve from the \(N\) problems so that it takes him no longer than \(T\) minutes in total to solve them.
+Find the longest possible time it takes him to solve his choice of problems. + +## Constraints +- All values in input are integers. +- \(1 \le N \le 40\) +- \(1 \le T \le 10^9\) +- \(1 \le A_i \le 10^9\) + +## Input +Input is given from Standard Input in the following format: +``` +N T +A_1 ... A_N +``` + +## Output +Print the answer as an integer. + +## Sample Input 1 +``` +5 17 +2 3 5 7 11 +``` + +### Sample Output 1 +``` +17 +``` +>If he chooses the 1-st, 2-nd, 3-rd, and 4-th problems, it takes him \(2+3+5+7=17\) minutes in total to solve them, which is the longest possible time not exceeding \(T=17\) minutes. + +--- + +## Sample Input 2 +``` +6 100 +1 2 7 5 8 10 +``` + +### Sample Output 2 +``` +33 +``` +>It is optimal to solve all the problems. + +--- + +## Sample Input 3 +``` +6 100 +101 102 103 104 105 106 +``` + +### Sample Output 3 +``` +0 +``` +>He cannot solve any of the problems. + +--- + +## Sample Input 4 +``` +7 273599681 +6706927 91566569 89131517 71069699 75200339 98298649 92857057 +``` + +### Sample Output 4 +``` +273555143 +``` +>If he chooses the 2-nd, 3-rd, and 7-th problems, it takes him \(273555143\) minutes in total to solve them. diff --git a/problems/PA-05/Atcoder/abc196_d/data/1.in b/problems/PA-05/Atcoder/abc196_d/data/1.in new file mode 100644 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0000000000000000000000000000000000000000..dbc2c13911b1ddfc282993e5ac01a540bf227522 --- /dev/null +++ b/problems/PA-05/Atcoder/abc196_d/editorial.md @@ -0,0 +1,13 @@ +# D - Hanjo (editorial) + +Source: https://atcoder.jp/contests/abc196/editorial/954 by en_translator + +Since the constraints are small, where \(HW \le 16\), so let us consider searching the way of covering exhaustively. + Starting from the top-left cell, perform DFS of trying placing a square tatami mat, a rectangular tatami mat that sticks out either to the left or below; then we can exhaustively search all the ways of covering. + +Now let’s estimate the time complexity. + We have the choices of the top-left cell and the direction of tatami mats, so we have the upper bound of answer of \(2^A\binom{HW}A\). + Since \(HW2^A\binom{HW}A\) is at most \(5 \times 10^7\), so it is fast enough. + In fact, the maximum answer is \(3388\), so it works much faster than estimated. + +BONUS: Solve for \(HW ≤ 200\)! diff --git a/problems/PA-05/Atcoder/abc196_d/judge.json b/problems/PA-05/Atcoder/abc196_d/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..58ff3019207ec11dc97ade6dd27e9057b9266512 --- /dev/null +++ b/problems/PA-05/Atcoder/abc196_d/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 50, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc196_d/meta.json b/problems/PA-05/Atcoder/abc196_d/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..e617274f5f866e82e79ae481c5936dbbd5c9c9fa --- /dev/null +++ b/problems/PA-05/Atcoder/abc196_d/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc196_d", + "contest": "abc196", + "type": "ABC", + "letter": "D", + "title": "Hanjo", + "url": "https://atcoder.jp/contests/abc196/tasks/abc196_d", + "difficulty": 1277, + "score": 400, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 50, + "editorial_url": "https://atcoder.jp/contests/abc196/editorial/954", + "code_source": "editorial", + "pid": "A1552" +} diff --git a/problems/PA-05/Atcoder/abc196_d/solution.cpp b/problems/PA-05/Atcoder/abc196_d/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..899a33a7fb9e2360e520c7ff97dc0eabea417dae --- /dev/null +++ b/problems/PA-05/Atcoder/abc196_d/solution.cpp @@ -0,0 +1,22 @@ +// abc196_d: D - Hanjo +// Code from the official editorial, https://atcoder.jp/contests/abc196/editorial/954 (by en_translator) +// Judged AC on all 50 tests, max 0.00 s of the 2 s limit. + +#include +using namespace std; + +int H, W, A, B, ans = 0; +void dfs(int i, int bit, int A, int B){ + if(i == H * W) return (void)ans++; + if(bit & 1 << i) return dfs(i + 1, bit, A, B); + if(B) dfs(i + 1, bit | 1 << i, A, B - 1); + if(A){ + if(i % W != W - 1 && ~bit & 1 << (i + 1)) dfs(i + 1, bit | 1 << i | 1 << (i + 1), A - 1, B); + if(i + W < H * W) dfs(i + 1, bit | 1 << i | 1 << (i + W), A - 1, B); + } +} +int main(){ + cin >> H >> W >> A >> B; + dfs(0, 0, A, B); + cout << ans << endl; +} diff --git a/problems/PA-05/Atcoder/abc196_d/statement.md b/problems/PA-05/Atcoder/abc196_d/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..6e9bc458972552db1d6626fd8668d2f0261ad7bb --- /dev/null +++ b/problems/PA-05/Atcoder/abc196_d/statement.md @@ -0,0 +1,62 @@ +# D - Hanjo + +**Score:** 400 points + +## Problem Statement +We have a rectangular room that is \(H\) meters long and \(W\) meters wide. +We will cover this room with \(A\) indistinguishable \(2\) meters \(\times\) \(1\) meters rectangular tatami mats and \(B\) indistinguishable \(1\) meter \(\times\) \(1\) meter square tatami mats. +The rectangular mats can be used in either direction: they can be \(2\) meters long and \(1\) meter wide, or \(1\) meter long and \(2\) meters wide. +How many ways are there to do this? +Here, it is guaranteed that \(2A + B = HW\), and two ways are distinguished if they match only after rotation, reflection, or both. + +## Constraints +- All values in input are integers. +- \(1 \leq H, W\) +- \(HW \leq 16\) +- \(0 \leq A, B\) +- \(2A + B = HW\) + +## Input +Input is given from Standard Input in the following format: +``` +H W A B +``` + +## Output +Print the answer. + +## Sample Input 1 +``` +2 2 1 2 +``` + +## Sample Output 1 +``` +4 +``` +There are four ways as follows: + +![Sample Output 1](https://img.atcoder.jp/ghi/d01b63c75c91bd87a73e9a4cc43dda28.png) + +## Sample Input 2 +``` +3 3 4 1 +``` + +## Sample Output 2 +``` +18 +``` +There are six ways as follows, and their rotations. + +![Sample Output 2](https://img.atcoder.jp/ghi/b7a492abe22e30683e8f9a7b309acd52.png) + +## Sample Input 3 +``` +4 4 8 0 +``` + +## Sample Output 3 +``` +36 +``` diff --git a/problems/PA-05/Atcoder/abc199_d/editorial.md b/problems/PA-05/Atcoder/abc199_d/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..bbbc9fb3d960977b53bd7a609487ffa0b381160b --- /dev/null +++ b/problems/PA-05/Atcoder/abc199_d/editorial.md @@ -0,0 +1,22 @@ +# D - RGB Coloring 2 (editorial) + +Source: https://atcoder.jp/contests/abc199/editorial/1169 by en_translator + +It is difficult to exhaustively search all the \(3^N\) ways of painting within the execution time limit. + The basic idea is as follows. + +- When the color of a vertex \(x\) has been determined, there are at most two candidate colors of a vertex connected to \(x\) with an edge, so if the graph is connected, we only have to search \(3 \times 2^{N - 1}\) ways of painting. + +This time, the graph is not necessarily connected, but the overall number of ways of painting is equal to the product of the number of ways of painting of each connected component, so we can boil down to the case where the graph is connected if we process for each connected component. + +There are various routes of implementation; the following is an example. + +- Start a DFS (Depth-First Search) from an arbitrary vertex (here we use vertex \(1\)) without visiting the same vertex twice, and create an array \(s\) where the vertices are stored in the order of visi +- Since \(s_i (i \ge 2)\) is connected to at least one of \(s_1, s_2, s_3, \dots, s_{i - 1}\), so we can enumerate the ways of painting as follows: + + - Enumerate the three colors of \(s_1\) + - Enumerate at most two colors of \(s_2\) (since there is at least one vertex \(s_i (i \lt 2)\) connected to \(s_2\), so there are at most two colors different to it) + - Enumerate at most two colors of \(s_3\) (since there is at least one vertex \(s_i (i \lt 3)\) connected to \(s_3\), so there are at most two colors different to it) + - \(\vdots\) + +This algorithm works within a complexity like \(O(2^NN)\). diff --git a/problems/PA-05/Atcoder/abc199_d/judge.json b/problems/PA-05/Atcoder/abc199_d/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..672fb6916751b32554ed293d1ca1d29cd79aa29f --- /dev/null +++ b/problems/PA-05/Atcoder/abc199_d/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 49, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc199_d/meta.json b/problems/PA-05/Atcoder/abc199_d/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..37a3bd129d138231c6992c98c58b14487914478d --- /dev/null +++ b/problems/PA-05/Atcoder/abc199_d/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc199_d", + "contest": "abc199", + "type": "ABC", + "letter": "D", + "title": "RGB Coloring 2", + "url": "https://atcoder.jp/contests/abc199/tasks/abc199_d", + "difficulty": 1804, + "score": 400, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 49, + "editorial_url": "https://atcoder.jp/contests/abc199/editorial/1169", + "code_source": "model", + "pid": "A1570" +} diff --git a/problems/PA-05/Atcoder/abc199_d/solution.cpp b/problems/PA-05/Atcoder/abc199_d/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..a336d493726768679e4afc2911059936e963ddf8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc199_d/solution.cpp @@ -0,0 +1,69 @@ +// abc199_d: D - RGB Coloring 2 +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc199/editorial/1169 +// Judged AC on all 49 tests, max 0.01 s of the 2 s limit (submission 1). + +#include +using namespace std; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int N, M; + cin >> N >> M; + vector> adj(N); + for (int i = 0; i < M; ++i) { + int a, b; + cin >> a >> b; + --a; + --b; + adj[a].push_back(b); + adj[b].push_back(a); + } + + vector visited(N, false); + vector color(N, -1); + long long answer = 1; + + for (int root = 0; root < N; ++root) { + if (visited[root]) continue; + + vector order; + auto dfs = [&](auto&& self, int v) -> void { + visited[v] = true; + order.push_back(v); + for (int u : adj[v]) { + if (!visited[u]) self(self, u); + } + }; + dfs(dfs, root); + + // By color symmetry, fix the first vertex's color and multiply by 3. + color[root] = 0; + auto count_colorings = [&](auto&& self, int pos) -> long long { + if (pos == static_cast(order.size())) return 1; + + int v = order[pos]; + int forbidden = 0; + for (int u : adj[v]) { + if (color[u] != -1) forbidden |= 1 << color[u]; + } + + long long ways = 0; + for (int c = 0; c < 3; ++c) { + if (forbidden & (1 << c)) continue; + color[v] = c; + ways += self(self, pos + 1); + } + color[v] = -1; + return ways; + }; + + answer *= 3LL * count_colorings(count_colorings, 1); + color[root] = -1; + if (answer == 0) break; + } + + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/abc199_d/statement.md b/problems/PA-05/Atcoder/abc199_d/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..7e56ee5534e6a14299bd284279a1ec5ff6a87b80 --- /dev/null +++ b/problems/PA-05/Atcoder/abc199_d/statement.md @@ -0,0 +1,105 @@ +# D - RGB Coloring 2 + +**Score :** 400 points + +## Problem Statement + +We have a simple undirected graph with \(N\) vertices and \(M\) edges. The vertices are numbered \(1\) through \(N\), and the edges are numbered \(1\) through \(M\). +Edge \(i\) connects Vertex \(A_i\) and Vertex \(B_i\). +Find the number of ways to paint each vertex in this graph red, green, or blue so that the following condition is satisfied: +- two vertices directly connected by an edge are always painted in different colors. + +Here, it is not mandatory to use all the colors. + +## Constraints +- \(1 \le N \le 20\) +- \(0 \le M \le \frac{N(N - 1)}{2}\) +- \(1 \le A_i \le N\) +- \(1 \le B_i \le N\) +- The given graph is simple (that is, has no multi-edges and no self-loops). + +--- + +## Input +Input is given from Standard Input in the following format: +``` +N M +A_1 B_1 +A_2 B_2 +A_3 B_3 +... +A_M B_M +``` + +## Output +Print the answer. + +--- + +## Sample Input 1 +``` +3 3 +1 2 +2 3 +3 1 +``` + +## Sample Output 1 +``` +6 +``` +Let \(c_1, c_2, c_3\) be the colors of Vertices \(1, 2, 3\) and \(R\), \(G\), \(B\) denote red, green, blue, respectively. There are six ways to satisfy the condition: +- \(c_1c_2c_3 = RGB\) +- \(c_1c_2c_3 = RBG\) +- \(c_1c_2c_3 = GRB\) +- \(c_1c_2c_3 = GBR\) +- \(c_1c_2c_3 = BRG\) +- \(c_1c_2c_3 = BGR\) + +--- + +## Sample Input 2 +``` +3 0 +``` + +## Sample Output 2 +``` +27 +``` + +Since the graph has no edge, we can freely choose the colors of the vertices. + +--- + +## Sample Input 3 +``` +4 6 +1 2 +2 3 +3 4 +2 4 +1 3 +1 4 +``` + +## Sample Output 3 +``` +0 +``` + +There may be no way to satisfy the condition. + +--- + +## Sample Input 4 +``` +20 0 +``` + +## Sample Output 4 +``` +3486784401 +``` + +The answer may not fit into the \(32\)-bit signed integer type. diff --git a/problems/PA-05/Atcoder/abc211_e/data/1.in b/problems/PA-05/Atcoder/abc211_e/data/1.in new file mode 100644 index 0000000000000000000000000000000000000000..4f8fd1615b585311d11a0e273a97f922ed3d0d73 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/1.in @@ -0,0 +1,10 @@ +8 +1 +.......# +....#... +...##... +..#..... +.#...... +.......# +........ +........ diff --git a/problems/PA-05/Atcoder/abc211_e/data/1.out b/problems/PA-05/Atcoder/abc211_e/data/1.out new file mode 100644 index 0000000000000000000000000000000000000000..e1617e842ad9a038ec2d8332c6084300a43cb09d --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/1.out @@ -0,0 +1 @@ +57 diff --git a/problems/PA-05/Atcoder/abc211_e/data/10.in b/problems/PA-05/Atcoder/abc211_e/data/10.in new file mode 100644 index 0000000000000000000000000000000000000000..fc4e950459d1e22bf60006eb96782650b03fc841 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/10.in @@ -0,0 +1,10 @@ +8 +3 +........ +.......# +.#...... +.......# +........ +......#. +...##... +........ diff --git a/problems/PA-05/Atcoder/abc211_e/data/10.out b/problems/PA-05/Atcoder/abc211_e/data/10.out new file mode 100644 index 0000000000000000000000000000000000000000..0d389107a348aa189d7b862d7834120e53f2e3a5 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/10.out @@ -0,0 +1 @@ +212 diff --git a/problems/PA-05/Atcoder/abc211_e/data/11.in b/problems/PA-05/Atcoder/abc211_e/data/11.in new file mode 100644 index 0000000000000000000000000000000000000000..108218960be034af55b46abef77c0ff39b0cfd44 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/11.in @@ -0,0 +1,10 @@ +8 +3 +........ +........ +........ +........ +........ +........ +....##.. +..#..... diff --git a/problems/PA-05/Atcoder/abc211_e/data/11.out b/problems/PA-05/Atcoder/abc211_e/data/11.out new file mode 100644 index 0000000000000000000000000000000000000000..f1aaa90592d0bda201d0977f17666ab4167bfbd8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/11.out @@ -0,0 +1 @@ +254 diff --git a/problems/PA-05/Atcoder/abc211_e/data/12.in b/problems/PA-05/Atcoder/abc211_e/data/12.in new file mode 100644 index 0000000000000000000000000000000000000000..82f0b2f0bbed70637e6a7980f7467544f8c1b0e7 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/12.in @@ -0,0 +1,10 @@ +8 +3 +..##.... +#.##.##. +.#...#.. +.####.## +....#.## +##.#.##. +######.. +..##.#.. diff --git a/problems/PA-05/Atcoder/abc211_e/data/12.out b/problems/PA-05/Atcoder/abc211_e/data/12.out new file mode 100644 index 0000000000000000000000000000000000000000..2bd5a0a98a36cc08ada88b804d3be047e6aa5b8a --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/12.out @@ -0,0 +1 @@ +22 diff --git a/problems/PA-05/Atcoder/abc211_e/data/13.in b/problems/PA-05/Atcoder/abc211_e/data/13.in new file mode 100644 index 0000000000000000000000000000000000000000..99cfd08f44d99bf9fb4fcd33c77ddf84aada8481 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/13.in @@ -0,0 +1,10 @@ +8 +3 +.....##. +#.#..#.# +####...# +..#..... +.#.#...# +#.#....# +.##.##.. +........ diff --git a/problems/PA-05/Atcoder/abc211_e/data/13.out b/problems/PA-05/Atcoder/abc211_e/data/13.out new file mode 100644 index 0000000000000000000000000000000000000000..85322d0b541fcf94328661c2a0cc60e5490d8c3b --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/13.out @@ -0,0 +1 @@ +79 diff --git a/problems/PA-05/Atcoder/abc211_e/data/14.in b/problems/PA-05/Atcoder/abc211_e/data/14.in new file mode 100644 index 0000000000000000000000000000000000000000..c4508fe5d4423d24f912217bf41e96473b0206b3 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/14.in @@ -0,0 +1,10 @@ +8 +3 +####..## +.####.## +###...#. +#.##.### +.##..#.# +#.###### +.#..#.#. +#...##.. diff --git a/problems/PA-05/Atcoder/abc211_e/data/14.out b/problems/PA-05/Atcoder/abc211_e/data/14.out new file mode 100644 index 0000000000000000000000000000000000000000..60d3b2f4a4cd5f1637eba020358bfe5ecb5edcf2 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/14.out @@ -0,0 +1 @@ +15 diff --git a/problems/PA-05/Atcoder/abc211_e/data/15.in b/problems/PA-05/Atcoder/abc211_e/data/15.in new file mode 100644 index 0000000000000000000000000000000000000000..bf7f754bee977dfe951a7fc64f485f301dcf766e --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/15.in @@ -0,0 +1,10 @@ +8 +3 +##.###.. +######.# +####.### +#.###.## +.#.##.#. +#.###### +.###.### +#...##.# diff --git a/problems/PA-05/Atcoder/abc211_e/data/15.out b/problems/PA-05/Atcoder/abc211_e/data/15.out new file mode 100644 index 0000000000000000000000000000000000000000..0cfbf08886fca9a91cb753ec8734c84fcbe52c9f --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/15.out @@ -0,0 +1 @@ +2 diff --git a/problems/PA-05/Atcoder/abc211_e/data/16.in b/problems/PA-05/Atcoder/abc211_e/data/16.in new file mode 100644 index 0000000000000000000000000000000000000000..48a3133b141fe8990c258f291d6da826108c57b6 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/16.in @@ -0,0 +1,10 @@ +8 +8 +#....... +........ +........ +.....#.. +.#...#.. +........ +........ +......#. diff --git a/problems/PA-05/Atcoder/abc211_e/data/16.out b/problems/PA-05/Atcoder/abc211_e/data/16.out new file mode 100644 index 0000000000000000000000000000000000000000..d2c5519c6c8f39a10acca870c21c54a8abb90d91 --- /dev/null +++ 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b/problems/PA-05/Atcoder/abc211_e/data/9.out new file mode 100644 index 0000000000000000000000000000000000000000..d6b24041cf04154f8f902651969675021f4d93a5 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/data/9.out @@ -0,0 +1 @@ +19 diff --git a/problems/PA-05/Atcoder/abc211_e/editorial.md b/problems/PA-05/Atcoder/abc211_e/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..5ea2ef8c9017377ab29b8cf14d11cfad8e3c7734 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/editorial.md @@ -0,0 +1,17 @@ +# E - Red Polyomino (editorial) + +Source: https://atcoder.jp/contests/abc211/editorial/2331 by en_translator + +The number of choices of \(K\) squares out of \(N^2\) squares is \(_{N^2}C_{K}\), and \(_{64}C_{8} =4426165368 \gt 4 \times 10^9\), so a naive brute force won’t fit in time. + However, since the red squares are mutually connected, we can predict that there are fairly small number of combinations that satisfies the conditions. We may evaluate its upper bound in some mathematical way, but this time we can estimate it from the Sample Code \(3\). + Therefore, it is sufficient to perform a DFS (Depth-First Search) that enumerates the patterns in which the red squares are connected. + If you find it difficult to implement DFS, please refer to the following example as well. + +Note that the code above is a bit slow for the sake of readability. + (541ms) + +One can manage the state of board with a 64-bit integer (like `unsigned long long`) instead of `vector` to speed up. + (85ms) + +With a little twist, one can avoid visiting the same board twice in the DFS. + (25ms) diff --git a/problems/PA-05/Atcoder/abc211_e/judge.json b/problems/PA-05/Atcoder/abc211_e/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..568377d939512aec9f59076716fe0b00e9ea80ad --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 32, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc211_e/meta.json b/problems/PA-05/Atcoder/abc211_e/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..f61213bd80225916fd2a921a759b79b64709881e --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc211_e", + "contest": "abc211", + "type": "ABC", + "letter": "E", + "title": "Red Polyomino", + "url": "https://atcoder.jp/contests/abc211/tasks/abc211_e", + "difficulty": 1823, + "score": 500, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 32, + "editorial_url": "https://atcoder.jp/contests/abc211/editorial/2331", + "code_source": "editorial", + "pid": "A1643" +} diff --git a/problems/PA-05/Atcoder/abc211_e/solution.cpp b/problems/PA-05/Atcoder/abc211_e/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..c1b32abd44d84afca60efeee612245fa874899a8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc211_e/solution.cpp @@ -0,0 +1,80 @@ +// abc211_e: E - Red Polyomino +// Code from the official editorial, https://atcoder.jp/contests/abc211/editorial/2331 (by en_translator) +// Judged AC on all 32 tests, max 0.31 s of the 2 s limit. + +#include +using namespace std; +typedef long long ll; +typedef pairl_l; + +set>used;//boards already searched +vectors;//current board +ll answer=0;//resulting answer +ll n; +vector dx={1,0,-1,0}; +vector dy={0,1,0,-1}; + +bool valid(ll x,ll y){ + return 0<=x && x<=n-1 && 0<=y && y<=n-1;//If (x,y) is in the board +} + +void dfs(ll num){ + //num: how many red squares do we have to place? + + if(used.find(s)!=used.end()){ + return;//We've already searched this, skipping + } + used.insert(s);//Add the current board to the set of already-searched boards + + if(num==0){ + answer++;//This board satisfies the condition; add 1 to answer + return; //We don't place red squares anymore + } + + vectornext;//candidates of squares to place a red square next + + for(int i=0;i<=n-1;i++){ + for(int j=0;j<=n-1;j++){ + //We can put a new red square to (i, j) if and only if + //s[i][j] is a white square ('.') and there is a red square ('@') next to s[i][j] + + if(s[i][j]=='.'){ + bool flag=false; + for(int z=0;z<=3;z++){ + ll nxt_i=i+dx[z]; + ll nxt_j=j+dy[z]; + if(valid(nxt_i,nxt_j) && s[nxt_i][nxt_j]=='@'){ + flag=true;//If there is a red square next to (i, j), set flag to true + } + } + if(flag){ + next.push_back({i,j});//add (i, j) as the next candidate of red square + } + } + } + } + for(l_l pos:next){ + s[pos.first][pos.second]='@';//paint red + dfs(num-1); //advance DFS + s[pos.first][pos.second]='.';//clean up + } +} + +signed main(){ + ll k;cin>>n>>k; + s.resize(n); + for(int i=0;i<=n-1;i++)cin>>s[i]; + + for(int i=0;i<=n-1;i++){ + for(int j=0;j<=n-1;j++){ + //We paint Square (i, J) first + if(s[i][j]=='.'){ + s[i][j]='@';//paint red + dfs(k-1); //start DFS + s[i][j]='.';//clean up + } + } + } + cout< +using namespace std; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int required = 0; + for (int r = 0; r < 4; ++r) { + for (int c = 0; c < 4; ++c) { + int x; + cin >> x; + if (x) required |= 1 << (4 * r + c); + } + } + + int answer = 0; + const int dr[4] = {-1, 1, 0, 0}; + const int dc[4] = {0, 0, -1, 1}; + + for (int mask = 1; mask < (1 << 16); ++mask) { + if ((mask & required) != required) continue; + + // The extra border represents the unbounded exterior. + bool inside[6][6] = {}; + int sr = -1, sc = -1; + for (int r = 0; r < 4; ++r) { + for (int c = 0; c < 4; ++c) { + if (mask & (1 << (4 * r + c))) { + inside[r + 1][c + 1] = true; + sr = r + 1; + sc = c + 1; + } + } + } + + // A diagonal checkerboard would make the boundary touch itself. + bool valid = true; + for (int r = 0; r < 5; ++r) { + for (int c = 0; c < 5; ++c) { + if (inside[r][c] == inside[r + 1][c + 1] && + inside[r + 1][c] == inside[r][c + 1] && + inside[r][c] != inside[r + 1][c]) { + valid = false; + } + } + } + if (!valid) continue; + + bool visited[6][6] = {}; + auto flood = [&](int start_r, int start_c) { + queue> q; + q.emplace(start_r, start_c); + visited[start_r][start_c] = true; + bool color = inside[start_r][start_c]; + while (!q.empty()) { + auto [r, c] = q.front(); + q.pop(); + for (int d = 0; d < 4; ++d) { + int nr = r + dr[d], nc = c + dc[d]; + if (nr < 0 || nr >= 6 || nc < 0 || nc >= 6) continue; + if (visited[nr][nc] || inside[nr][nc] != color) continue; + visited[nr][nc] = true; + q.emplace(nr, nc); + } + } + }; + + flood(sr, sc); + flood(0, 0); + for (int r = 0; r < 6; ++r) { + for (int c = 0; c < 6; ++c) { + if (!visited[r][c]) valid = false; + } + } + if (valid) ++answer; + } + + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/abc219_e/statement.md b/problems/PA-05/Atcoder/abc219_e/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..08425cc430532d9767e95dbfd85bc4208ec9a594 --- /dev/null +++ b/problems/PA-05/Atcoder/abc219_e/statement.md @@ -0,0 +1,82 @@ +# E - Moat + +**Score**: 500 points + +## Problem Statement +There are villages at some number of points in the \( xy \)-plane. +Takahashi will construct a moat to protect these villages from enemies such as civil armies and witches. + +You are given a \( 4 \times 4 \) matrix \( A = (A_{i, j}) \) consisting of \( 0 \) and \( 1 \). +For each pair of integers \( (i, j) \) \( (1 \leq i, j \leq 4) \) such that \( A_{i, j} = 1 \), there is a village at the coordinates \( (i-0.5, j-0.5) \). + +The moat will be a polygon in the plane. +Takahashi will construct it so that the following conditions will be satisfied. (See also the annotation at Sample Input/Output 1.) +1. There is no self-intersection. +2. All villages are contained in the interior of the polygon. +3. The \( x \)- and \( y \)-coordinates of every vertex are integers between \( 0 \) and \( 4 \) (inclusive). +4. Every edge is parallel to the \( x \)- or \( y \)-axis. +5. Every inner angle is \( 90 \) or \( 270 \) degrees. + +Print the number of ways in which Takahashi can construct the moat. + +## Constraints +- \( A_{i, j} \in \lbrace 0, 1\rbrace \) +- There is at least one pair \( (i, j) \) such that \( A_{i, j} = 1 \). + + +## Input +Input is given from Standard Input in the following format: +``` +A_{1, 1} A_{1, 2} A_{1, 3} A_{1, 4} +A_{2, 1} A_{2, 2} A_{2, 3} A_{2, 4} +A_{3, 1} A_{3, 2} A_{3, 3} A_{3, 4} +A_{4, 1} A_{4, 2} A_{4, 3} A_{4, 4} +``` + +## Output +Print the number of ways in which Takahashi can construct the moat. + +--- + +### Sample Input 1 +``` +1 0 0 0 +0 0 1 0 +0 0 0 0 +1 0 0 0 +``` + +### Sample Output 1 +``` +1272 +``` +The two ways to construct the moat shown in the images below are **valid**. +![valid1](https://img.atcoder.jp/ghi/7b3181deb4e1df72e4c0661b1137db4d.png) +![valid2](https://img.atcoder.jp/ghi/a1e46c7db32d63942caa7119a4f3a593.png) + +The four ways to construct the moat shown in the images below are **invalid**. +![invalid1](https://img.atcoder.jp/ghi/335053c01a13eb99e55767a3dc02eb38.png) +![invalid2](https://img.atcoder.jp/ghi/c4df3d1fa24557b0d4d94ac0eaa8b9ab.png) +![invalid3](https://img.atcoder.jp/ghi/be93de595e9222d5e20c90bd28d24563.png) +![invalid4](https://img.atcoder.jp/ghi/37dac3af065c013ce0b8c0ee7591b97a.png) + +Here are the reasons the above ways are invalid. +1. The first way violates the condition: "There is no self-intersection." +2. The second way violates the condition: "All villages are contained in the interior of the polygon." +3. The third way violates the condition: "The \( x \)- and \( y \)-coordinates of every vertex are integers between \( 0 \) and \( 4 \)." (Some vertices have non-integer coordinates.) +4. The fourth way violates the condition: "Every edge is parallel to the \( x \)- or \( y \)-axis." + +--- + +### Sample Input 2 +``` +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +``` + +### Sample Output 2 +``` +1 +``` diff --git a/problems/PA-05/Atcoder/abc226_f/data/1.in b/problems/PA-05/Atcoder/abc226_f/data/1.in new file mode 100644 index 0000000000000000000000000000000000000000..78454aabfb2f18959434287fece91a9070ffa075 --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/data/1.in @@ -0,0 +1 @@ +2 2 diff --git a/problems/PA-05/Atcoder/abc226_f/data/1.out b/problems/PA-05/Atcoder/abc226_f/data/1.out new file mode 100644 index 0000000000000000000000000000000000000000..7ed6ff82de6bcc2a78243fc9c54d3ef5ac14da69 --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/data/1.out @@ -0,0 +1 @@ +5 diff --git a/problems/PA-05/Atcoder/abc226_f/data/10.in b/problems/PA-05/Atcoder/abc226_f/data/10.in new file mode 100644 index 0000000000000000000000000000000000000000..872a6acbbcaefaa51c923a7dc32f8c4bc3fb912a 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b/problems/PA-05/Atcoder/abc226_f/editorial.md @@ -0,0 +1,35 @@ +# F - Score of Permutations (editorial) + +Source: https://atcoder.jp/contests/abc226/editorial/2892 by en_translator + +First, consider computing the score for a fixed permutation \(P\). Choose arbitrarily some number of \(P\)’s, and you will see the fact from the observation: + +- If you write numbers from \(1\) through \(N\), and an arrow to whom each person is going to pass balls, then it can be divided into loops. + Then, the score for the permutation is the least common multiplier (LCM) of the size of all loops. + + - For example, when \(N = 5\) and \(P = (2,5,4,3,1)\), it is divided into two loops, \(3 \to 4 \to 3\) and \(1 \to 2 \to 5 \to 1\), and the score is \(\mathrm{lcm}(2, 3) = 6\). + +The proof follows from the property of LCM. Also, such a loop is called a **cyclic permutation**. + +Next, consider how to find the sum of \(S(P)^K\). Of course, if you try every \(P\), it requires a exponential time of \(N\), so we should reduce the time complexity. Here, we consider solving this problem with DP (Dynamic Programming). + +Before getting down to business, let us state the general rule, not limited to this problem: when trying to reduce the time complexity with DP, it is important to focus on “what aspect can be identified.” It is a very common technique to equate multiple states into one to reduce the computational complexity, enabling us to solve more problems. + +Now, in this problem, it can be easily understood that the important factor for the score is “the lengths of the loops,” but not person-specific information like “who passes balls to whom, and which loop they belong.” Therefore, in this problem, it is a good idea to do DP in which only the lengths of loops are stored, and process as “there are this number of permutations that generate this kind of loop, and they yield this score.” + +As a subproblem for computing DP, let us consider how many permutations \(P\) of length \(N\) can be represented as cyclic permutations of lengths \(k_1, k_2, \dots, k_m\). + +From now on, this problem requires a complex observation using binomial coefficients, and some of you might have dropped out because of this. When encountered to such situation, in my experience it can often be barely overcome by “**make some specific examples and construct a generic formula that matches them**”, which is recommended for other people who are not good at it as well. Also, if you have no idea why it is wrong, it is typical to **compare with a naive solution and look for the mistakes** like the code below. + +I will tell you the answer first. Transform the lengths of cyclic permutations \(k_1, k_2, \dots, k_m\) to the sequence of occurrences in the format of “there are \(F_i\) loops of length \(L_i\).” Then, the answer is + +\[ N! \times \prod_{L,F} \frac{1}{(L!)^F \cdot F!} \times ((L-1)!)^F.\] + +The formula can be explained as follows. (It may make easier to understand to check for some specific examples.) + +- \(N!\) and \(\frac{1}{(L!)^F \cdot F!}\) inside \(\prod\) represents how to divide \(N\) vertices into groups. +- The final \(((L-1)!)\) part correspond to the number of ways to add arrows to \(L\) vertices so that they form a cyclic permutation. Since there are \(F\) loops of length \(L\), we multiply its \(F\)-th power. + +Once we found it, all that left is to enumerate the ways of divisions by DP , with the techniques like DFS (Depth-First Search). + +We can see that the complexity of DFS depends on the number of ways to divide \(N\) into some number of integers. The number is fairly as small as \(204226\) even for \(N = 50\), so we can see that the DFS works fast enough. (By the way, this is called “partition number”.) Hence, the problem has been solved. diff --git a/problems/PA-05/Atcoder/abc226_f/judge.json b/problems/PA-05/Atcoder/abc226_f/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..32e7bd655019ee2c3fe9486e8769efbc397a9148 --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 41, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc226_f/meta.json b/problems/PA-05/Atcoder/abc226_f/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..4c29f45d20fe36e2ba0dd5a69bf987228434874c --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc226_f", + "contest": "abc226", + "type": "ABC", + "letter": "F", + "title": "Score of Permutations", + "url": "https://atcoder.jp/contests/abc226/tasks/abc226_f", + "difficulty": 2086, + "score": 500, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 41, + "editorial_url": "https://atcoder.jp/contests/abc226/editorial/2892", + "code_source": "model", + "pid": "A1762" +} diff --git a/problems/PA-05/Atcoder/abc226_f/solution.cpp b/problems/PA-05/Atcoder/abc226_f/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..f2412208264490dee94518466c12e07345385403 --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/solution.cpp @@ -0,0 +1,42 @@ +// abc226_f: F - Score of Permutations +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc226/editorial/2892 +// Judged AC on all 41 tests, max 0.04 s of the 2 s limit (submission 1). + +#include +#include +using namespace std; +using mint = atcoder::modint998244353; + +int N, K; +mint inv[51]; +mint answer = 0; + +void dfs(int remaining, int last, int multiplicity, long long score, mint weight) { + if (remaining == 0) { + answer += weight * mint(score).pow(K); + return; + } + for (int length = last; length <= remaining; ++length) { + int count = (length == last ? multiplicity + 1 : 1); + long long next_score = std::lcm(score, static_cast(length)); + dfs(remaining - length, length, count, next_score, + weight * inv[length] * inv[count]); + } +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + cin >> N >> K; + mint factorial = 1; + for (int i = 1; i <= N; ++i) { + inv[i] = mint(i).inv(); + factorial *= i; + } + + dfs(N, 1, 0, 1, mint(1)); + answer *= factorial; + cout << answer.val() << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/abc226_f/statement.md b/problems/PA-05/Atcoder/abc226_f/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..58aca5d62363ab1a5b8e89deec0d53a8391e3f7c --- /dev/null +++ b/problems/PA-05/Atcoder/abc226_f/statement.md @@ -0,0 +1,98 @@ +# F - Score of Permutations + +**Score**: 500 points + +## Problem Statement + +For a permutation \(P = (p_1, p_2, \dots, p_N)\) of \((1,2, \dots, N)\), let us define the score \(S(P)\) of \(P\) as follows. + +- There are \(N\) people, numbered \(1,2,\dots,N\). Additionally, Snuke is there. Initially, Person \(i\) \( (1 \leq i \leq N)\) has Ball \(i\). + Each time Snuke screams, every Person \(i\) such that \(i \neq p_i\) gives their ball to Person \(p_i\) simultaneously. + If, after screaming at least once, every Person \(i\) has Ball \(i\), Snuke stops screaming. + The score is the number of times Snuke screams until he stops. Here, it is guaranteed that the score will be a finite value. + +There are \(N!\) permutations \(P\) of \((1,2, \dots, N)\). Find the sum, modulo \(998244353\), of the scores of those permutations, each raised to the \(K\)-th power. + +- Formally, let \(S_N\) be the set of the permutations of \((1,2, \dots, N)\). Compute the following: +\[ +\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353} +\] + +### Constraints +- \(2 \leq N \leq 50\) +- \(1 \leq K \leq 10^4\) +- All values in input are integers. + +--- + +## Input +Input is given from Standard Input in the following format: +``` +N K +``` + +## Output +Print +\[ +\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353} +\] + +--- + +## Sample Input 1 +``` +2 2 +``` + +## Sample Output 1 +``` +5 +``` +When \(N = 2\), there are two possible permutations \(P\): \((1,2),(2,1)\). +The score of the permutation \((1,2)\) is found as follows. +- Initially, Person \(1\) has Ball \(1\), and Person \(2\) has Ball \(2\). + After Snuke's first scream, Person \(1\) has Ball \(1\), and Person \(2\) has Ball \(2\). + Here, every Person \(i\) has Ball \(i\), so he stops screaming. + Thus, the score is \(1\). + +The score of the permutation \((2,1)\) is found as follows. +- Initially, Person \(1\) has Ball \(1\), and Person \(2\) has Ball \(2\). + After Snuke's first scream, Person \(1\) has Ball \(2\), and Person \(2\) has Ball \(1\). + After Snuke's second scream, Person \(1\) has Ball \(1\), and Person \(2\) has Ball \(2\). + Here, every Person \(i\) has Ball \(i\), so he stops screaming. + Thus, the score is \(2\). + +Therefore, the answer in this case is \(1^2 + 2^2 = 5\). + +--- + +## Sample Input 2 +``` +3 3 +``` + +## Sample Output 2 +``` +79 +``` +All permutations and their scores are listed below. +- \((1,2,3)\): The score is \(1\). +- \((1,3,2)\): The score is \(2\). +- \((2,1,3)\): The score is \(2\). +- \((2,3,1)\): The score is \(3\). +- \((3,1,2)\): The score is \(3\). +- \((3,2,1)\): The score is \(2\). + +Thus, we should print \(1^3 + 2^3 + 2^3 + 3^3 + 3^3 + 2^3 = 79\). + +--- + +## Sample Input 3 +``` +50 10000 +``` + +## Sample Output 3 +``` +77436607 +``` diff --git a/problems/PA-05/Atcoder/abc270_g/data/1.in b/problems/PA-05/Atcoder/abc270_g/data/1.in new file mode 100644 index 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minimum \(n\) such that \(f^n(x)=y\). + We answer this problem by, for an arbitrary constant \(M\), solving the problem “what is the minimum value of \(0\leq j< M\) such that \(f^{iM}(x)=(f^{-1})^j(y)\), if exists?” + If \(f^{-1}\), this subproblem can be rephrased to “what is the minimum value of \(0\leq j< M\) such that \(f^{iM}(x)=(f^{-1})^j(y)\), if exists?” The idea of this algorithm that, once we enumerate the candidates for the left hand side, we can determine the existence fast for each \(i\). + +#### The solution for this problem + +If \(A=0\), it is easy to find the answer. We consider when \(A\neq 0\). Here, if there is an \(n\) such that \(X_n=T\), then that is at most \(P\). Therefore, we consider applying baby-step giant-step with \(M=\lfloor\sqrt{P}\rfloor\). + The inverse function of \(f(x)=Ax+B\) can be explicitly found in the form \(f^{-1}(x)=Cx+D\), so we can find \(f^M\) explicitly in the form \(f^M(x)=A'x+B'\). Thus, by enumerating \(M\) values \(T,f^{-1}(T),f^{-1}(f^{-1}(T)),\ldots\) and checking if any of \(S,f^M(S),f^M(f^M(S)),\ldots\) coincides with them, in this order, so that the problem is solved in a total of \(O(M\log{M})\) time (or expected \(O(M)\) time using a hash map). + When implementing, beware of the cases where \(T,f^{-1}(T),f^{-1}(f^{-1}(T)),\ldots\) have duplicates when the sequence \((X_i)\) has a period less than \(M\). (In that case, the answer is \(M\) anyway, so one solution to search exhaustively for these candidates firsthand.) + +#### Another solution + +If \(A=1\), it is easy to find the answer. We consider when \(A\neq 1\). Let \(\alpha=\frac{B}{A-1}\) and \(Y_i=X_i+\alpha\), then we have \(Y_i=AY_{i-1}\), so \(Y_i=Y_0 A^i\). Thus the original problem is boiled down to a discrete logarithm problem, which can be solved in a total of \(O(\sqrt{P}\log{P})\) time (or expected \(O(\sqrt{P})\) time using a hash map). + This solution is by en_translator. + +Bonus: solve for a non-prime \(P\) in an \(O(\sqrt{P}\log{P})\) time. diff --git a/problems/PA-05/Atcoder/abc270_g/judge.json b/problems/PA-05/Atcoder/abc270_g/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..36beaf05a28b7b058452412964ef68ddbcc1ce1a --- /dev/null +++ b/problems/PA-05/Atcoder/abc270_g/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 4.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 27, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc270_g/meta.json b/problems/PA-05/Atcoder/abc270_g/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..f8e299f2a0adea6dda2509cdb9bc89f3028867df --- /dev/null +++ b/problems/PA-05/Atcoder/abc270_g/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc270_g", + "contest": "abc270", + "type": "ABC", + "letter": "G", + "title": "Sequence in mod P", + "url": "https://atcoder.jp/contests/abc270/tasks/abc270_g", + "difficulty": 2220, + "score": 600, + "time_limit_sec": 4.0, + "memory_limit_mb": 1024, + "n_tests": 27, + "editorial_url": "https://atcoder.jp/contests/abc270/editorial/4882", + "code_source": "model", + "pid": "A2116" +} diff --git a/problems/PA-05/Atcoder/abc270_g/solution.cpp b/problems/PA-05/Atcoder/abc270_g/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..03cc3f754eeaaba28273786cf3a5e668e1ea4ce4 --- /dev/null +++ b/problems/PA-05/Atcoder/abc270_g/solution.cpp @@ -0,0 +1,68 @@ +// abc270_g: G - Sequence in mod P +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc270/editorial/4882 +// Judged AC on all 27 tests, max 0.33 s of the 4 s limit (submission 1). + +#include +using namespace std; +using int64 = long long; + +int64 mod_pow(int64 a, int64 e, int64 mod) { + int64 r = 1; + while (e > 0) { + if (e & 1) r = r * a % mod; + a = a * a % mod; + e >>= 1; + } + return r; +} + +int64 solve(int64 p, int64 a, int64 b, int64 s, int64 g) { + if (s == g) return 0; + if (a == 0) return b == g ? 1 : -1; + + int64 m = sqrtl((long double)p); + while (m * m < p) ++m; + while ((m - 1) * (m - 1) >= p) --m; + + int64 inv_a = mod_pow(a, p - 2, p); + int64 inv_b = (p - inv_a * b % p) % p; + vector> baby; + baby.reserve(m); + + int64 y = g; + int64 step_a = 1, step_b = 0; + for (int j = 0; j < m; ++j) { + baby.emplace_back((int)y, j); + y = (inv_a * y + inv_b) % p; + step_a = a * step_a % p; + step_b = (a * step_b + b) % p; + } + // Equal values are ordered by j, so lower_bound gives the smallest j. + sort(baby.begin(), baby.end()); + + int64 x = s; + for (int64 base = 0; base < p; base += m) { + auto it = lower_bound(baby.begin(), baby.end(), + make_pair((int)x, -1)); + if (it != baby.end() && it->first == x) { + int64 answer = base + it->second; + if (answer < p) return answer; + } + x = (step_a * x + step_b) % p; + } + return -1; +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int t; + cin >> t; + while (t--) { + int64 p, a, b, s, g; + cin >> p >> a >> b >> s >> g; + cout << solve(p, a, b, s, g) << '\n'; + } + return 0; +} diff --git a/problems/PA-05/Atcoder/abc270_g/statement.md b/problems/PA-05/Atcoder/abc270_g/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..2974b8da97982ec7aede4c7657468ac9ce01d9f8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc270_g/statement.md @@ -0,0 +1,63 @@ +# G - Sequence in mod P + +Score: **600** points + +## Problem Statement +There is a sequence \(X=(X_0, X_1, \ldots)\) defined by the following recurrence relation. + +\[ +X_i = +\begin{cases} +S & \text{(i = 0)}\\ +(A X_{i-1}+B) \mod P & \text{(i \geq 1)} +\end{cases} +\] + +Determine whether there is an \(i\) such that \(X_i=G\). If it exists, find the smallest such \(i\). +Here, \(x \mod y\) denotes the remainder when \(x\) is divided by \(y\) (the least non-negative residue). + +You are given \(T\) test cases for each input file. + +## Constraints +- \(1 \leq T \leq 100\) +- \(2 \leq P \leq 10^9\) +- \(P\) is a prime. +- \(0 \leq A, B, S, G < P\) +- All values in the input are integers. + +## Input +The input is given from Standard Input in the following format: + +``` +T +case_1 +case_2 +... +case_T +``` +Each test case is in the following format: + +``` +P A B S G +``` + +## Output +Print \(T\) lines. +The \(t\)-th line should contain the smallest \(i\) such that \(X_i=G\) for \(case_t\), or `-1` if there is no such \(i\). + +## Sample Input 1 +``` +3 +5 2 1 1 0 +5 2 2 3 0 +11 1 1 0 10 +``` + +## Sample Output 1 +``` +3 +-1 +10 +``` +For the first test case, we have \(X=(1,3,2,0,\ldots)\), so the smallest \(i\) such that \(X_i=0\) is \(3\). +For the second test case, we have \(X=(3,3,3,3,\ldots)\), so there is no \(i\) such that \(X_i=0\). diff --git a/problems/PA-05/Atcoder/abc271_f/editorial.md b/problems/PA-05/Atcoder/abc271_f/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..176699ade18be086f3f0fa4751170fbad99c15ae --- /dev/null +++ b/problems/PA-05/Atcoder/abc271_f/editorial.md @@ -0,0 +1,14 @@ +# F - XOR on Grid Path (editorial) + +Source: https://atcoder.jp/contests/abc271/editorial/4941 by en_translator + +Let us call the exclusive logical sum XOR and denote the XOR of non-negative integers \(x\) and \(y\) by \(x \oplus y\). + +There are \(\binom{2N - 2}{N - 1}\) ways to travel from square \((1, 1)\) to \((N, N)\); in this problem, there are at most about \(3.5 × 10^{10}\) ways. Thus, trying all of them does not finish in the time limit. + Instead, we use the trick called **meet-in-the-middle**. Roughly, we try to optimize the program by corresponding \(S\) objects into the direct product set of two sets of sizes about \(\sqrt{S}\) and process them. + +When traveling from square \((1, 1)\) to \((N, N)\), you pass through a unique square on the squares on the anti-diagonal line, that is, a square \((x, y)\) such that \(x + y = N + 1\). For one path, let \(p\) be the XOR on the squares in the path from square \((1, 1)\) to \((x, y)\), and \(q\) be that from square \((x, y)\) to \((N, N)\); then the path satisfies the condition in the problem statement if and only if \(p \oplus a_{x, y} \oplus q = 0\). + +Therefore, let \(P\) be the multiset of the XORs of the integers written on each path from square \((1, 1)\) to \((x, y)\), and \(Q\) be the counterpart from \((x, y)\) to \((N, N)\), then the number of paths satisfying the conditions in the problem statement passing through square \((x, y)\) equals the number of pairs \((p, q)\) such that \(p \in P, q \in Q\), and \(p \oplus a_{x, y} \oplus q = 0\). This can be found by sorting or binary search in a total of \(O((|P| + |Q|) \log (|P| + |Q|))\) time. + +Since \(|P| = |Q|\), and the sum of \(|P|\) over all squares \((x, y)\) on the anti-diagonal line is \(2^{N - 1}\), the problem has been solved in a total of \(O(N2^{N})\). diff --git a/problems/PA-05/Atcoder/abc271_f/judge.json b/problems/PA-05/Atcoder/abc271_f/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..b403f7babaf78166430306734cdd7648e75809e1 --- /dev/null +++ b/problems/PA-05/Atcoder/abc271_f/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 3.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 44, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc271_f/meta.json b/problems/PA-05/Atcoder/abc271_f/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..412c6a25e0eada696231ff0d4e2e226076853825 --- /dev/null +++ b/problems/PA-05/Atcoder/abc271_f/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc271_f", + "contest": "abc271", + "type": "ABC", + "letter": "F", + "title": "XOR on Grid Path", + "url": "https://atcoder.jp/contests/abc271/tasks/abc271_f", + "difficulty": 1669, + "score": 500, + "time_limit_sec": 3.0, + "memory_limit_mb": 1024, + "n_tests": 44, + "editorial_url": "https://atcoder.jp/contests/abc271/editorial/4941", + "code_source": "model", + "pid": "A2123" +} diff --git a/problems/PA-05/Atcoder/abc271_f/solution.cpp b/problems/PA-05/Atcoder/abc271_f/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..4a787ce250dfb214bec88db75197f61a46ca8105 --- /dev/null +++ b/problems/PA-05/Atcoder/abc271_f/solution.cpp @@ -0,0 +1,50 @@ +// abc271_f: F - XOR on Grid Path +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc271/editorial/4941 +// Judged AC on all 44 tests, max 0.10 s of the 3 s limit (submission 1). + +#include +using namespace std; + +int n; +int a[20][20]; +vector pref[20]; +long long answer = 0; + +void enumerate_front(int r, int c, int x) { + x ^= a[r][c]; + if (r + c == n - 1) { + pref[r].push_back(x); + return; + } + if (r + 1 < n) enumerate_front(r + 1, c, x); + if (c + 1 < n) enumerate_front(r, c + 1, x); +} + +void enumerate_back(int r, int c, int x) { + x ^= a[r][c]; + if (r + c == n - 1) { + int target = x ^ a[r][c]; + auto range = equal_range(pref[r].begin(), pref[r].end(), target); + answer += range.second - range.first; + return; + } + if (r > 0) enumerate_back(r - 1, c, x); + if (c > 0) enumerate_back(r, c - 1, x); +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + cin >> n; + for (int i = 0; i < n; ++i) { + for (int j = 0; j < n; ++j) cin >> a[i][j]; + } + + enumerate_front(0, 0, 0); + for (int i = 0; i < n; ++i) sort(pref[i].begin(), pref[i].end()); + enumerate_back(n - 1, n - 1, 0); + + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/abc271_f/statement.md b/problems/PA-05/Atcoder/abc271_f/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..2085ddb76d2ae72f2824e7cea7a1fc4390b546ce --- /dev/null +++ b/problems/PA-05/Atcoder/abc271_f/statement.md @@ -0,0 +1,89 @@ +# F - XOR on Grid Path + +**Score:** 500 points + +## Problem Statement + +There is a grid with \(N\) rows and \(N\) columns. We denote by \((i, j)\) the square at the \(i\)-th \((1 \leq i \leq N)\) row from the top and \(j\)-th \((1 \leq j \leq N)\) column from the left.
+Square \((i, j)\) has a non-negative integer \(a_{i, j}\) written on it. + +When you are at square \((i, j)\), you can move to either square \((i+1, j)\) or \((i, j+1)\). Here, you are not allowed to go outside the grid. + +Find the number of ways to travel from square \((1, 1)\) to square \((N, N)\) such that the exclusive logical sum of the integers written on the squares visited (including \((1, 1)\) and \((N, N)\)) is \(0\). + +
+What is the exclusive logical sum? +The exclusive logical sum \(a \oplus b\) of two integers \(a\) and \(b\) is defined as follows. +
    +
  • The \(2^k\)'s place \((k \geq 0)\) in the binary notation of \(a \oplus b\) is \(1\) if exactly one of the \(2^k\)'s places in the binary notation of \(a\) and \(b\) is \(1\); otherwise, it is \(0\). +
    • +
+For example, \(3 \oplus 5 = 6\) (In binary notation: \(011 \oplus 101 = 110\)).
+In general, the exclusive logical sum of \(k\) integers \(p_1, \dots, p_k\) is defined as \((\cdots ((p_1 \oplus p_2) \oplus p_3) \oplus \cdots \oplus p_k)\). We can prove that it is independent of the order of \(p_1, \dots, p_k\). +
+ +## Constraints +- \(2 \leq N \leq 20\) +- \(0 \leq a_{i, j} < 2^{30} \, (1 \leq i, j \leq N)\) +- All values in the input are integers. + +## Input +The input is given from Standard Input in the following format: + +``` +N +a_{1, 1} a_{1, 2} ... a_{1, N} +... +a_{N, 1} a_{N, 2} ... a_{N, N} +``` + +## Output +Print the answer. + +## Sample Input 1 +``` +3 +1 5 2 +7 0 5 +4 2 3 +``` + +## Sample Output 1 +``` +2 +``` +The following two ways satisfy the condition: +- \((1, 1) \rightarrow (1, 2) \rightarrow (1, 3) \rightarrow (2, 3) \rightarrow (3, 3)\); +- \((1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3)\). + +## Sample Input 2 +``` +2 +1 2 +2 1 +``` + +## Sample Output 2 +``` +0 +``` + +## Sample Input 3 +``` +10 +1 0 1 0 0 1 0 0 0 1 +0 0 0 1 0 1 0 1 1 0 +1 0 0 0 1 0 1 0 0 0 +0 1 0 0 0 1 1 0 0 1 +0 0 1 1 0 1 1 0 1 0 +1 0 0 0 1 0 0 1 1 0 +1 1 1 0 0 0 1 1 0 0 +0 1 1 0 0 1 1 0 1 0 +1 0 1 1 0 0 0 0 0 0 +1 0 1 1 0 0 1 1 1 0 +``` + +## Sample Output 3 +``` +24307 +``` diff --git a/problems/PA-05/Atcoder/abc300_g/editorial.md b/problems/PA-05/Atcoder/abc300_g/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..6cf75fd3a0fe848bb584fb2f1ca8f408382ad864 --- /dev/null +++ b/problems/PA-05/Atcoder/abc300_g/editorial.md @@ -0,0 +1,25 @@ +# G - P-smooth number (editorial) + +Source: https://atcoder.jp/contests/abc300/editorial/6290 by en_translator + +Hint: the last sample of this problem is the maximum case. + If the sample includes the maximum case, you can observe something or check if your program works correctly for the maximum case (before submitting.) + +In case of this problem, the answer to this problem turns out to be about \(2 \times 10^9\). Thus, we can observe that: + +- enumerating all conforming integers is a bit reckless; +- if we can enumerate the answer effectively, for example with the meet-in-the-middle trick, it will finish in the execution time limit. + +We consider adopting the meet-in-the-middle trick. How can we do that? + +- Prepare arrays \(U=(1)\) and \(V=(1)\). +- For all primes \(q\) between \(2\) and \(P\), repeat the following. + + - Apply the following operation against \(W\), which is the smaller of \(U\) and \(V\). + + - Insert all integers not exceeding \(N\) obtained by multiplying each element of \(W\) by a power of \(q\). +- Finally, count the number of pairs taken from \(U\) and \(V\) whose product do not exceed \(N\), just as the merging process of the meet-in-the-middle trick. + +In other words, we divide the prime factors up to \(P\) into two groups, enumerate the integers whose prime factors are only in one of them, and then merge them to count the integers whose prime factors are only in the sum set of them. + +While it is difficult to estimate the complexity of this solution by hand, we can actually implement and try the maximum case to see that \(|U|,|V| \le 5 \times 10^6\) approximately, to see that sorting them and merging is still fast enough. diff --git a/problems/PA-05/Atcoder/abc300_g/judge.json b/problems/PA-05/Atcoder/abc300_g/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..82f54878b13b4d049ef3b1e6f6449106dfd28f92 --- /dev/null +++ b/problems/PA-05/Atcoder/abc300_g/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 4.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 33, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc300_g/meta.json b/problems/PA-05/Atcoder/abc300_g/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..0b4c86c9093dea15c5ff04890c9094bd35c417a4 --- /dev/null +++ b/problems/PA-05/Atcoder/abc300_g/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc300_g", + "contest": "abc300", + "type": "ABC", + "letter": "G", + "title": "P-smooth number", + "url": "https://atcoder.jp/contests/abc300/tasks/abc300_g", + "difficulty": 2343, + "score": 600, + "time_limit_sec": 4.0, + "memory_limit_mb": 1024, + "n_tests": 33, + "editorial_url": "https://atcoder.jp/contests/abc300/editorial/6290", + "code_source": "editorial", + "pid": "A2356" +} diff --git a/problems/PA-05/Atcoder/abc300_g/solution.cpp b/problems/PA-05/Atcoder/abc300_g/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..323d3c2c9416c4eea822897a40de8b9fa82d1c7a --- /dev/null +++ b/problems/PA-05/Atcoder/abc300_g/solution.cpp @@ -0,0 +1,45 @@ +// abc300_g: G - P-smooth number +// Code from the official editorial, https://atcoder.jp/contests/abc300/editorial/6290 (by en_translator) +// Judged AC on all 33 tests, max 0.51 s of the 4 s limit. + +#include + +using namespace std; + +void step(vector &a,long long p,long long mxm){ + long long sz=a.size(); + for(long long i=0;imxm){break;} + a.push_back(ad); + } + } +} + +int main(){ + vector pl={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}; + long long n,p; + cin >> n >> p; + while(p u={1},v={1}; + for(auto &nx : pl){ + if(u.size()=0 && mv + +using namespace std; +#define rep(i, n) for(int i = 0; i < n; ++i) + +int main(void){ + string a[3][10]; + char tmp[10][11]; + int h[3],w[3]; + bool flag; + for(int k=0;k<3;k++){ + cin>>h[k]>>w[k]; + for(int i=0;i>a[k][i]; + } + + for(int dh0=-h[0]+1;dh0=h[2])flag=false; + if(j+dw0<0)flag=false; + if(j+dw0>=w[2])flag=false; + if(flag)tmp[i+dh0][j+dw0]='#'; + } + } + } + for(int i=0;i=h[2])flag=false; + if(j+dw1<0)flag=false; + if(j+dw1>=w[2])flag=false; + if(flag)tmp[i+dh1][j+dw1]='#'; + } + } + } + for(int i=0;i +using namespace std; + +static int owner[100][100][100]; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int N; + cin >> N; + for (int id = 1; id <= N; ++id) { + int x1, y1, z1, x2, y2, z2; + cin >> x1 >> y1 >> z1 >> x2 >> y2 >> z2; + for (int x = x1; x < x2; ++x) + for (int y = y1; y < y2; ++y) + for (int z = z1; z < z2; ++z) + owner[x][y][z] = id; + } + + vector pairs; + pairs.reserve(3000000); + auto add_pair = [&](int a, int b) { + if (b == 0 || a == b) return; + if (a > b) swap(a, b); + pairs.push_back((uint64_t(a) << 32) | uint64_t(b)); + }; + + for (int x = 0; x < 100; ++x) { + for (int y = 0; y < 100; ++y) { + for (int z = 0; z < 100; ++z) { + int a = owner[x][y][z]; + if (a == 0) continue; + if (x + 1 < 100) add_pair(a, owner[x + 1][y][z]); + if (y + 1 < 100) add_pair(a, owner[x][y + 1][z]); + if (z + 1 < 100) add_pair(a, owner[x][y][z + 1]); + } + } + } + + sort(pairs.begin(), pairs.end()); + pairs.erase(unique(pairs.begin(), pairs.end()), pairs.end()); + vector answer(N + 1, 0); + for (uint64_t p : pairs) { + int a = int(p >> 32); + int b = int(p & 0xffffffffULL); + ++answer[a]; + ++answer[b]; + } + for (int id = 1; id <= N; ++id) + cout << answer[id] << '\n'; + + return 0; +} diff --git a/problems/PA-05/Atcoder/abc312_e/statement.md b/problems/PA-05/Atcoder/abc312_e/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..13328bc15c3303534d86d1b2b7a4d4355ecf7610 --- /dev/null +++ b/problems/PA-05/Atcoder/abc312_e/statement.md @@ -0,0 +1,91 @@ +# E - Tangency of Cuboids + +**Score :** 500 points + +## Problem Statement +There are \(N\) rectangular cuboids in a three-dimensional space. +These cuboids do not overlap. Formally, for any two different cuboids among them, their intersection has a volume of \(0\). + +The diagonal of the \(i\)-th cuboid is a segment that connects two points \((X_{i,1},Y_{i,1},Z_{i,1})\) and \((X_{i,2},Y_{i,2},Z_{i,2})\), and its edges are all parallel to one of the coordinate axes. + +For each cuboid, find the number of other cuboids that share a face with it. +Formally, for each \(i\), find the number of \(j\) with \(1 \leq j \leq N\) and \(j \neq i\) such that the intersection of the surfaces of the \(i\)-th and \(j\)-th cuboids has a positive area. + +## Constraints +- \(1 \leq N \leq 10^5\) +- \(0 \leq X_{i,1} < X_{i,2} \leq 100\) +- \(0 \leq Y_{i,1} < Y_{i,2} \leq 100\) +- \(0 \leq Z_{i,1} < Z_{i,2} \leq 100\) +- Cuboids do not have an intersection with a positive volume. +- All input values are integers. + +## Input +The input is given from Standard Input in the following format: +``` +N +X_{1,1} Y_{1,1} Z_{1,1} X_{1,2} Y_{1,2} Z_{1,2} +... +X_{N,1} Y_{N,1} Z_{N,1} X_{N,2} Y_{N,2} Z_{N,2} +``` + +## Output +Print the answer. + +## Sample Input 1 +``` +4 +0 0 0 1 1 1 +0 0 1 1 1 2 +1 1 1 2 2 2 +3 3 3 4 4 4 +``` + +## Sample Output 1 +``` +1 +1 +0 +0 +``` +>The 1-st and 2-nd cuboids share a rectangle whose diagonal is the segment connecting two points \((0,0,1)\) and \((1,1,1)\). +>The 1-st and 3-rd cuboids share a point \((1,1,1)\), but do not share a surface. + +## Sample Input 2 +``` +3 +0 0 10 10 10 20 +3 4 1 15 6 10 +0 9 6 1 20 10 +``` + +## Sample Output 2 +``` +2 +1 +1 +``` + +## Sample Input 3 +``` +8 +0 0 0 1 1 1 +0 0 1 1 1 2 +0 1 0 1 2 1 +0 1 1 1 2 2 +1 0 0 2 1 1 +1 0 1 2 1 2 +1 1 0 2 2 1 +1 1 1 2 2 2 +``` + +## Sample Output 3 +``` +3 +3 +3 +3 +3 +3 +3 +3 +``` diff --git a/problems/PA-05/Atcoder/abc322_d/data/1.in b/problems/PA-05/Atcoder/abc322_d/data/1.in new file mode 100644 index 0000000000000000000000000000000000000000..45be949b2d7b976da949d928fefcc0a7bf12f601 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/1.in @@ -0,0 +1,12 @@ +#### +...# +...# +.... +.... +.... +..#. +.##. +#### +###. +#... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/1.out b/problems/PA-05/Atcoder/abc322_d/data/1.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/1.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/10.in b/problems/PA-05/Atcoder/abc322_d/data/10.in new file mode 100644 index 0000000000000000000000000000000000000000..3a40f1d3c4dfb0648d32939c7482bbf85514e81c --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/10.in @@ -0,0 +1,12 @@ +.... +..#. +.... +.... +.... +..#. +.... +.... +.... +..#. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/10.out b/problems/PA-05/Atcoder/abc322_d/data/10.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/10.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/11.in b/problems/PA-05/Atcoder/abc322_d/data/11.in new file mode 100644 index 0000000000000000000000000000000000000000..0f46365fb022c21db3f46eb5719441db6619db2e --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/11.in @@ -0,0 +1,12 @@ +###. +###. +.### +.... +...# +.... +.... +.... +.... +#### +...# +...# diff --git a/problems/PA-05/Atcoder/abc322_d/data/11.out b/problems/PA-05/Atcoder/abc322_d/data/11.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/11.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/12.in b/problems/PA-05/Atcoder/abc322_d/data/12.in new file mode 100644 index 0000000000000000000000000000000000000000..be558511462df3101d777367f76791db5d89829d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/12.in @@ -0,0 +1,12 @@ +.... +#### +#... +#... +.... +#### +...# +..## +.... +..## +..#. +..## diff --git a/problems/PA-05/Atcoder/abc322_d/data/12.out b/problems/PA-05/Atcoder/abc322_d/data/12.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/12.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/13.in b/problems/PA-05/Atcoder/abc322_d/data/13.in new file mode 100644 index 0000000000000000000000000000000000000000..0a2bbdbc3af963f4d3159d5d8ade9acaf0247586 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/13.in @@ -0,0 +1,12 @@ +..#. +..#. +.... +.... +.... +.... +#### +#### +.... +.### +.### +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/13.out b/problems/PA-05/Atcoder/abc322_d/data/13.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/13.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/14.in b/problems/PA-05/Atcoder/abc322_d/data/14.in new file mode 100644 index 0000000000000000000000000000000000000000..37313867a4c4441564a02fafe4bfa9c281bebcf1 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/14.in @@ -0,0 +1,12 @@ +#### +###. +..#. +.... +..#. +..#. +..#. +.... +##.. +##.. +#... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/14.out b/problems/PA-05/Atcoder/abc322_d/data/14.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/14.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/15.in b/problems/PA-05/Atcoder/abc322_d/data/15.in new file mode 100644 index 0000000000000000000000000000000000000000..e9d62bff0e992b01e9cbf8cfc11a997ab94d912d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/15.in @@ -0,0 +1,12 @@ +###. +.##. +..#. +.### +.... +...# +..## +...# +.... +#... +#... +#... diff --git a/problems/PA-05/Atcoder/abc322_d/data/15.out b/problems/PA-05/Atcoder/abc322_d/data/15.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/15.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/16.in b/problems/PA-05/Atcoder/abc322_d/data/16.in new file mode 100644 index 0000000000000000000000000000000000000000..015a4e5d5577fd8747cf5a56cb8664e3053fbe0b --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/16.in @@ -0,0 +1,12 @@ +..#. +..## +#### +.... +##.. +##.. +.... +.... +#### +.### +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/16.out b/problems/PA-05/Atcoder/abc322_d/data/16.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/16.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/17.in b/problems/PA-05/Atcoder/abc322_d/data/17.in new file mode 100644 index 0000000000000000000000000000000000000000..183048e5cf61e12e80e5d3c66a1ccc7696d44d56 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/17.in @@ -0,0 +1,12 @@ +#### +#..# +...# +.### +...# +.... +.... +.... +.... +..## +.### +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/17.out b/problems/PA-05/Atcoder/abc322_d/data/17.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/17.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/18.in b/problems/PA-05/Atcoder/abc322_d/data/18.in new file mode 100644 index 0000000000000000000000000000000000000000..87d9f6920405d52f05ab1f0b1eb82d8b5618bee7 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/18.in @@ -0,0 +1,12 @@ +.... +###. +.... +.... +###. +###. +#... +.... +.... +..#. +..#. +..#. diff --git a/problems/PA-05/Atcoder/abc322_d/data/18.out b/problems/PA-05/Atcoder/abc322_d/data/18.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/18.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/19.in b/problems/PA-05/Atcoder/abc322_d/data/19.in new file mode 100644 index 0000000000000000000000000000000000000000..7dc43b7269af125feee64eff7553a94752c507b9 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/19.in @@ -0,0 +1,12 @@ +.... +##.. +###. +.#.. +..## +.... +.... +.... +###. +.##. +..#. +.##. diff --git a/problems/PA-05/Atcoder/abc322_d/data/19.out b/problems/PA-05/Atcoder/abc322_d/data/19.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/19.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/2.in b/problems/PA-05/Atcoder/abc322_d/data/2.in new file mode 100644 index 0000000000000000000000000000000000000000..ca255b51b0898c7e100bf0239475976a975852db --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/2.in @@ -0,0 +1,12 @@ +.... +.... +.... +...# +.### +#..# +##.# +#### +.... +.... +.##. +.#.. diff --git a/problems/PA-05/Atcoder/abc322_d/data/2.out b/problems/PA-05/Atcoder/abc322_d/data/2.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/2.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/20.in b/problems/PA-05/Atcoder/abc322_d/data/20.in new file mode 100644 index 0000000000000000000000000000000000000000..55d6be8cdeb5f4bc2f788288d141705e4eb2ab6c --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/20.in @@ -0,0 +1,12 @@ +##.. +.#.. +##.. +.... +.... +.... +..## +.##. +.#.. +##.. +.#.. +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/20.out b/problems/PA-05/Atcoder/abc322_d/data/20.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/20.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/21.in b/problems/PA-05/Atcoder/abc322_d/data/21.in new file mode 100644 index 0000000000000000000000000000000000000000..551403a8bf458ac42291518928c157afeae9f463 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/21.in @@ -0,0 +1,12 @@ +.... +#... +#... +#... +..#. +..#. +..## +..#. +.... +##.. +###. +###. diff --git a/problems/PA-05/Atcoder/abc322_d/data/21.out b/problems/PA-05/Atcoder/abc322_d/data/21.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/21.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/22.in b/problems/PA-05/Atcoder/abc322_d/data/22.in new file mode 100644 index 0000000000000000000000000000000000000000..3f1a1cd8ca164fe243d437ea0a0a677f22f97c5f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/22.in @@ -0,0 +1,12 @@ +.... +.#.. +##.. +.#.. +#### +.... +.... +.... +.... +.### +.### +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/22.out b/problems/PA-05/Atcoder/abc322_d/data/22.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/22.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/23.in b/problems/PA-05/Atcoder/abc322_d/data/23.in new file mode 100644 index 0000000000000000000000000000000000000000..d1062c5d58da56adc47c150564cc400aa7a04b32 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/23.in @@ -0,0 +1,12 @@ +.... +.... +...# +...# +..#. +..#. +.##. +.### +..#. +..## +..## +..## diff --git a/problems/PA-05/Atcoder/abc322_d/data/23.out b/problems/PA-05/Atcoder/abc322_d/data/23.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/23.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/24.in b/problems/PA-05/Atcoder/abc322_d/data/24.in new file mode 100644 index 0000000000000000000000000000000000000000..d9dcee1605e386c66e6ee976383ac36f7947c637 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/24.in @@ -0,0 +1,12 @@ +.#.. +.##. +.#.. +.... +.#.. +.#.. +.#.. +##.. +.... +.#.. +.#.. +.##. diff --git a/problems/PA-05/Atcoder/abc322_d/data/24.out b/problems/PA-05/Atcoder/abc322_d/data/24.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/24.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/25.in b/problems/PA-05/Atcoder/abc322_d/data/25.in new file mode 100644 index 0000000000000000000000000000000000000000..4dccc6d2b351597bf9e3649d1f3eb5ef59c4ceda --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/25.in @@ -0,0 +1,12 @@ +#### +...# +.... +.... +.... +.... +###. +##.. +.... +.... +###. +###. diff --git a/problems/PA-05/Atcoder/abc322_d/data/25.out b/problems/PA-05/Atcoder/abc322_d/data/25.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/25.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/26.in b/problems/PA-05/Atcoder/abc322_d/data/26.in new file mode 100644 index 0000000000000000000000000000000000000000..c50097054dd9e0dc8edae337b96472c51948d258 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/26.in @@ -0,0 +1,12 @@ +##.. +.... +.... +.... +...# +..## +..## +.... +#... +###. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/26.out b/problems/PA-05/Atcoder/abc322_d/data/26.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/26.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/27.in b/problems/PA-05/Atcoder/abc322_d/data/27.in new file mode 100644 index 0000000000000000000000000000000000000000..a0a67ea3ba5b560080fce72d5d4ab82f901cf27b --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/27.in @@ -0,0 +1,12 @@ +..#. +#### +.... +.... +.... +###. +###. +.... +###. +##.. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/27.out b/problems/PA-05/Atcoder/abc322_d/data/27.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/27.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/28.in b/problems/PA-05/Atcoder/abc322_d/data/28.in new file mode 100644 index 0000000000000000000000000000000000000000..61b475944ef20db1b09f5eb36aea17d980129148 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/28.in @@ -0,0 +1,12 @@ +..#. +.##. +.##. +.##. +.... +.... +.### +.##. +##.. +##.. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/28.out b/problems/PA-05/Atcoder/abc322_d/data/28.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/28.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/29.in b/problems/PA-05/Atcoder/abc322_d/data/29.in new file mode 100644 index 0000000000000000000000000000000000000000..7a62bad64897856a9b698aa64b093e8292f5e9d8 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/29.in @@ -0,0 +1,12 @@ +.... +..## +.... +.... +.... +.... +.... +#### +#### +###. +###. +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/29.out b/problems/PA-05/Atcoder/abc322_d/data/29.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/29.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/3.in b/problems/PA-05/Atcoder/abc322_d/data/3.in new file mode 100644 index 0000000000000000000000000000000000000000..0908ca4c69a3858486d1f49eb1c86a60fb05facd --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/3.in @@ -0,0 +1,12 @@ +.... +###. +.#.. +.... +.... +.### +.##. +.... +..#. +.##. +.##. +.##. diff --git a/problems/PA-05/Atcoder/abc322_d/data/3.out b/problems/PA-05/Atcoder/abc322_d/data/3.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/3.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/30.in b/problems/PA-05/Atcoder/abc322_d/data/30.in new file mode 100644 index 0000000000000000000000000000000000000000..bdc5260c0051433945dfa21327c6f12a8b523b0b --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/30.in @@ -0,0 +1,12 @@ +##.. +##.. +##.. +.#.. +...# +#### +.... +.... +.... +#### +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/30.out b/problems/PA-05/Atcoder/abc322_d/data/30.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/30.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/31.in b/problems/PA-05/Atcoder/abc322_d/data/31.in new file mode 100644 index 0000000000000000000000000000000000000000..1c7dfa5873edbb4d171b3495593d638aa54201b2 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/31.in @@ -0,0 +1,12 @@ +#### +#.## +.... +.... +.... +.... +##.. +.#.. +#### +##.. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/31.out b/problems/PA-05/Atcoder/abc322_d/data/31.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/31.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/32.in b/problems/PA-05/Atcoder/abc322_d/data/32.in new file mode 100644 index 0000000000000000000000000000000000000000..cfd3bca79b1ef1307bd920f1fc3890ccd8515745 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/32.in @@ -0,0 +1,12 @@ +..#. +..#. +..#. +.... +..## +..## +..## +.### +.... +.... +.... +#### diff --git a/problems/PA-05/Atcoder/abc322_d/data/32.out b/problems/PA-05/Atcoder/abc322_d/data/32.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ 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-0,0 +1,12 @@ +#... +##.. +#.## +#### +.... +.... +.... +#... +###. +##.. +.... +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/42.out b/problems/PA-05/Atcoder/abc322_d/data/42.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/42.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/5.in b/problems/PA-05/Atcoder/abc322_d/data/5.in new file mode 100644 index 0000000000000000000000000000000000000000..c5f66408ce3437f5f3fcfc6a472a26264262b4b1 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/5.in @@ -0,0 +1,12 @@ +.... +.... +..#. +.... +.### +#..# +##.# +#### +.... +.... +.##. +.#.. diff --git a/problems/PA-05/Atcoder/abc322_d/data/5.out b/problems/PA-05/Atcoder/abc322_d/data/5.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ 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+1,12 @@ +.... +.... +##.. +###. +..#. +..## +.### +.... +.##. +.#.. +.##. +.#.. diff --git a/problems/PA-05/Atcoder/abc322_d/data/7.out b/problems/PA-05/Atcoder/abc322_d/data/7.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/7.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/data/8.in b/problems/PA-05/Atcoder/abc322_d/data/8.in new file mode 100644 index 0000000000000000000000000000000000000000..d92d2822ae088685fc2694b37e887d984870be33 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/8.in @@ -0,0 +1,12 @@ +.... +.... +.### +.#.. +.### +#### +###. +#... +.... +.... +..#. +.... diff --git a/problems/PA-05/Atcoder/abc322_d/data/8.out b/problems/PA-05/Atcoder/abc322_d/data/8.out new file mode 100644 index 0000000000000000000000000000000000000000..dcd7a5d6d55bef30b5619e90266d7af48a1c239f --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/8.out @@ -0,0 +1 @@ +Yes diff --git a/problems/PA-05/Atcoder/abc322_d/data/9.in b/problems/PA-05/Atcoder/abc322_d/data/9.in new file mode 100644 index 0000000000000000000000000000000000000000..5359820d8cf682c36396d9a79d2fb30878aef82c --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/9.in @@ -0,0 +1,12 @@ +##.. +#..# +#### +.... +.... +##.. +.##. +.... +.#.. +.#.. +.#.. +.#.. diff --git a/problems/PA-05/Atcoder/abc322_d/data/9.out b/problems/PA-05/Atcoder/abc322_d/data/9.out new file mode 100644 index 0000000000000000000000000000000000000000..cf456979c3c1f62274f0a5b15ac4877ea042a50d --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/data/9.out @@ -0,0 +1 @@ +No diff --git a/problems/PA-05/Atcoder/abc322_d/editorial.md b/problems/PA-05/Atcoder/abc322_d/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..6197e1c844b7b2b6beb4e726b47c854c3dde5216 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/editorial.md @@ -0,0 +1,29 @@ +# D - Polyomino (editorial) + +Source: https://atcoder.jp/contests/abc322/editorial/7314 by en_translator + +We think that the hardness of the implementation of this problem largely depends on the choice of approaches. If you had trouble in implementing it, we recommend you seek for a better way reading this article and other contestant’s solutions. + +Since the constraints of this problem are small, we do not have to use an advanced algorithm, but only exhaustively enumerate all possible placements. Specifically, we can determine every element one-by-one as follows: + +- Fix the orientation of the first polyomino, and enumerate all possible \(4 \times 4 = 16\) orientations of the second and third. + + - Enumerate all position of the first polyomino. + + - Enumerate all position of the second polyomino. + + - Enumerate all position of the third polyomino. +- Print `Yes` if a conforming placement is found, and `No` otherwise. + +By properly implementing this algorithm, the problem can be solved. + The most difficult part of this algorithm may be enumerate all possible rotations of a polyomino, so we describe in detail how to do so. (We think that most people are relatively used to placing them, as there have been similar problems like [ABC307-C](https://atcoder.jp/contests/abc307/tasks/abc307_c).) + +Since this problem allows rotation of a polyomino, we have to brute-force all possible orientations of each polyomino. Since the grid is square-shaped, we can assume that the first polyomino are not rotated, so the rotation of the first is skipped, but still we need to inspect \(4 \times 4 = 16\) cases of orientations of the second and third. + This part may require a bunch of variables depending on approaches, which can bother; we introduce typical implementations in such problems. + +First, let us extract the rotation of polyomino as a function: + +Then, write the following double loop. + +In fact, this double loop inspects all \(4 \times 4 = 16\) cases. Specifically, the state where the second polyomino are rotated \(a\) times and third \(b\) times are inspected when the loop variables of the for statements are \(t = a\) and \(u = b\). + The technique of exhaustive search with rotations of variables is a bit technical but has a wide range of applications (especially when the reverse of a sequence matters), so we recommend you to learn it. diff --git a/problems/PA-05/Atcoder/abc322_d/judge.json b/problems/PA-05/Atcoder/abc322_d/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..0d9addb6ce0766878023b9486010595fdfdde086 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 42, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc322_d/meta.json b/problems/PA-05/Atcoder/abc322_d/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..5827d960959868534b1fe84a0319f26e493d3d97 --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc322_d", + "contest": "abc322", + "type": "ABC", + "letter": "D", + "title": "Polyomino", + "url": "https://atcoder.jp/contests/abc322/tasks/abc322_d", + "difficulty": 1310, + "score": 400, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 42, + "editorial_url": "https://atcoder.jp/contests/abc322/editorial/7314", + "code_source": "editorial", + "pid": "A2509" +} diff --git a/problems/PA-05/Atcoder/abc322_d/solution.cpp b/problems/PA-05/Atcoder/abc322_d/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..ece7e513724a18a62a500e1a2296eca394772eac --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/solution.cpp @@ -0,0 +1,75 @@ +// abc322_d: D - Polyomino +// Code from the official editorial, https://atcoder.jp/contests/abc322/editorial/7314 (by en_translator) +// Judged AC on all 42 tests, max 0.02 s of the 2 s limit. + +#include +#include +#include +using namespace std; + +#define rep(i, n) for (int i = 0; i < (n); i++) + +// Print `s` and terminate the program +void Print(const string& s) { + cout << s << "\n"; + exit(0); +} + +// Rotate clockwise by 90 degrees +void Rotate(vector& v) { + vector w = v; + rep(i, 4) rep(j, 4) w[3 - j][i] = v[i][j]; + v = w; +} + +// Determines if (i, j) is within the grid +bool in(int i, int j) { return 0 <= i and i < 4 and 0 <= j and j < 4; } + +// Can we put p onto the grid, so that the top-left position is (di, dj)? +bool can_put(vector>& exist, const vector& p, int di, + int dj) { + rep(i, 4) rep(j, 4) { + if (p[i][j] == '#') { + int ni = i + di; + int nj = j + dj; + if (!in(ni, nj)) return false; + if (exist[ni][nj] == 1) return false; + exist[ni][nj] = 1; + } + } + return true; +} + +// A recursive function that places polyominoes +void dfs(int i, const vector>& exist, + const vector>& ps) { + if (i == 3) { + int ok = 1; + rep(u, 4) rep(v, 4) ok &= exist[u][v]; + if (ok) Print("Yes"); + return; + } + for (int di = -3; di <= 3; di++) { + for (int dj = -3; dj <= 3; dj++) { + auto ex2 = exist; + bool flag = can_put(ex2, ps[i], di, dj); + if (flag) dfs(i + 1, ex2, ps); + } + } +} + +int main() { + vector> ps(3); + rep(i, 3) { + ps[i].resize(4); + for (auto& s : ps[i]) cin >> s; + } + rep(t, 4) { + rep(u, 4) { + dfs(0, vector(4, vector(4, 0)), ps); + Rotate(ps[2]); + } + Rotate(ps[1]); + } + Print("No"); +} diff --git a/problems/PA-05/Atcoder/abc322_d/statement.md b/problems/PA-05/Atcoder/abc322_d/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..74548079d08c43908d717035d4d1917f98ace2ea --- /dev/null +++ b/problems/PA-05/Atcoder/abc322_d/statement.md @@ -0,0 +1,179 @@ +# D - Polyomino + +**Score**: 400 points + +## Problem Statement + +A **polyomino** is a puzzle piece in the shape of a connected polygon made by connecting several squares by their edges. +There is a grid with four rows and four columns, and three polyominoes that fit within the grid. +The shape of the *i*-th polyomino is represented by *16* characters *P_{i,j,k}* (*1 ≤ j, k ≤ 4*). They describe the state of the grid when the *i*-th polyomino is placed on it. If *P_{i, j, k}* is `#`, the square at the *j*-th row from the top and *k*-th column from the left is occupied by the polyomino; if it is `.`, the square is not occupied. (Refer to the figures at Sample Input/Output *1*.) + +You want to fill the grid with all three polyominoes so that all of the following conditions are satisfied. +- All squares of the grid are covered by the polyominoes. +- The polyominoes must not overlap each other. +- The polyominoes must not stick out of the grid. +- The polyominoes may be freely translated and rotated but may not be flipped over. + +Can the grid be filled with the polyominoes to satisfy these conditions? + +## Constraints +- *P_{i, j, k}* is `#` or `.`. +- The given polyominoes are connected. In other words, the squares that make up a polyomino can be reached from each other by following only the squares up, down, left, and right. +- The given polyominoes are not empty. + +## Input +The input is given from Standard Input in the following format: +``` +P_{1,1,1}P_{1,1,2}P_{1,1,3}P_{1,1,4} +P_{1,2,1}P_{1,2,2}P_{1,2,3}P_{1,2,4} +P_{1,3,1}P_{1,3,2}P_{1,3,3}P_{1,3,4} +P_{1,4,1}P_{1,4,2}P_{1,4,3}P_{1,4,4} +P_{2,1,1}P_{2,1,2}P_{2,1,3}P_{2,1,4} +P_{2,2,1}P_{2,2,2}P_{2,2,3}P_{2,2,4} +P_{2,3,1}P_{2,3,2}P_{2,3,3}P_{2,3,4} +P_{2,4,1}P_{2,4,2}P_{2,4,3}P_{2,4,4} +P_{3,1,1}P_{3,1,2}P_{3,1,3}P_{3,1,4} +P_{3,2,1}P_{3,2,2}P_{3,2,3}P_{3,2,4} +P_{3,3,1}P_{3,3,2}P_{3,3,3}P_{3,3,4} +P_{3,4,1}P_{3,4,2}P_{3,4,3}P_{3,4,4} +``` + +## Output +If it is possible to fill the grid with the polyominoes to satisfy the conditions in the problem statement, print `Yes`; otherwise, print `No`. + +## Sample Input 1 +``` +.... +###. +.#.. +.... +.### +.##. +.... +..#. +.##. +.##. +.##. +``` + +## Sample Output 1 +``` +Yes +``` + +The figure below shows the shapes of the polyominoes corresponding to Sample Input *1*. +![image1](https://img.atcoder.jp/abc322/f0e25c2abcdbeade76fcb12eaee39f23.jpg) + +In this case, you can fill the grid with them to satisfy the conditions in the problem statement by placing them as shown in the figure below. +![image2](https://img.atcoder.jp/abc322/81e983f85e958e0d612063adcc455c71.jpg) + +Thus, the answer is `Yes`. + +## Sample Input 2 +``` +###. +#.#. +##.. +.... +.... +..#. +.... +.... +#### +##.. +#... +#... +``` + +## Sample Output 2 +``` +Yes +``` + +As in the first polyomino in Sample Input *2*, a polyomino may be in the shape of a polygon with a hole. + +## Sample Input 3 +``` +##.. +#..# +#### +.... +.... +##.. +.##. +.... +.#.. +.#.. +.#.. +.#.. +``` + +## Sample Output 3 +``` +No +``` + +Note that the polyominoes may not be flipped over when filling the grid. + +## Sample Input 4 +``` +.... +..#. +.... +.... +.... +..#. +.... +.... +.... +..#. +.... +.... +``` + +## Sample Output 4 +``` +No +``` + +## Sample Input 5 +``` +.... +#### +#... +#... +.... +#### +...# +..## +.... +..## +..#. +..## +``` + +## Sample Output 5 +``` +No +``` + +## Sample Input 6 +``` +###. +.##. +..#. +.### +.... +...# +..## +...# +.... +#... +#... +#... +``` + +## Sample Output 6 +``` +Yes +``` diff --git a/problems/PA-05/Atcoder/abc339_f/editorial.md b/problems/PA-05/Atcoder/abc339_f/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..721d923c78bd52bcfa69e6778ebbf05042771109 --- /dev/null +++ b/problems/PA-05/Atcoder/abc339_f/editorial.md @@ -0,0 +1,26 @@ +# F - Product Equality (editorial) + +Source: https://atcoder.jp/contests/abc339/editorial/9269 by en_translator + +There are \(N^2\) ways to choose two variables and multiply them. + However, since each variable is enormous, if you naively perform multiplication using big integers (without caring overflows), a computational cost of about \(O(N^2 d \log d)\) is predicted (where \(d\) is the number of digits), which makes it difficult to solve the problem in this way. + +For now, let us consider how to determine if three variables \(p,q\), and \(r\) satisfies \(p \times q = r\). + Here, we come up with somewhat daring criteria: + +- Choose an integer \(x\). If \(p \times q \equiv r\) ( \(\rm{mod}\) \(x\) ), then determine that \(p \times q = r\). + +This method fails when \((p \times q - r) \equiv 0\) ( \(\rm{mod}\) \(x\) ). + +Here, consider choosing a prime greater than \(10^9\) as \(x\). + Then, the criteria misbehaves only for at most \(222\) primes \(x\) (because \(|p \times q - r| \le 10^{2000}\)), while there are tens of millions of primes between \(10^9\) and \(2 \times 10^9\). + +Thus, the following randomized algorithm is a valid solution. + +- Let \(S = \) a set of randomly chosen primes (between \(10^9\) and \(2 \times 10^9\)). +- If \(A_i \times A_j \equiv A_k\) for all elements \(x\) in \(S\), then determine that \(A_i \times A_j = A_k\). +- First evaluate \(A_k\) modulo \(x\) for all \(A_k\) and all elements \(x\) in \(S\), then determine if \(A_i \times A_j\) modulo \(x\) is contained in them. + +This way, the problem can be solved in \(O(|S| \times N^2 \log N)\) time. While we do not precisely evaluate success rate in this editorial, it would be safe to choose \(|S|\) to be around \(20\). + +Instead of determining if a picked \(x\) is a prime, you can just choose a random large integer. (It is very difficult to hack such a solution (prepare a test case for which such a solution misbehaves). diff --git a/problems/PA-05/Atcoder/abc339_f/judge.json b/problems/PA-05/Atcoder/abc339_f/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..8d2b7f3724d4980cc941753316d41ea02eb3dd72 --- /dev/null +++ b/problems/PA-05/Atcoder/abc339_f/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 80, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/abc339_f/meta.json b/problems/PA-05/Atcoder/abc339_f/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..37e002396f4ecf26f770de57bd639a5e7829f3f5 --- /dev/null +++ b/problems/PA-05/Atcoder/abc339_f/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "abc339_f", + "contest": "abc339", + "type": "ABC", + "letter": "F", + "title": "Product Equality", + "url": "https://atcoder.jp/contests/abc339/tasks/abc339_f", + "difficulty": 1716, + "score": 550, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 80, + "editorial_url": "https://atcoder.jp/contests/abc339/editorial/9269", + "code_source": "editorial", + "pid": "A2630" +} diff --git a/problems/PA-05/Atcoder/abc339_f/solution.cpp b/problems/PA-05/Atcoder/abc339_f/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..5c08efc91f41855be8539c7fea238c8e3876e857 --- /dev/null +++ b/problems/PA-05/Atcoder/abc339_f/solution.cpp @@ -0,0 +1,73 @@ +// abc339_f: F - Product Equality +// Code from the official editorial, https://atcoder.jp/contests/abc339/editorial/9269 (by en_translator) +// Judged AC on all 80 tests, max 0.32 s of the 2 s limit. + +#include + +using namespace std; + +long long takemod(string &s,long long mod){ + long long res=0; + for(auto &nx : s){ + res*=10;res%=mod; + res+=(nx-'0');res%=mod; + } + return res; +} + +int lb(vector &x,vector> &lis){ + int l=0,r=lis.size()-1; + while(l<=r){ + int te=(l+r)/2; + if(lis[te] < x){l=te+1;} + else{r=te-1;} + } + return l; +} + +int ub(vector &x,vector> &lis){ + int l=0,r=lis.size()-1; + while(l<=r){ + int te=(l+r)/2; + if(lis[te] <= x){l=te+1;} + else{r=te-1;} + } + return l; +} + +int main(){ + std::random_device seed_gen; + std::mt19937_64 engine(seed_gen()); + + int n; + cin >> n; + int ss=20; + vector S; + for(int i=0;i> md; + for(int i=0;i> a; + vector cv; + for(int j=0;j cp; + for(int k=0;k + +using namespace std; + +#define rep(i, n) for(int i = 0; i < n; ++i) + +#define MAX_N 7 +#define MAX_H 10 +#define MAX_W 10 + +int n,h,w; +int a[MAX_N]; +int b[MAX_N]; +int c[MAX_H][MAX_W]; +bool ans; + +void solve(int unused,int curi,int curj){ + bool can; + while(c[curi][curj]>=0){ + curj++; + if(curj>=w){curi++;curj=0;} + if(curi>=h)break; + } + if(curi>=h){ans=true;return;} + rep(i,n){ + if(unused&(1<>n>>h>>w; + rep(i,n)cin>>a[i]>>b[i]; + rep(i,h)rep(j,w)c[i][j]=-1; + ans=false; + solve((1< +#include +using namespace std; +using mint = atcoder::modint1000000007; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int N; + cin >> N; + vector A(N); + mint total = 1; + for (int &a : A) { + cin >> a; + total *= a; + } + + mint inv[7]; + for (int i = 1; i <= N; ++i) inv[i] = mint(i).inv(); + + int patterns = 1; + for (int i = 0; i < N; ++i) patterns *= N; + mint answer = 0; + + for (int code = 0; code < patterns; ++code) { + int rank[6], mask = 0, x = code; + for (int i = 0; i < N; ++i) { + rank[i] = x % N; + x /= N; + mask |= 1 << rank[i]; + } + // The ranks used must be exactly 0, 1, ..., k-1. + if ((mask & (mask + 1)) != 0) continue; + int k = __builtin_popcount((unsigned)mask); + + int M[6]; + fill(M, M + k, INT_MAX); + for (int i = 0; i < N; ++i) { + M[rank[i]] = min(M[rank[i]], A[i]); + } + + int lis[6], length = 0; + for (int i = 0; i < N; ++i) { + lis[i] = 1; + for (int j = 0; j < i; ++j) { + if (rank[j] < rank[i]) lis[i] = max(lis[i], lis[j] + 1); + } + length = max(length, lis[i]); + } + + vector endpoints(M, M + k); + sort(endpoints.begin(), endpoints.end()); + endpoints.erase(unique(endpoints.begin(), endpoints.end()), endpoints.end()); + + // dp[i]: the first i distinct values have been chosen in processed segments. + array dp{}; + dp[0] = 1; + int left = 0; + for (int right : endpoints) { + int width = right - left; + mint choose[7]; + choose[0] = 1; + for (int t = 1; t <= k; ++t) { + if (t > width) choose[t] = 0; + else choose[t] = choose[t - 1] * (width - t + 1) * inv[t]; + } + + array next = dp; // Choose no values from this segment. + for (int i = 0; i < k; ++i) { + for (int t = 1; i + t <= k; ++t) { + if (M[i + t - 1] < right) break; + next[i + t] += dp[i] * choose[t]; + } + } + dp = next; + left = right; + } + answer += dp[k] * length; + } + + cout << (answer / total).val() << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/arc104_e/statement.md b/problems/PA-05/Atcoder/arc104_e/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..6e463a5dee26e191319af32f96b93797c2fa670d --- /dev/null +++ b/problems/PA-05/Atcoder/arc104_e/statement.md @@ -0,0 +1,91 @@ +# E - Random LIS + +**Score**: 700 points + +## Problem Statement +Given is an integer sequence of length \(N\): \(A_1, A_2, \cdots, A_N\). + +An integer sequence \(X\), which is also of length \(N\), will be chosen randomly by independently choosing \(X_i\) from a uniform distribution on the integers \(1, 2, \ldots, A_i\) for each \(i (1 \leq i \leq N)\). + +Compute the expected value of the length of the longest increasing subsequence of this sequence \(X\), modulo \(1000000007\). + +More formally, under the constraints of the problem, we can prove that the expected value can be represented as a rational number, that is, an irreducible fraction \(\frac{P}{Q}\), and there uniquely exists an integer \(R\) such that \(R \times Q \equiv P \pmod {1000000007}\) and \(0 \leq R < 1000000007\), so print this integer \(R\). + +## Notes +A subsequence of a sequence \(X\) is a sequence obtained by extracting some of the elements of \(X\) and arrange them without changing the order. The longest increasing subsequence of a sequence \(X\) is the sequence of the greatest length among the strictly increasing subsequences of \(X\). + +## Constraints +- \(1 \leq N \leq 6\) +- \(1 \leq A_i \leq 10^9\) +- All values in input are integers. + +--- + +## Input +Input is given from Standard Input in the following format: +``` +N +A_1 A_2 \cdots A_N +``` + +--- + +## Output +Print the expected value modulo \(1000000007\). + +--- + +## Sample Input 1 +``` +3 +1 2 3 +``` + +### Sample Output 1 +``` +2 +``` + +\(X\) becomes one of the following, with probability \(1/6\) for each: +- \(X = (1,1,1)\), for which the length of the longest increasing subsequence is \(1\); +- \(X = (1,1,2)\), for which the length of the longest increasing subsequence is \(2\); +- \(X = (1,1,3)\), for which the length of the longest increasing subsequence is \(2\); +- \(X = (1,2,1)\), for which the length of the longest increasing subsequence is \(2\); +- \(X = (1,2,2)\), for which the length of the longest increasing subsequence is \(2\); +- \(X = (1,2,3)\), for which the length of the longest increasing subsequence is \(3\). + +Thus, the expected value of the length of the longest increasing subsequence is \((1 + 2 + 2 + 2 + 2 + 3) / 6 \equiv 2 \pmod {1000000007}\). + +--- + +## Sample Input 2 +``` +3 +2 1 2 +``` + +### Sample Output 2 +``` +500000005 +``` + +\(X\) becomes one of the following, with probability \(1/4\) for each: +- \(X = (1,1,1)\), for which the length of the longest increasing subsequence is \(1\); +- \(X = (1,1,2)\), for which the length of the longest increasing subsequence is \(2\); +- \(X = (2,1,1)\), for which the length of the longest increasing subsequence is \(1\); +- \(X = (2,1,2)\), for which the length of the longest increasing subsequence is \(2\). + +Thus, the expected value of the length of the longest increasing subsequence is \((1 + 2 + 1 + 2) / 4 = 3 / 2\). Since \(2 \times 500000005 \equiv 3 \pmod {1000000007}\), we should print \(500000005\). + +--- + +## Sample Input 3 +``` +6 +8 6 5 10 9 14 +``` + +### Sample Output 3 +``` +959563502 +``` diff --git a/problems/PA-05/Atcoder/arc105_c/data/1.in b/problems/PA-05/Atcoder/arc105_c/data/1.in new file mode 100644 index 0000000000000000000000000000000000000000..8a9f6933d0e289449ea2591b145310370702f0ec --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/data/1.in @@ -0,0 +1,100002 @@ +8 100000 +3 8 5 1 9 1 7 6 +73943961 99 +5483423 60 +10762957 54 +55474813 83 +12280585 34 +55245162 75 +50937378 41 +27066716 72 +35568208 52 +7336428 37 +62537904 68 +65591001 69 +44796097 10 +50561157 35 +53976296 79 +33222774 54 +85310700 92 +21443286 10 +38061946 11 +53210100 97 +65271970 92 +97441473 39 +73879384 65 +81857364 66 +37764084 23 +75292775 78 +56228530 15 +23289303 35 +39292211 34 +35820327 22 +59085306 69 +66757630 100 +28926960 73 +74960804 63 +20133601 24 +10324617 39 +31780861 82 +93366719 26 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58 +94301554 99 +83904608 38 +16204086 100 +39786348 11 +95787009 89 diff --git a/problems/PA-05/Atcoder/arc105_c/data/8.out b/problems/PA-05/Atcoder/arc105_c/data/8.out new file mode 100644 index 0000000000000000000000000000000000000000..193a61018de87a423d9cd33b138404c0e5cd03c8 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/data/8.out @@ -0,0 +1 @@ +399903742 diff --git a/problems/PA-05/Atcoder/arc105_c/data/9.in b/problems/PA-05/Atcoder/arc105_c/data/9.in new file mode 100644 index 0000000000000000000000000000000000000000..874b7915c36eb6d3e2012d25db6d3b09830b8667 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/data/9.in @@ -0,0 +1,3 @@ +8 1 +1 1 1 1 1 1 1 1 +100000000 1 diff --git a/problems/PA-05/Atcoder/arc105_c/data/9.out b/problems/PA-05/Atcoder/arc105_c/data/9.out new file mode 100644 index 0000000000000000000000000000000000000000..ff50ba38c346411e11abfd24018c33be37e48669 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/data/9.out @@ -0,0 +1 @@ +700000000 diff --git a/problems/PA-05/Atcoder/arc105_c/editorial.md b/problems/PA-05/Atcoder/arc105_c/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..28c7117a78e43fe1bac740a626c3ded3e80bf5a4 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/editorial.md @@ -0,0 +1,9 @@ +# C - Camels and Bridge (editorial) + +Source: https://atcoder.jp/contests/arc105/editorial/201 by evima + +The collapse of the bridge is unavoidable only if \(\max(w) > \min(v)\). From now on, we assume \(\max(w) \leq \min(v)\). + +Let us consider the case in which the camels must be in the order Camel \(1\), \(2\), \(\ldots\), \(N\). Then, we will have \(N(N-1)/2\) restrictions in this form: for \(i\) and \(j\) such that \(1 \leq i < j \leq N\), the distance between Camel \(i\) and \(j\) must be at least \(x_{i,j}\). \(x_{i,j}\) can be found in \(O(\log M)\) after precomputation. Here, the answer (for the fixed order of camels) corresponds to the length of the longest path from Vertex \(1\) \(N\) in the graph where there are directed edges of length \(x_{i,j}\) from Vertex \(i\) to \(j\). This graph does not have cycles, so we can find the length of the longest path in \(O(N^2)\) with dynamic programming. + +We can try all possible orders of camels and find the length of the longest path in each case to find the final answer. This can be done in \(O(N! N^2 \log M)\) time, which is fast enough. diff --git a/problems/PA-05/Atcoder/arc105_c/judge.json b/problems/PA-05/Atcoder/arc105_c/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..91448978fa47079c0c57be249cafab4d9f10d3f5 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 2.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 64, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/arc105_c/meta.json b/problems/PA-05/Atcoder/arc105_c/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..c6a80556dbd55130195364cebda5901a818be1e6 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "arc105_c", + "contest": "arc105", + "type": "ARC", + "letter": "C", + "title": "Camels and Bridge", + "url": "https://atcoder.jp/contests/arc105/tasks/arc105_c", + "difficulty": 1964, + "score": 500, + "time_limit_sec": 2.0, + "memory_limit_mb": 1024, + "n_tests": 64, + "editorial_url": "https://atcoder.jp/contests/arc105/editorial/201", + "code_source": "model", + "pid": "A3575" +} diff --git a/problems/PA-05/Atcoder/arc105_c/solution.cpp b/problems/PA-05/Atcoder/arc105_c/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..b44b29d5f56f47aedd4415e276dd1145dd55cded --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/solution.cpp @@ -0,0 +1,53 @@ +// arc105_c: C - Camels and Bridge +// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/arc105/editorial/201 +// Judged AC on all 64 tests, max 0.05 s of the 2 s limit (submission 1). + +#include +using namespace std; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int N, M; + cin >> N >> M; + vector w(N); + for (auto &x : w) cin >> x; + + vector> bridges(M); + for (int i = 0; i < M; ++i) { + long long l, v; + cin >> l >> v; + bridges[i] = {v, l}; + } + sort(bridges.begin(), bridges.end()); + if (*max_element(w.begin(), w.end()) > bridges[0].first) { + cout << -1 << '\n'; + return 0; + } + + vector capacities(M), prefix_max(M + 1, 0); + for (int i = 0; i < M; ++i) { + capacities[i] = bridges[i].first; + prefix_max[i + 1] = max(prefix_max[i], bridges[i].second); + } + + sort(w.begin(), w.end()); + long long answer = LLONG_MAX; + do { + vector dp(N, 0); + for (int j = 1; j < N; ++j) { + long long sum = w[j]; + for (int i = j - 1; i >= 0; --i) { + sum += w[i]; + int k = lower_bound(capacities.begin(), capacities.end(), sum) + - capacities.begin(); + dp[j] = max(dp[j], dp[i] + prefix_max[k]); + } + } + answer = min(answer, dp[N - 1]); + } while (next_permutation(w.begin(), w.end())); + + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/Atcoder/arc105_c/statement.md b/problems/PA-05/Atcoder/arc105_c/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..792ba1153d14f55b5e1b1b546caad17f1f7ac2b3 --- /dev/null +++ b/problems/PA-05/Atcoder/arc105_c/statement.md @@ -0,0 +1,114 @@ +# C - Camels and Bridge + +**Score:** 500 points + +## Problem Statement +There are \(N\) camels numbered \(1\) through \(N\). +The weight of Camel \(i\) is \(w_i\). +You will arrange the camels in a line and make them cross a bridge consisting of \(M\) parts. +Before they cross the bridge, you can choose their order in the line - it does not have to be Camel \(1, 2, \ldots, N\) from front to back - and specify the distance between each adjacent pair of camels to be any non-negative real number. +The camels will keep the specified distances between them while crossing the bridge. +The \(i\)-th part of the bridge has length \(l_i\) and weight capacity \(v_i\). +If the sum of the weights of camels inside a part (excluding the endpoints) exceeds \(v_i\), the bridge will collapse. +Determine whether it is possible to make the camels cross the bridge without it collapsing. If it is possible, find the minimum possible distance between the first and last camels in the line in such a case. +It can be proved that the answer is always an integer, so print an integer. + +## Constraints +- All values in input are integers. +- \(2 \leq N \leq 8\) +- \(1 \leq M \leq 10^5\) +- \(1 \leq w_i, l_i, v_i \leq 10^8\) + +--- + +## Input +Input is given from Standard Input in the following format: +``` +N M +w_1 w_2 ... w_N +l_1 v_1 +... +l_M v_M +``` + +## Output +If the bridge will unavoidably collapse when the camels cross the bridge, print `-1`. +Otherwise, print the minimum possible distance between the first and last camels in the line when the camels cross the bridge without it collapsing. + +--- + +## Sample Input 1 +``` +3 2 +1 4 2 +10 4 +2 6 +``` + +## Sample Output 1 +``` +10 +``` +*It is possible to make the camels cross the bridge without it collapsing by, for example, arranging them in the order 1, 3, 2 from front to back, and setting the distances between them to be 0, 10.* + +--- + +## Sample Input 2 +``` +2 1 +12 345 +1 1 +``` + +## Sample Output 2 +``` +-1 +``` +*Print -1 if the bridge will unavoidably collapse.* + +--- + +## Sample Input 3 +``` +8 1 +1 1 1 1 1 1 1 1 +100000000 1 +``` + +## Sample Output 3 +``` +700000000 +``` + +--- + +## Sample Input 4 +``` +8 20 +57 806 244 349 608 849 513 857 +778 993 +939 864 +152 984 +308 975 +46 860 +123 956 +21 950 +850 876 +441 899 +249 949 +387 918 +34 965 +536 900 +875 889 +264 886 +583 919 +88 954 +845 869 +208 963 +511 975 +``` + +## Sample Output 4 +``` +3802 +``` diff --git a/problems/PA-05/Atcoder/arc177_e/editorial.md b/problems/PA-05/Atcoder/arc177_e/editorial.md new file mode 100644 index 0000000000000000000000000000000000000000..9f70edeb13c004afe61c8567066fe03418ef845c --- /dev/null +++ b/problems/PA-05/Atcoder/arc177_e/editorial.md @@ -0,0 +1,178 @@ +# E - Wrong Scoreboard (editorial) + +Source: https://atcoder.jp/contests/arc177/editorial/9992 by evima + +### Solution to the problem + +In fact, there are only \(113\) scoring distributions for the contest, shown below, that need to be considered. + +Moreover, once the scoring distribution is determined, if we arrange the penalties so that the smaller \(R_i\) values correspond to smaller penalties (within ties), the squared error will always be minimized. Therefore, by using something like `std::sort`, we can solve this in approximately \(113N \log N\) computations. + +``` +[1, 1, 1, 1, 1] +[1, 1, 1, 1, 2] +[1, 1, 1, 1, 3] +[1, 1, 1, 1, 4] +[1, 1, 1, 2, 2] +[1, 1, 1, 2, 3] +[1, 1, 1, 2, 4] +[1, 1, 1, 2, 5] +[1, 1, 1, 3, 3] +[1, 1, 1, 3, 4] +[1, 1, 1, 3, 5] +[1, 1, 1, 3, 6] +[1, 1, 2, 2, 2] +[1, 1, 2, 2, 3] +[1, 1, 2, 2, 4] +[1, 1, 2, 2, 5] +[1, 1, 2, 2, 6] +[1, 1, 2, 3, 3] +[1, 1, 2, 3, 4] +[1, 1, 2, 3, 5] +[1, 1, 2, 3, 6] +[1, 1, 2, 3, 7] +[1, 1, 2, 4, 4] +[1, 1, 2, 4, 5] +[1, 1, 2, 4, 6] +[1, 1, 2, 4, 7] +[1, 1, 2, 4, 8] +[1, 1, 3, 3, 4] +[1, 1, 3, 3, 5] +[1, 1, 3, 4, 5] +[1, 1, 3, 4, 6] +[1, 1, 3, 5, 6] +[1, 1, 3, 5, 7] +[1, 1, 4, 4, 6] +[1, 1, 4, 5, 7] +[1, 1, 4, 6, 8] +[1, 2, 2, 2, 3] +[1, 2, 2, 2, 5] +[1, 2, 2, 3, 3] +[1, 2, 2, 3, 4] +[1, 2, 2, 3, 5] +[1, 2, 2, 3, 6] +[1, 2, 2, 3, 7] +[1, 2, 2, 3, 8] +[1, 2, 2, 4, 5] +[1, 2, 2, 4, 7] +[1, 2, 2, 5, 6] +[1, 2, 2, 5, 8] +[1, 2, 3, 3, 4] +[1, 2, 3, 3, 5] +[1, 2, 3, 3, 7] +[1, 2, 3, 4, 4] +[1, 2, 3, 4, 5] +[1, 2, 3, 4, 6] +[1, 2, 3, 4, 7] +[1, 2, 3, 4, 8] +[1, 2, 3, 4, 9] +[1, 2, 3, 4, 10] +[1, 2, 3, 5, 6] +[1, 2, 3, 5, 7] +[1, 2, 3, 5, 9] +[1, 2, 3, 6, 7] +[1, 2, 3, 6, 8] +[1, 2, 3, 6, 10] +[1, 2, 4, 4, 5] +[1, 2, 4, 4, 7] +[1, 2, 4, 5, 6] +[1, 2, 4, 5, 7] +[1, 2, 4, 5, 8] +[1, 2, 4, 6, 7] +[1, 2, 4, 6, 9] +[1, 2, 4, 7, 8] +[1, 2, 4, 7, 10] +[1, 2, 5, 6, 8] +[1, 2, 5, 6, 9] +[1, 2, 6, 7, 10] +[1, 3, 3, 4, 5] +[1, 3, 3, 5, 7] +[1, 3, 4, 4, 6] +[1, 3, 4, 5, 6] +[1, 3, 4, 5, 7] +[1, 3, 4, 6, 8] +[1, 3, 4, 7, 9] +[1, 3, 4, 8, 10] +[1, 3, 5, 6, 7] +[1, 3, 5, 7, 9] +[1, 3, 6, 8, 10] +[1, 4, 5, 6, 8] +[2, 2, 2, 3, 3] +[2, 2, 3, 3, 4] +[2, 2, 3, 4, 5] +[2, 2, 3, 5, 6] +[2, 2, 3, 6, 7] +[2, 2, 3, 7, 8] +[2, 3, 3, 4, 4] +[2, 3, 3, 4, 5] +[2, 3, 3, 4, 8] +[2, 3, 4, 4, 5] +[2, 3, 4, 5, 6] +[2, 3, 4, 5, 8] +[2, 3, 4, 6, 7] +[2, 3, 4, 7, 8] +[2, 3, 4, 8, 9] +[2, 3, 4, 9, 10] +[2, 3, 5, 6, 7] +[2, 3, 6, 7, 8] +[2, 4, 5, 6, 7] +[2, 4, 5, 7, 8] +[2, 5, 6, 8, 9] +[3, 4, 4, 5, 6] +[3, 4, 5, 6, 7] +[3, 4, 5, 6, 8] +[4, 5, 6, 7, 8] +``` + +### How to come up with the solution? + +Now, how can we come up with the above solution? In the latter part of this explanation, I would like to discuss how to think of the solution, also serving as a proof. + +#### Step 1 + +First, let’s consider what kind of scoring distributions we should investigate, using the following example: + +- The contest has \(3\) problems. +- Contestant \(1\) solved problem \(1\). +- Contestant \(2\) solved problem \(3\). +- Contestant \(3\) solved problems \(1\) and \(2\). + +Let the scores for problems \(1\), \(2\), and \(3\) be \(x_1\), \(x_2\), and \(x_3\), respectively. As we vary \(x_1, x_2, x_3\), the rankings of the contestants change at the following moments: + +- When \(x_3 = x_1\): The ranks of contestants \(1\) and \(2\) switch. +- When \(x_3 = x_1 + x_2\): The ranks of contestants \(2\) and \(3\) switch. + +Therefore, assuming the score for problem \(1\) is \(1\) point, the relationship between scores and rankings can be represented as shown in the diagram on the left below. Moreover, reading the problem statement carefully, we see that the scores must be non-decreasing \((x_1 \leq x_2 \leq x_3)\). Thus, the part of the diagram that satisfies this constraint looks like the diagram on the right. + +Since the ranks among contestants with the same total score can be freely swapped, we only need to investigate the boundaries of the regions, particularly the **intersections**. Therefore, in the example above, we only need to consider the following two scoring distributions: + +- \((x_1, x_2, x_3) = (1, 1, 1)\) +- \((x_1, x_2, x_3) = (1, 1, 2)\) + +[The Japanese texts below mean the following. 配点と順位の関係: relation between scores and ranks, 問 \(i\) の配点 \(x_i\): the score \(x_i\) of problem \(i\), 選手: contestants, 制約を満たす部分: the part satisfying the constraints] + +![ ](https://img.atcoder.jp/arc177/d5414bc63f0402d6b8deab35358c977e.png) + +#### Step 2 + +The considerations from Step 1 also apply when the contest has \(5\) problems. Specifically, the moments when the ranks of two contestants switch, as well as the constraint conditions \((1 =) \ x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5\), can be represented as several hyperplanes in a \(5\)-dimensional space. However, we only need to consider the points where four of these hyperplanes intersect. Note that we are assuming \(x_1 = 1\), as in Step 1. + +Now, how many hyperplanes are there? The emerging hyperplanes are always represented by equations of the form \(a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5 = 0\), where the values of \(a_1, a_2, a_3, a_4, a_5\) are \(-1\), \(0\), or \(+1\). Therefore, we only need to consider at most \(3^5 = 243\) hyperplanes. In reality, we do not need to consider those with all coefficients zero or those that are the same when signs are reversed, so there are only \(121\) hyperplanes to consider. + +Thus, the number of scoring distributions to consider is narrowed down to at most \(C(121, 4) = 8495410\). + +#### Step 3 + +With the considerations up to Step 2, we could solve this problem with approximately \(8495410 N \log N\) computations. However, given the constraint \(N \leq 3 \times 10^5\), this would unfortunately result in a TLE. So, let’s use the following idea: + +- Among the \(8495410\) cases, aren’t there many with exactly the same scoring distribution? + +Indeed, this idea is correct. For example, the following five “intersection points of four hyperplanes” all result in the same scoring distribution \((x_1, x_2, x_3, x_4, x_5) = (1, 1, 1, 1, 2)\): + +- Intersection of hyperplanes \(x_1 = x_2, x_2 = x_3, x_3 = x_4, x_1 + x_2 = x_5\) +- Intersection of hyperplanes \(x_1 = x_2, x_2 = x_3, x_2 = x_4, x_1 + x_3 = x_5\) +- Intersection of hyperplanes \(x_1 = x_2, x_1 = x_3, x_3 = x_4, x_1 + x_4 = x_5\) +- Intersection of hyperplanes \(x_1 = x_3, x_2 = x_3, x_3 = x_4, x_2 + x_3 = x_5\) +- Intersection of hyperplanes \(x_1 = x_3, x_2 = x_4, x_3 = x_4, x_2 + x_4 = x_5\) + +So, how many cases are there once duplicates and the ones without non-decreasing scores are removed? When checked with the following program, it turns out there are only \(113\) cases. Given the constraint \(N \leq 3 \times 10^5\), this allows us to achieve AC. diff --git a/problems/PA-05/Atcoder/arc177_e/judge.json b/problems/PA-05/Atcoder/arc177_e/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..accf7701b5b93648345e3b2f4faee406319a5744 --- /dev/null +++ b/problems/PA-05/Atcoder/arc177_e/judge.json @@ -0,0 +1,15 @@ +{ + "time_limit_s": 9.0, + "time_limit_source": "task page", + "memory_mb": 1024, + "memory_source": "task page", + "mode": "exact", + "validator_flags": "case_sensitive", + "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".out", + "n_sample": null, + "n_secret": 15, + "note": "no sample/secret split: the statement's samples are among the numbered tests" +} diff --git a/problems/PA-05/Atcoder/arc177_e/meta.json b/problems/PA-05/Atcoder/arc177_e/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..43792603d6734db87ecd6903757356b505d8a5d8 --- /dev/null +++ b/problems/PA-05/Atcoder/arc177_e/meta.json @@ -0,0 +1,16 @@ +{ + "task_id": "arc177_e", + "contest": "arc177", + "type": "ARC", + "letter": "E", + "title": "Wrong Scoreboard", + "url": "https://atcoder.jp/contests/arc177/tasks/arc177_e", + "difficulty": 3227, + "score": 1000, + "time_limit_sec": 9.0, + "memory_limit_mb": 1024, + "n_tests": 15, + "editorial_url": "https://atcoder.jp/contests/arc177/editorial/9992", + "code_source": "editorial", + "pid": "A4004" +} diff --git a/problems/PA-05/Atcoder/arc177_e/solution.cpp b/problems/PA-05/Atcoder/arc177_e/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..a337ad09b7a793247cc460911c22f2b41ddaba95 --- /dev/null +++ b/problems/PA-05/Atcoder/arc177_e/solution.cpp @@ -0,0 +1,92 @@ +// arc177_e: E - Wrong Scoreboard +// Code from the official editorial, https://atcoder.jp/contests/arc177/editorial/9992 (by evima) +// Judged AC on all 15 tests, max 3.14 s of the 9 s limit. + +#include +using namespace std; + +vector> Candidates = { + vector{1, 1, 1, 1, 1}, vector{1, 1, 1, 1, 2}, vector{1, 1, 1, 1, 3}, + vector{1, 1, 1, 1, 4}, vector{1, 1, 1, 2, 2}, vector{1, 1, 1, 2, 3}, + vector{1, 1, 1, 2, 4}, vector{1, 1, 1, 2, 5}, vector{1, 1, 1, 3, 3}, + vector{1, 1, 1, 3, 4}, vector{1, 1, 1, 3, 5}, vector{1, 1, 1, 3, 6}, + vector{1, 1, 2, 2, 2}, vector{1, 1, 2, 2, 3}, vector{1, 1, 2, 2, 4}, + vector{1, 1, 2, 2, 5}, vector{1, 1, 2, 2, 6}, vector{1, 1, 2, 3, 3}, + vector{1, 1, 2, 3, 4}, vector{1, 1, 2, 3, 5}, vector{1, 1, 2, 3, 6}, + vector{1, 1, 2, 3, 7}, vector{1, 1, 2, 4, 4}, vector{1, 1, 2, 4, 5}, + vector{1, 1, 2, 4, 6}, vector{1, 1, 2, 4, 7}, vector{1, 1, 2, 4, 8}, + vector{1, 1, 3, 3, 4}, vector{1, 1, 3, 3, 5}, vector{1, 1, 3, 4, 5}, + vector{1, 1, 3, 4, 6}, vector{1, 1, 3, 5, 6}, vector{1, 1, 3, 5, 7}, + vector{1, 1, 4, 4, 6}, vector{1, 1, 4, 5, 7}, vector{1, 1, 4, 6, 8}, + vector{1, 2, 2, 2, 3}, vector{1, 2, 2, 2, 5}, vector{1, 2, 2, 3, 3}, + vector{1, 2, 2, 3, 4}, vector{1, 2, 2, 3, 5}, vector{1, 2, 2, 3, 6}, + vector{1, 2, 2, 3, 7}, vector{1, 2, 2, 3, 8}, vector{1, 2, 2, 4, 5}, + vector{1, 2, 2, 4, 7}, vector{1, 2, 2, 5, 6}, vector{1, 2, 2, 5, 8}, + vector{1, 2, 3, 3, 4}, vector{1, 2, 3, 3, 5}, vector{1, 2, 3, 3, 7}, + vector{1, 2, 3, 4, 4}, vector{1, 2, 3, 4, 5}, vector{1, 2, 3, 4, 6}, + vector{1, 2, 3, 4, 7}, vector{1, 2, 3, 4, 8}, vector{1, 2, 3, 4, 9}, + vector{1, 2, 3, 4,10}, vector{1, 2, 3, 5, 6}, vector{1, 2, 3, 5, 7}, + vector{1, 2, 3, 5, 9}, vector{1, 2, 3, 6, 7}, vector{1, 2, 3, 6, 8}, + vector{1, 2, 3, 6,10}, vector{1, 2, 4, 4, 5}, vector{1, 2, 4, 4, 7}, + vector{1, 2, 4, 5, 6}, vector{1, 2, 4, 5, 7}, vector{1, 2, 4, 5, 8}, + vector{1, 2, 4, 6, 7}, vector{1, 2, 4, 6, 9}, vector{1, 2, 4, 7, 8}, + vector{1, 2, 4, 7,10}, vector{1, 2, 5, 6, 8}, vector{1, 2, 5, 6, 9}, + vector{1, 2, 6, 7,10}, vector{1, 3, 3, 4, 5}, vector{1, 3, 3, 5, 7}, + vector{1, 3, 4, 4, 6}, vector{1, 3, 4, 5, 6}, vector{1, 3, 4, 5, 7}, + vector{1, 3, 4, 6, 8}, vector{1, 3, 4, 7, 9}, vector{1, 3, 4, 8,10}, + vector{1, 3, 5, 6, 7}, vector{1, 3, 5, 7, 9}, vector{1, 3, 6, 8,10}, + vector{1, 4, 5, 6, 8}, vector{2, 2, 2, 3, 3}, vector{2, 2, 3, 3, 4}, + vector{2, 2, 3, 4, 5}, vector{2, 2, 3, 5, 6}, vector{2, 2, 3, 6, 7}, + vector{2, 2, 3, 7, 8}, vector{2, 3, 3, 4, 4}, vector{2, 3, 3, 4, 5}, + vector{2, 3, 3, 4, 8}, vector{2, 3, 4, 4, 5}, vector{2, 3, 4, 5, 6}, + vector{2, 3, 4, 5, 8}, vector{2, 3, 4, 6, 7}, vector{2, 3, 4, 7, 8}, + vector{2, 3, 4, 8, 9}, vector{2, 3, 4, 9,10}, vector{2, 3, 5, 6, 7}, + vector{2, 3, 6, 7, 8}, vector{2, 4, 5, 6, 7}, vector{2, 4, 5, 7, 8}, + vector{2, 5, 6, 8, 9}, vector{3, 4, 4, 5, 6}, vector{3, 4, 5, 6, 7}, + vector{3, 4, 5, 6, 8}, vector{4, 5, 6, 7, 8} +}; + +long long GetError(int N, vector> &A, vector &R, vector Score) { + vector> Sorted; + + // Determine Order + for (int i = 0; i < N; i++) { + int sum = 0; + for (int j = 0; j < 5; j++) sum += A[i][j] * Score[j]; + Sorted.push_back(make_pair(sum, -R[i])); + } + sort(Sorted.begin(), Sorted.end()); + reverse(Sorted.begin(), Sorted.end()); + + // Calculate Error + long long Error = 0; + for (int i = 0; i < N; i++) { + int idx1 = -Sorted[i].second; + int idx2 = i + 1; + Error += 1LL * (idx1 - idx2) * (idx1 - idx2); + } + return Error; +} + +int main() { + int T; cin >> T; + for (int t = 1; t <= T; t++) { + // Input + int N; cin >> N; + vector> A(N, vector(5, 0)); + vector R(N, 0); + for (int i = 0; i < N; i++) { + for (int j = 0; j < 5; j++) cin >> A[i][j]; + } + for (int i = 0; i < N; i++) cin >> R[i]; + + // Solve + long long Answer = (1LL << 60); + for (vector cand : Candidates) { + long long ret = GetError(N, A, R, cand); + Answer = min(Answer, ret); + } + cout << Answer << endl; + } + return 0; +} diff --git a/problems/PA-05/Atcoder/arc177_e/statement.md b/problems/PA-05/Atcoder/arc177_e/statement.md new file mode 100644 index 0000000000000000000000000000000000000000..a9f1e42974573400eff9b9e66c68562b9d07862c --- /dev/null +++ b/problems/PA-05/Atcoder/arc177_e/statement.md @@ -0,0 +1,142 @@ +# E - Wrong Scoreboard + +**Score:** 1000 points + +## Problem Statement + +In the AtCoder World Tour Finals 2800, \(N\) contestants participated, and a total of five problems were presented. Each problem is assigned an integer score of at least \(1\) point, and the problems are numbered so that the scores are **non-decreasing** from problem \(1\) to problem \(5\). There are no partial points. Similar to the usual AtCoder rules, ranking is done as follows. **In this problem, we do not consider the situation where multiple contestants have the same total score and penalty.** + +
Ranking +The contestant with the higher total score ranks higher. In case of a tie, the one with the smaller penalty ranks higher. +
+ +Now, Aoki, a reporter covering the finals, noted the following information: +1. The number of participants \(N\). +2. Which problems each contestant solved. \(A_{i,j}=1\) means the \(i\)-th contestant \((1 \leq i \leq N)\) correctly solved problem \(j\), and \(A_{i,j}=0\) means they did not. +3. The rank of each contestant. The \(i\)-th contestant \((1 \leq i \leq N)\) was ranked \(R_i\)-th. + +However, when he started writing the article, he realized he did not note the scores and penalties. Furthermore, he realized there might be inconsistencies in the information he noted. Now, solve the following problem. + +> Assume that he correctly noted information 1 and 2. Let \(D_i\) be the actual rank of contestant \(i\) \((1 \leq i \leq N)\), and find the minimum possible total squared error \((D_1 - R_1)^2 + (D_2 - R_2)^2 + \dots + (D_N - R_N)^2\). + +You have \(T\) test cases to process. + +## Constraints +- \(1 \leq T \leq 10^5\) +- \(2 \leq N \leq 3 \times 10^5\) +- Each of \(A_{i,1}, A_{i,2}, A_{i,3}, A_{i,4}, A_{i,5}\) is \(0\) or \(1\) \((1 \leq i \leq N)\). +- The sum of \(A_{i,1}, A_{i,2}, A_{i,3}, A_{i,4}, A_{i,5}\) is at least \(1\) \((1 \leq i \leq N)\). +- \(1 \leq R_i \leq N\) \((1 \leq i \leq N)\). +- \(R_1, R_2, \dots, R_N\) are distinct. +- **The total value of \(N\) across all test cases is at most \(3 \times 10^5\).** +- All input values are integers. + +## Input +The input is given from Standard Input in the following format. Here \(\mathrm{case}_i\) represents the \(i\)-th test case \((1 \leq i \leq T)\). + +``` +T +\mathrm{case}_1 +\mathrm{case}_2 +\vdots +\mathrm{case}_T +``` +Each test case is given in the following format: + +``` +N +A_{1,1} A_{1,2} A_{1,3} A_{1,4} A_{1,5} +A_{2,1} A_{2,2} A_{2,3} A_{2,4} A_{2,5} +\vdots +A_{N,1} A_{N,2} A_{N,3} A_{N,4} A_{N,5} +R_1 R_2 \cdots R_N +``` + +## Output +Print the answers. + +## Sample Input 1 +``` +6 +4 +0 1 1 0 0 +1 0 0 1 0 +1 1 0 1 0 +1 0 1 0 0 +1 2 3 4 +8 +0 1 0 0 0 +1 1 0 1 0 +0 1 1 0 1 +1 0 0 0 0 +1 1 0 1 0 +0 1 0 0 0 +0 0 0 1 0 +0 1 1 1 1 +7 4 2 8 3 6 5 1 +6 +1 1 0 0 0 +0 0 1 0 0 +1 1 1 0 0 +0 0 0 1 0 +1 1 1 1 0 +0 0 0 0 1 +1 2 3 4 5 6 +6 +1 1 0 0 0 +0 0 1 0 0 +1 1 1 0 0 +0 0 0 1 0 +1 1 1 1 0 +0 0 0 0 1 +6 5 4 3 2 1 +20 +0 0 0 0 1 +0 0 1 0 0 +1 1 0 0 1 +1 0 1 0 1 +0 0 0 1 1 +0 0 1 1 1 +1 1 1 1 0 +1 1 0 1 0 +0 0 1 1 0 +1 0 1 0 0 +0 1 0 0 1 +0 1 1 1 1 +1 1 1 1 1 +0 1 0 1 0 +1 0 0 0 1 +1 1 1 0 0 +0 1 1 1 0 +0 0 0 1 0 +1 1 1 0 1 +1 1 0 1 1 +7 18 3 5 19 11 13 2 4 10 14 15 17 6 16 9 8 12 1 20 +15 +0 0 1 1 0 +0 0 0 1 0 +0 0 0 0 1 +0 0 1 1 1 +1 1 0 0 1 +0 1 1 1 0 +1 1 1 1 1 +0 1 1 0 1 +1 1 0 1 0 +1 0 0 1 1 +1 0 1 0 0 +1 1 0 1 1 +0 1 0 1 0 +1 1 0 0 0 +0 1 0 0 1 +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 +``` + +## Sample Output 1 +``` +6 +0 +26 +0 +1054 +428 +``` diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.ans b/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.ans new file mode 100644 index 0000000000000000000000000000000000000000..56f36300e8f4527ae03c0d74906d140c3f8ebc54 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.ans @@ -0,0 +1,179 @@ +Case 1: equivalent solutions +Case 2: inconsistent solutions +Case 3: no solution +Case 4: inconsistent solutions +Case 5: equivalent solutions +Case 6: inconsistent solutions +Case 7: inconsistent solutions +Case 8: inconsistent solutions +Case 9: equivalent solutions +Case 10: inconsistent solutions +Case 11: no solution +Case 12: equivalent solutions +Case 13: inconsistent solutions +Case 14: equivalent solutions +Case 15: inconsistent solutions +Case 16: inconsistent solutions +Case 17: no solution +Case 18: equivalent solutions +Case 19: inconsistent solutions +Case 20: equivalent solutions +Case 21: equivalent solutions +Case 22: inconsistent solutions +Case 23: no solution +Case 24: equivalent solutions +Case 25: equivalent solutions +Case 26: inconsistent solutions +Case 27: equivalent solutions +Case 28: inconsistent solutions +Case 29: no solution +Case 30: equivalent solutions +Case 31: equivalent solutions +Case 32: inconsistent solutions +Case 33: equivalent solutions +Case 34: inconsistent solutions +Case 35: no solution +Case 36: equivalent solutions +Case 37: equivalent solutions +Case 38: equivalent solutions +Case 39: equivalent solutions +Case 40: inconsistent solutions +Case 41: no solution +Case 42: inconsistent solutions +Case 43: inconsistent solutions +Case 44: equivalent solutions +Case 45: equivalent solutions +Case 46: inconsistent solutions +Case 47: no solution +Case 48: inconsistent solutions +Case 49: equivalent solutions +Case 50: equivalent solutions +Case 51: equivalent solutions +Case 52: inconsistent solutions +Case 53: no solution +Case 54: no solution +Case 55: no solution +Case 56: inconsistent solutions +Case 57: inconsistent solutions +Case 58: inconsistent solutions +Case 59: no solution +Case 60: equivalent solutions +Case 61: no solution +Case 62: inconsistent solutions +Case 63: equivalent solutions +Case 64: inconsistent solutions +Case 65: no solution +Case 66: equivalent solutions +Case 67: no solution +Case 68: inconsistent solutions +Case 69: equivalent solutions +Case 70: inconsistent solutions +Case 71: no solution +Case 72: no solution +Case 73: no solution +Case 74: equivalent solutions +Case 75: equivalent solutions +Case 76: inconsistent solutions +Case 77: no solution +Case 78: no solution +Case 79: inconsistent solutions +Case 80: equivalent solutions +Case 81: no solution +Case 82: equivalent solutions +Case 83: equivalent solutions +Case 84: equivalent solutions +Case 85: equivalent solutions +Case 86: equivalent solutions +Case 87: inconsistent solutions +Case 88: inconsistent solutions +Case 89: inconsistent solutions +Case 90: inconsistent solutions +Case 91: no solution +Case 92: no solution +Case 93: no solution +Case 94: no solution +Case 95: no solution +Case 96: inconsistent solutions +Case 97: inconsistent solutions +Case 98: inconsistent solutions +Case 99: inconsistent solutions +Case 100: inconsistent solutions +Case 101: inconsistent solutions +Case 102: equivalent solutions +Case 103: equivalent solutions +Case 104: equivalent solutions +Case 105: equivalent solutions +Case 106: equivalent solutions +Case 107: equivalent solutions +Case 108: equivalent solutions +Case 109: equivalent solutions +Case 110: no solution +Case 111: equivalent solutions +Case 112: no solution +Case 113: equivalent solutions +Case 114: equivalent solutions +Case 115: no solution +Case 116: inconsistent solutions +Case 117: inconsistent solutions +Case 118: inconsistent solutions +Case 119: no solution +Case 120: equivalent solutions +Case 121: equivalent solutions +Case 122: equivalent solutions +Case 123: equivalent solutions +Case 124: equivalent solutions +Case 125: equivalent solutions +Case 126: inconsistent solutions +Case 127: inconsistent solutions +Case 128: inconsistent solutions +Case 129: inconsistent solutions +Case 130: inconsistent solutions +Case 131: equivalent solutions +Case 132: no solution +Case 133: no solution +Case 134: inconsistent solutions +Case 135: inconsistent solutions +Case 136: inconsistent solutions +Case 137: inconsistent solutions +Case 138: equivalent solutions +Case 139: equivalent solutions +Case 140: equivalent solutions +Case 141: equivalent solutions +Case 142: equivalent solutions +Case 143: equivalent solutions +Case 144: equivalent solutions +Case 145: equivalent solutions +Case 146: equivalent solutions +Case 147: equivalent solutions +Case 148: equivalent solutions +Case 149: equivalent solutions +Case 150: equivalent solutions +Case 151: equivalent solutions +Case 152: equivalent solutions +Case 153: equivalent solutions +Case 154: equivalent solutions +Case 155: equivalent solutions +Case 156: equivalent solutions +Case 157: equivalent solutions +Case 158: equivalent solutions +Case 159: inconsistent solutions +Case 160: inconsistent solutions +Case 161: inconsistent solutions +Case 162: inconsistent solutions +Case 163: inconsistent solutions +Case 164: inconsistent solutions +Case 165: inconsistent solutions +Case 166: inconsistent solutions +Case 167: no solution +Case 168: equivalent solutions +Case 169: inconsistent solutions +Case 170: inconsistent solutions +Case 171: inconsistent solutions +Case 172: equivalent solutions +Case 173: inconsistent solutions +Case 174: equivalent solutions +Case 175: inconsistent solutions +Case 176: equivalent solutions +Case 177: inconsistent solutions +Case 178: inconsistent solutions +Case 179: no solution diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.in b/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.in new file mode 100644 index 0000000000000000000000000000000000000000..46d99a2fa850491c2f85fd632ba715d74b715473 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/data/secret-1.in @@ -0,0 +1,359 @@ +3 0 4 0 1 4 +-2 -4 -1 3 3 -4 +0 1 1 1 2 1 +1 2 2 2 3 2 +1 0 2 0 3 0 +3 3 1 1 2 2 +1 0 2 0 3 0 +3 2 1 1 2 2 +2 3 0 6 1 2 +2 3 0 6 1 2 +33 -60 33 -56 32 -58 +177 11 -128 -19 177 -49 +-27 7 -27 8 -27 6 +-201 -30 -201 -329 -201 269 +-6 0 -3 -1 0 -2 +134 -14 -14 -194 60 -122 +4 7 8 -6 15 0 +15 0 4 7 8 -6 +-10 -57 -2 -30 6 -3 +-10 -57 6 -3 -2 -30 +41 49 49 58 32 51 +421 -99 -148 -44 -179 160 +-2 3 0 -1 -1 0 +201 129 201 93 -36 57 +0 -3 0 2 0 1 +110 90 -115 -48 -115 -94 +0 10 9 -4 18 -18 +-286 89 314 -95 14 1 +-12 10 -10 -3 3 0 +-10 -3 3 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new file mode 100644 index 0000000000000000000000000000000000000000..f1ef8ef7fb4dba12ed5a28b889b921cd9b22cae9 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/judge.json @@ -0,0 +1,14 @@ +{ + "time_limit_s": 10.0, + "time_limit_source": "default (10 s; the statement gives none)", + "memory_mb": 2048, + "memory_source": "default (the statement gives none)", + "mode": "exact", + "validator_flags": null, + "comparison": "Kattis default output validator: token-wise, whitespace- and case-insensitive", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".ans", + "n_sample": 0, + "n_secret": 1 +} diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/meta.json b/problems/PA-05/ICPC_WF/2011/B-affine-mess/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..4f76c04a179e4c50799096d18f404e1dd1b5605e --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/meta.json @@ -0,0 +1,9 @@ +{ + "year": "2011", + "letter": "B", + "title": "Affine Mess", + "slug": "affine-mess", + "source_pdf": "contest_problems.pdf", + "page_start": 3, + "page_end": 4 +} diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/page.html b/problems/PA-05/ICPC_WF/2011/B-affine-mess/page.html new file mode 100644 index 0000000000000000000000000000000000000000..dbd4ed1bdd169422d91f9e0188e6f0cb2c97936d --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/page.html @@ -0,0 +1,203 @@ + B. Affine Mess | Ethan Pham
EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2011
ICPC 2011

B. Affine Mess

Problem ID: affine Tess L. Ation ran into a little problem last week when she demonstrated the beta version of her new drawing software. On the screen she had an elegant demonstration design that illustrated every feature of her program; it had taken her ho...

Updated May 21, 2026
Track ICPC
Year 2011
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

+Statement text + +Statement PDF +

Tess L. Ation ran into a little problem last week when she demonstrated the beta version of her new drawing software. On the screen she had an elegant demonstration design that illustrated every feature of her program; it had taken her hours to produce it. She was just putting the finishing touches on it as a group of potential investors entered the room to see the demonstration. The presentation went well. Near the end, Tess clicked on a control panel button and told her audience, “This is the ‘snap to grid’ control. It forces control points, such as vertices, to jump to the nearest grid point. Here, let me show you,” and she placed three bright red dots on the screen. Each one appeared at the grid point nearest to where she clicked. (“Luckily all control points in my demo design were already at integer coordinates. But I will have to remember to delete these three red dots before I save my diagram,” she thought to herself.) “Now I’ll step into the next room and get out of your way so you can discuss the system among yourselves and get a closer look at the screen, but please don’t touch anything, since I haven’t saved that file yet.” A few minutes later, the group joined Tess. One of the visitors stepped up to Tess and said, “I hope you don’t mind, but I wanted to try it myself. Don’t worry, I just played with the x-scale and y-scale controls a little bit.” The next person said, “Sorry if this is a problem, but I really wanted to get a feel for the speed of display, so I just played around with the translation tool.” And a third person said, “I couldn’t resist just one tiny test: I rotated the image just so I could see all of the vertices snap to the nearest grid points after the rotation.” The person who played with the rotation tool remembered going first, but the other two could not recall their order. The three remembered only a few details of the changes. The x- and y-scaling factors had been (possibly negative) non- zero integers; the center of scaling was the origin (0, 0). The x- and y-translation amounts had been integers. Rotation had been specified by a point with integer coordinates (x, y) on the perimeter of a square of width 20 centered at the origin (hence, −10 ≤ x, y ≤ 10 and the absolute value of x or y or both was 10). The tool rotated the drawing around the origin such that the positive x-axis would pass through (x, y) afterwards. Snapping took place after this rotation (coordinates with a fractional part of 0.5 were rounded away from zero). After they left, Tess looked at her design – it was completely changed! She had not yet implemented the “undo” feature, and she had not saved the diagram prior to giving the demonstration. However, the three identical red dots were still there (transformed to other integer grid locations, of course), and Tess could remember the integer coordinates where she had originally placed them. Obviously, someone else might have altered the drawing without saying anything to her, but she could write a program to see if it was possible to reconstruct the sequence of alterations. Can you too?

Input

The input contains several test cases. Each test case consists of six pairs of integers xi and yi (−500 ≤ xi , yi ≤ 500 for 1 ≤ i ≤ 6), three pairs per input line. The first three pairs represent the distinct initial locations of the three red dots. The last three pairs represent the distinct final locations of the three dots. The indexing of the pairs in each group of three is not significant: for example, (x1 , y1 ) could have been mapped to any of (x4 , y4 ), (x5 , y5 ) or (x6 , y6 ). The last test case is followed by a line with six zeros.

+

ICPC 2011 World Finals Problem B: Affine Mess

Output

For each test case, display its case number followed by one of the following three messages:

+
  • “equivalent solutions” to indicate that there are one or more valid transformations, and all of them have the same effect on the whole drawing (no matter what the whole drawing looks like).
  • “inconsistent solutions” to indicate that there are several valid transformations, but in general not all of them map the entire drawing in the same way (some drawing is mapped differently by two valid transforma- tions).
  • “no solution” to indicate that neither of the first two cases occurs.
+

A valid transformation is a combination of rotation, translation and scaling (or rotation, scaling and translation) which satisfies the restrictions described above and maps the initial set of red dots to the final set (occupying all three final locations). Follow the format of the sample output.

Sample Tests

Sample
Sample Input 
  3 0 4    0 1 4
+  -2 -4    -1 3 3 -4
+  0 1 1    1 2 1
+  1 2 2    2 3 2
+  1 0 2    0 3 0
+  3 3 1    1 2 2
+  1 0 2    0 3 0
+  3 2 1    1 2 2
+  2 3 0    6 1 2
+  2 3 0    6 1 2
+  0 0 0    0 0 0
+
+ICPC 2011 World Finals Problem B: Affine Mess
Sample Output 
Case    1:   equivalent solutions
+Case    2:   inconsistent solutions
+Case    3:   no solution
+Case    4:   inconsistent solutions
+Case    5:   equivalent solutions
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

+

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

+

Algorithm

+

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

+

Correctness Proof

+

We prove that the algorithm returns the correct answer.

+

Lemma 1.

+

State the first key claim.

+

Proof.

+

Provide a concise proof.

+

Lemma 2.

+

State the next claim if needed.

+

Proof.

+

Provide a concise proof.

+

Theorem.

+

The algorithm outputs the correct answer for every valid input.

+

Proof.

+

Combine the lemmas and finish the argument.

+

Complexity Analysis

+

State the running time and memory usage in terms of the input size.

+

Implementation Notes

+

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#include <bits/stdc++.h>
+using namespace std;
+
+namespace {
+
+void solve() {
+    // Fill in the full solution logic for the problem here.
+}
+
+}  // namespace
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    solve();
+    return 0;
+}
+
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TeX write-upcompetitive_programming/icpc/2011/B-affine-mess/solution.texC++ implementationcompetitive_programming/icpc/2011/B-affine-mess/solution.cppStatement textcompetitive_programming/icpc/2011/B-affine-mess/statement.txtStatement PDFcompetitive_programming/icpc/2011/B-affine-mess/statement.pdfMetadatacompetitive_programming/icpc/2011/B-affine-mess/meta.jsonYear packetcompetitive_programming/icpc/2011/contest_problems.pdf
Endnote

+Personal site by Ethan Pham (Pham Van Nghia). Project Euler, competitive programming, essays, + and code notes from Ho Chi Minh City, Vietnam. +

Project Euler, competitive programming, and notes on the things I keep studying.

\ No newline at end of file diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.cpp b/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..285a62d10c7cfd0d427b3d09ee36fb2c03523a13 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.cpp @@ -0,0 +1,126 @@ +#include +using namespace std; + +using ll = long long; + +struct Point { + int x, y; +}; + +struct Rotation { + int a, b, d; + double inv; + + // Round n / sqrt(d), with ties away from zero. + int snap(int n) const { + ll v = abs(n); + ll square = 4 * v * v; + ll k = (ll)(v * inv + 0.5); + + // The floating-point estimate is checked with exact integer arithmetic. + while (square >= (2 * k + 1) * (2 * k + 1) * d) ++k; + while (k > 0 && square < (2 * k - 1) * (2 * k - 1) * d) --k; + return int(n < 0 ? -k : k); + } +}; + +struct Fit { + bool valid; + ll scale, shift; + // For a valid fit, scale == 0 denotes an unrestricted nonzero scale. +}; + +Fit fit(const array& from, const array& to) { + int j = 1; + while (j < 3 && from[j] == from[0]) ++j; + + if (j == 3) { + return {to[0] == to[1] && to[0] == to[2], 0, to[0]}; + } + + ll d = from[j] - from[0]; + ll e = to[j] - to[0]; + if (e == 0 || e % d != 0) return {false, 0, 0}; + + ll scale = e / d; + ll shift = to[0] - scale * from[0]; + for (int i = 0; i < 3; ++i) { + if (scale * from[i] + shift != to[i]) return {false, 0, 0}; + } + return {true, scale, shift}; +} + +string solve(const array& initial, + const array& finalPoints, + const vector& rotations) { + bool found = false; + array first{}; + + for (int id = 0; id < (int)rotations.size(); ++id) { + const Rotation& r = rotations[id]; + array x, y; + for (int i = 0; i < 3; ++i) { + x[i] = r.snap(r.a * initial[i].x - r.b * initial[i].y); + y[i] = r.snap(r.b * initial[i].x + r.a * initial[i].y); + } + + array perm{0, 1, 2}; + do { + array targetX, targetY; + for (int i = 0; i < 3; ++i) { + targetX[i] = finalPoints[perm[i]].x; + targetY[i] = finalPoints[perm[i]].y; + } + + Fit fx = fit(x, targetX); + if (!fx.valid) continue; + Fit fy = fit(y, targetY); + if (!fy.valid) continue; + + // A free scale gives genuinely different maps of the whole plane. + if (fx.scale == 0 || fy.scale == 0) { + return "inconsistent solutions"; + } + + array current{ + id, fx.scale, fy.scale, fx.shift, fy.shift + }; + if (found && first != current) return "inconsistent solutions"; + first = current; + found = true; + } while (next_permutation(perm.begin(), perm.end())); + } + + return found ? "equivalent solutions" : "no solution"; +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + vector rotations; + // Opposite rotation directions are equivalent after negating both scales. + for (int a = 0; a <= 10; ++a) { + for (int b = -10; b <= 10; ++b) { + if (max(abs(a), abs(b)) != 10 || (a == 0 && b < 0)) continue; + int d = a * a + b * b; + rotations.push_back({a, b, d, 1.0 / sqrt((double)d)}); + } + } + + array initial, finalPoints; + int caseNumber = 0; + while (cin >> initial[0].x >> initial[0].y + >> initial[1].x >> initial[1].y + >> initial[2].x >> initial[2].y) { + bool finished = true; + for (Point p : initial) { + if (p.x != 0 || p.y != 0) finished = false; + } + if (finished) break; + + for (Point& p : finalPoints) cin >> p.x >> p.y; + cout << "Case " << ++caseNumber << ": " + << solve(initial, finalPoints, rotations) << '\n'; + } +} diff --git a/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.tex b/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.tex new file mode 100644 index 0000000000000000000000000000000000000000..81ca3c21275bf32d86b33031438211d5bbd1d2b5 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2011/B-affine-mess/solution.tex @@ -0,0 +1,98 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{enumitem} + +\title{ICPC World Finals 2011\\B. Affine Mess} +\author{} +\date{} + +\begin{document} +\maketitle + +\section*{Problem Summary} + +Given unordered triples of distinct integer points before and after an unknown transformation, classify the possible transformations. First rotate about the origin toward an integer point $(a,b)$ with $\max(|a|,|b|)=10$ and snap coordinates to integers, with ties away from zero. Then apply nonzero signed integer scales on the axes and an integer translation, in either order. + +Report whether no transformation works, all valid transformations agree on every integer-grid drawing, or two valid transformations can act differently. + +\section*{Key Observations} + +Let $\rho$ denote coordinatewise rounding and define +\[ + d=a^2+b^2,\qquad + R=\frac{1}{\sqrt d}\begin{pmatrix}a&-b\\b&a\end{pmatrix}, + \qquad D=\operatorname{diag}(s_x,s_y). +\] +All maps have the form $F(p)=D\rho(Rp)+t$. A translation $u$ before scaling becomes $Du$ afterward, while scaling then translation permits every integer $t$. Therefore searching this order suffices. + +Opposite boundary points give opposite rotated coordinates. Since $\rho(-z)=-\rho(z)$, negating both scales compensates for choosing the opposite point. Of the $80$ directions, keep only the $40$ with $a>0$, or $a=0$ and $b>0$. + +Fixing a rotation and a dot correspondence leaves independent equations $v_i=su_i+t$ on each axis. Two unequal source coordinates determine $s,t$; constant source and target triples leave $s$ free. + +Two canonical tuples $(\text{rotation identifier},s_x,s_y,t_x,t_y)$ describe the same grid map exactly when equal, as proved below. + +\section*{Algorithm} + +Precompute the $40$ retained rotations. For each test case: +\begin{enumerate}[leftmargin=*] + \item For each rotation, rotate and snap the three initial points, storing their two coordinate triples in arrays. + \item Enumerate all six permutations of the final points. For each correspondence, fit each axis separately, using snapped coordinates $u_i$ and matched final coordinates $v_i$, for $i=0,1,2$: + \begin{itemize}[leftmargin=*] + \item If all $u_i$ equal $u_0$, accept the axis exactly when all $v_i$ equal $v_0$, and mark the scale as free. Any nonzero integer $s$ then works with $t=v_0-su_0$. + \item Otherwise choose the first $j$ with $u_j\ne u_0$. Let $\delta=u_j-u_0$ and $e=v_j-v_0$. Reject if $e=0$ or $\delta$ does not divide $e$. Otherwise set $s=e/\delta$, $t=v_0-su_0$, and check $su_i+t=v_i$ for all three indices. + \end{itemize} + \item Discard the correspondence unless both axes fit. If either valid fit has a free scale, report \texttt{inconsistent solutions} immediately. + \item Otherwise form the canonical five-entry tuple. Store the first valid tuple. If a later tuple differs, report \texttt{inconsistent solutions} immediately. +\end{enumerate} + +If the search finishes, report \texttt{no solution} when no tuple was found, and \texttt{equivalent solutions} otherwise. Only one tuple and a Boolean flag are stored. + +\section*{Correctness Proof} + +\paragraph{Lemma 1.} +The reduced search represents exactly all legal maps. + +\paragraph{Proof.} +Moving translation before scaling to afterward replaces $u$ by $Du$. Replacing an omitted rotation by its opposite and negating both scales preserves the map because rounding is odd. Conversely, every retained rotation with nonzero integer scales and an integer translation afterward is a legal sequence. \qed + +\paragraph{Lemma 2.} +For a fixed rotation and correspondence, the fitting procedure describes exactly all valid scale and shift choices on each axis. + +\paragraph{Proof.} +If $u_j\ne u_0$, subtracting the two image equations forces $s=(v_j-v_0)/(u_j-u_0)$. It must be a nonzero integer, and then $t=v_0-su_0$ is forced. Checking all three equations is necessary and sufficient. If the source coordinates coincide, their images must also coincide. When they do, every nonzero integer $s$, paired with $t=v_0-su_0$, satisfies the equations. \qed + +\paragraph{Lemma 3.} +Two representations using retained rotations agree on every integer point if and only if their tuples agree. + +\paragraph{Proof.} +Equal tuples clearly give equal maps. Conversely, suppose $F(p)=D\rho(Rp)+t$ and $F'(p)=D'\rho(R'p)+t'$ agree on every integer point. At the origin this gives $t=t'$. Since rounding changes each coordinate by at most $1/2$, for either standard basis vector $e_j$, +\[ + \lim_{n\to\infty}\frac{F(ne_j)}{n}=DR e_j. +\] +Equality with the corresponding limits for $F'$ implies $DR=D'R'$. + +The first row is $s_x$ times the unit vector $(a,-b)/\sqrt d$. Equality of the first rows, with nonzero scales, forces the two unit vectors to be equal or opposite. Opposite directions cannot both be retained, so the directions agree. Each direction has a unique boundary point, hence the rotation identifiers and $s_x$ agree. The second rows then force equality of $s_y$. Together with $t=t'$, all tuple entries agree. \qed + +\paragraph{Theorem.} +The algorithm returns the required classification. + +\paragraph{Proof.} +Lemma 1 covers every possible map. Each valid map induces one of the six correspondences, since the final dots are distinct, and Lemma 2 ensures its scale and shift choices are found. The same lemma ensures every accepted pair of fits maps all three dots correctly. + +A free scale allows two different nonzero values with their corresponding shifts, producing valid but different tuples. By Lemma 3 these maps are inequivalent. That lemma also justifies reporting inconsistency for two different concrete tuples. Otherwise, finding no tuple means no valid map exists; finding one means every valid map has that tuple and acts identically on every drawing. Thus each reported classification is correct. \qed + +\section*{Complexity Analysis} + +Each of $40\cdot6$ candidates needs constant work on three points. With the fixed numeric bounds, snapping also takes constant time. Thus time and space per case are $O(1)$; processing $T$ cases takes $O(T)$ time. The rotation table, coordinate triples, and saved tuple all have constant size. + +\section*{Implementation Notes} + +\begin{itemize}[leftmargin=*] + \item For snapping $n/\sqrt d$, let $m=|n|$. The code estimates $k=\lfloor m/\sqrt d+1/2\rfloor$ using a stored floating-point reciprocal, then checks half-integer boundaries exactly. It increments $k$ while $4m^2\ge d(2k+1)^2$, then decrements it while $k>0$ and $4m^2 D. Factors | Ethan Pham
EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2013
ICPC 2013

D. Factors

2013 World Finals St. Petersburg HOSTED BY ITMO Problem D Factors Time Limit: 2 seconds The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely repre- sented as a product of one or more primes. While unique, several...

Updated May 21, 2026
Track ICPC
Year 2013
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

+Statement text + +Statement PDF +
Time limit 2 seconds

The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely repre- sented as a product of one or more primes. While unique, several arrangements of the prime factors may be possible. For example:

+
10 = 2 · 5                            20 = 2 · 2 · 5
+   =5·2                                   =2·5·2
+                                          =5·2·2
+

Let f (k) be the number of different arrangements of the prime factors of k. So f (10) = 2 and f (20) = 3. Given a positive number n, there always exists at least one number k such that f (k) = n. We want to know the smallest such k.

Input

The input consists of at most 1 000 test cases, each on a separate line. Each test case is a positive integer n < 263 .

Output

For each test case, display its number n and the smallest number k > 1 such that f (k) = n. The numbers in the input are chosen such that k < 263 .

Sample Tests

Sample 1
Sample Input 
 1
+ 2
+ 3
+ 105
Sample Output 
1 2
+2 6
+3 12
+105 720
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

+

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

+

Algorithm

+

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

+

Correctness Proof

+

We prove that the algorithm returns the correct answer.

+

Lemma 1.

+

State the first key claim.

+

Proof.

+

Provide a concise proof.

+

Lemma 2.

+

State the next claim if needed.

+

Proof.

+

Provide a concise proof.

+

Theorem.

+

The algorithm outputs the correct answer for every valid input.

+

Proof.

+

Combine the lemmas and finish the argument.

+

Complexity Analysis

+

State the running time and memory usage in terms of the input size.

+

Implementation Notes

+

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#include <bits/stdc++.h>
+using namespace std;
+
+namespace {
+
+void solve() {
+    // Fill in the full solution logic for the problem here.
+}
+
+}  // namespace
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    solve();
+    return 0;
+}
+
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TeX write-upcompetitive_programming/icpc/2013/D-factors/solution.texC++ implementationcompetitive_programming/icpc/2013/D-factors/solution.cppStatement textcompetitive_programming/icpc/2013/D-factors/statement.txtStatement PDFcompetitive_programming/icpc/2013/D-factors/statement.pdfMetadatacompetitive_programming/icpc/2013/D-factors/meta.jsonYear packetcompetitive_programming/icpc/2013/contest_problems.pdf
Endnote

+Personal site by Ethan Pham (Pham Van Nghia). Project Euler, competitive programming, essays, + and code notes from Ho Chi Minh City, Vietnam. +

Project Euler, competitive programming, and notes on the things I keep studying.

\ No newline at end of file diff --git a/problems/PA-05/ICPC_WF/2013/D-factors/solution.cpp b/problems/PA-05/ICPC_WF/2013/D-factors/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..c206685b67ac946d3b6efd277137b3bc3b74cf18 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/D-factors/solution.cpp @@ -0,0 +1,55 @@ +#include +#include +#include +#include +using namespace std; + +using u64 = unsigned long long; +constexpr u64 LIMIT = static_cast(numeric_limits::max()); +constexpr int MAX_FACTORS = 62; +// The product of these 16 primes exceeds LIMIT, so this list is sufficient. +constexpr int primes[] = { + 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 +}; +u64 choose[MAX_FACTORS + 1][MAX_FACTORS + 1]; +unordered_map best; + +void enumerate(int pos, int maxExponent, int total, u64 value, u64 ways) { + if (total > 0) { + auto [it, inserted] = best.emplace(ways, value); + if (!inserted) it->second = min(it->second, value); + } + if (pos == 16) return; + + // Exponents on consecutive primes must be nonincreasing. + for (int exponent = 1; + exponent <= maxExponent && total + exponent <= MAX_FACTORS; + ++exponent) { + if (value > LIMIT / primes[pos]) break; + value *= primes[pos]; + + // Insert 'exponent' identical new factors among the existing ones. + u64 multiplier = choose[total + exponent][exponent]; + if (ways > LIMIT / multiplier) break; + enumerate(pos + 1, exponent, total + exponent, + value, ways * multiplier); + } +} + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + for (int i = 0; i <= MAX_FACTORS; ++i) { + choose[i][0] = choose[i][i] = 1; + for (int j = 1; j < i; ++j) + choose[i][j] = choose[i - 1][j - 1] + choose[i - 1][j]; + } + best.reserve(50000); + enumerate(0, MAX_FACTORS, 0, 1, 1); + + u64 n; + while (cin >> n) + cout << n << ' ' << best.at(n) << '\n'; + return 0; +} diff --git a/problems/PA-05/ICPC_WF/2013/D-factors/solution.tex b/problems/PA-05/ICPC_WF/2013/D-factors/solution.tex new file mode 100644 index 0000000000000000000000000000000000000000..f20bb00427d824da8bb9dfe294f5dc6e5c99ac38 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/D-factors/solution.tex @@ -0,0 +1,107 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{enumitem} + +\title{ICPC World Finals 2013\\D. Factors Time Limit: 2 seconds} +\author{} +\date{} + +\begin{document} +\maketitle + +\section*{Problem Summary} + +Let $f(k)$ count the distinct orderings of the prime factors of $k$, with repeated primes indistinguishable. For each positive input $n$, output $n$ and the smallest integer $k>1$ with $f(k)=n$. There are at most $1000$ queries; both $n$ and its answer are guaranteed to be less than $2^{63}$. + +\section*{Key Observations} + +For a prime factorization +\[ + k=p_0^{e_0}p_1^{e_1}\cdots p_{r-1}^{e_{r-1}}, + \qquad s=\sum_{j=0}^{r-1}e_j, +\] +the arrangement count is the multinomial coefficient +\[ + f(k)=\frac{s!}{e_0!e_1!\cdots e_{r-1}!}. +\] +Only the exponents matter, not the primes themselves. To minimize the product for fixed exponents, use consecutive primes starting at $2$ and assign larger exponents to smaller primes. Thus we need only enumerate +\[ + k=2^{e_0}3^{e_1}5^{e_2}\cdots, + \qquad e_0\ge e_1\ge e_2\ge\cdots\ge 1. +\] + +Let $L=2^{63}-1$. Any product at most $L$ has at most $62$ prime factors counted with multiplicity, since $63$ factors give a product at least $2^{63}$. The product of the first $16$ primes exceeds $L$, so at most $15$ distinct primes occur. + +We can update the arrangement count without factorials. If a prefix has $s$ factors and $w$ arrangements, appending $e$ copies of a new prime gives +\[ + w'=w\binom{s+e}{e}. +\] +Choose the positions for the new copies, then fill the remaining positions with an old arrangement. Both the product and the arrangement count are nondecreasing as an exponent grows or more primes are appended. Therefore exceeding $L$ in either quantity permits pruning. + +\section*{Algorithm} + +\begin{enumerate}[leftmargin=*] + \item Precompute \texttt{choose}, the binomial coefficients $\binom{t}{e}$ for $0\le e\le t\le 62$, using Pascal's recurrence. Store the first $16$ primes, through $53$. Initialize an empty \texttt{unordered\_map} named \texttt{best}, mapping arrangement counts to minimum products. + \item Search recursively with state $(i,a,s,v,w)$: the next prime index, its maximum allowed exponent, the number of factors already chosen, their product, and their arrangement count. Start at $(0,62,0,1,1)$. + \item At each state with $s>0$, insert $v$ under key $w$ in \texttt{best}, or replace the existing value by its minimum with $v$. Record every nonempty prefix, but not the empty initial state. + \item If $i=16$, return. Otherwise set a local running product $x=v$. Try $e=1,2,\ldots,a$ while $s+e\le 62$: + \begin{itemize}[leftmargin=*] + \item If $x>\lfloor L/p_i\rfloor$, stop the loop. Otherwise multiply $x$ by $p_i$, giving $x=vp_i^e$. + \item Let $m=\binom{s+e}{e}$. If $w>\lfloor L/m\rfloor$, stop the loop. + \item Recurse into $(i+1,e,s+e,x,wm)$. The new exponent limit $e$ enforces the nonincreasing order. + \end{itemize} + \item After precomputation, read queries until end of input. Output each $n$ followed by the value stored under key $n$ in \texttt{best}. +\end{enumerate} + +\section*{Correctness Proof} + +\paragraph{Lemma 1.} +The smallest integer with any fixed arrangement count uses consecutive primes starting at $2$, with nonincreasing positive exponents. + +\paragraph{Proof.} +Replacing a used prime by a smaller unused prime preserves the exponents and hence the arrangement count, while decreasing the product. A minimum therefore cannot skip a prime. + +If primes $p1$ with $f(k)=n$. + +\paragraph{Proof.} +Let $k^*$ be the minimum answer. The input guarantees $k^*\le L$, and Lemma 1 puts it in the searched form. Since $f(k^*)=n\le L$, Lemma 3 ensures it is recorded. By Lemma 2, every product stored under key $n$ is a valid answer. Taking the minimum under that key therefore leaves exactly $k^*$, which the algorithm outputs. \qed + +\section*{Complexity Analysis} + +Let $S$ be the number of visited search states, including the root, and $Q$ the number of queries. Each loop iteration either creates a child or is the single failed iteration ending that state's loop. With expected constant-time hash-table operations, the total expected time is $O(62^2+S+Q)$, including the binomial table. Space usage is $O(62^2+S)$; the recursion has at most $16$ active calls. + +For this fixed product limit, even without the arrangement-count cutoff there are only $43\,607$ product-feasible states. The precomputation is therefore small and is shared by all queries. + +\section*{Implementation Notes} + +\begin{itemize}[leftmargin=*] + \item Products, counts, and binomial coefficients use \texttt{unsigned long long}, but products and counts are capped at the signed $64$-bit maximum $L$. Test against $L$ divided by the multiplier before multiplying; checking afterward could be too late. + \item The largest needed binomial coefficient is $\binom{62}{31} 1 such that f (k) = n. The +numbers in the input are chosen such that k < 2^63. + + Sample Input 1 Sample Output 1 + 1 1 2 + 2 2 6 + 3 3 12 + 105 105 720 + +This page is intentionally left blank. diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.ans new file mode 100644 index 0000000000000000000000000000000000000000..7ed6ff82de6bcc2a78243fc9c54d3ef5ac14da69 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.ans @@ -0,0 +1 @@ +5 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.in new file mode 100644 index 0000000000000000000000000000000000000000..2647b264be38c40153d4a920fee0a87549534396 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-001.in @@ -0,0 +1,2 @@ +1 2 +V1 V2 V1 V1 V2 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.ans new file mode 100644 index 0000000000000000000000000000000000000000..1e8b314962144c26d5e0e50fd29d2ca327864913 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.ans @@ -0,0 +1 @@ +6 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.in new file mode 100644 index 0000000000000000000000000000000000000000..564e97e93c431d3c8487459925062df5b0a7ca46 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-002.in @@ -0,0 +1,2 @@ +2 1 +V1 V2 V1 V1 V2 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.ans new file mode 100644 index 0000000000000000000000000000000000000000..64bb6b746dceaf12b0ba8c08f310b0426babde44 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.ans @@ -0,0 +1 @@ +30 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.in new file mode 100644 index 0000000000000000000000000000000000000000..8041f2a312296fbada46a712af1fdc73e2bdf2a0 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-003.in @@ -0,0 +1,2 @@ +1 2 +R10 V1 V2 V1 E diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.ans new file mode 100644 index 0000000000000000000000000000000000000000..98d9bcb75a685dfbfd60f611c309410152935b3d --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.ans @@ -0,0 +1 @@ +17 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.in new file mode 100644 index 0000000000000000000000000000000000000000..684674e0da9df471e632dcd4acce7718bdf7bc2c --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/sample-harvard-004.in @@ -0,0 +1,2 @@ +4 1 +V1 R2 V2 V4 R2 V1 E V3 E diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.ans new file mode 100644 index 0000000000000000000000000000000000000000..d6ee9012f2ab8aa0961ad84e2066b4518eafea4f --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.ans @@ -0,0 +1 @@ +1000000000000 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.in new file mode 100644 index 0000000000000000000000000000000000000000..d95a473d04d60e67684ab40a17f9f901ed7a2764 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-001.in @@ -0,0 +1,2 @@ +13 1 +R1000000 R1000000 V1 E E diff --git 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V6 V6 V6 V6 V6 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.ans new file mode 100644 index 0000000000000000000000000000000000000000..fc9d0823b49e54250bead37f5e4c6d79ef821246 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.ans @@ -0,0 +1 @@ +998000000 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.in new file mode 100644 index 0000000000000000000000000000000000000000..cfee8eeef637aa47dd8abaa7dfc17f73b1ba6565 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-007.in @@ -0,0 +1,2 @@ +1 3 +R1000000 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 V3 E diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-008.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-008.ans new file mode 100644 index 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V11 V5 V4 V8 V1 V4 V11 V6 V9 V8 V12 V6 V7 V4 V10 V2 V8 V8 V11 V6 V12 V12 V2 V10 V12 V9 V2 V3 V13 V2 V13 V13 V11 V12 V1 V6 V4 V12 V9 V6 V2 V10 V13 V2 V7 V6 V7 V1 V2 V10 V12 V4 V12 V1 V4 V7 V2 V2 V6 V4 V9 V3 V10 V12 V7 V5 V2 V12 V3 V11 V6 V10 V11 V9 V1 V8 V12 V5 V12 V9 V6 V8 V12 V9 V6 V11 V1 V9 V1 V11 V13 V6 V5 V7 V11 V2 V9 V5 V8 V2 V13 V12 V10 V4 V10 V11 V8 V9 V9 R260 V1 V8 V10 V7 V12 V7 V11 V3 V2 V11 V3 V5 V5 V4 V5 V4 V7 V8 R72 V12 V4 V12 V3 V8 R458 V11 V9 V9 V4 V3 V7 V2 V9 V13 V1 V5 V5 V3 V9 V10 V2 V8 V11 V4 V8 V12 R343 V3 R311 V10 E V10 V3 V9 V12 V7 V13 V9 V9 V5 V7 V10 V4 V9 V8 V11 V10 V2 V3 V4 V3 V4 V7 V13 V1 V12 V9 V2 E V10 V4 V13 V7 V3 V4 V5 V2 V1 V9 V5 V9 V2 V3 V13 V6 V11 V11 V10 V13 V6 V11 V3 V5 V9 V13 V12 V6 V4 V10 V9 V10 E V8 R670 V10 V9 V5 V6 V8 R12 V12 V12 V7 V13 V3 V6 V6 V6 V6 V7 V2 V8 V2 V9 V7 V9 E R4 V3 V6 E V9 V9 V2 E V3 V11 R87 V1 V2 V10 E V8 V13 V6 V10 V6 R25 V6 E V13 V7 V1 R1 V12 E E V8 V2 V8 V7 V13 R5 R1 V6 E E V10 E V10 V10 V9 V9 V5 V13 V13 V10 V2 V7 V9 V10 V4 V11 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V12 V3 V9 V2 R676 V10 V1 V9 V13 V12 V13 R281 V1 V7 V7 V9 V13 V6 V1 V8 V8 V8 V7 V5 V3 V5 V4 V2 V3 V4 V6 V2 R104 V7 R5 V11 E E V8 V6 V1 V7 V6 V5 V13 V1 V12 R12 V3 V5 V7 V12 E V4 V2 V11 V9 V12 V3 V5 V4 V4 E V12 V8 R16 V10 V13 E V2 V2 V12 V1 V10 E V4 V9 V5 V4 V2 V1 V12 V7 R157 V13 E V3 V9 V8 V9 V5 V11 V5 V4 V11 V5 V1 R116 V6 E V4 V5 V8 V3 V6 V4 E V4 V12 V9 V10 V6 R89 V9 V9 V1 V7 V2 E V10 V4 V7 V5 V13 V11 V11 R2 V7 V11 V11 V10 V11 V10 V6 V7 R4 V5 E V1 R2 V3 E E R1 V9 E diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.ans new file mode 100644 index 0000000000000000000000000000000000000000..77be5c17bfa5b1de5ce046ee574e5ba38f02a471 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.ans @@ -0,0 +1 @@ +1005073621717 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.in new file mode 100644 index 0000000000000000000000000000000000000000..ebbd031115652ec7d8354089a031c358c8b68e2c --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-036.in @@ -0,0 +1,2 @@ +13 3 +V2 R9 V7 V5 V3 V9 R997 V9 V11 V4 R789 V6 V9 V2 V7 V4 V3 V5 V4 V12 V1 V4 V5 V1 V9 E V3 V12 V11 V4 E V9 V6 V7 V8 V9 V7 V11 V12 V7 V8 V9 V7 V10 E V5 R71 V7 V2 V1 V13 V5 V11 V1 V4 R961 V7 V7 V9 V10 V2 V5 V9 V3 V6 V3 R444 V13 V5 V6 V8 V3 V13 V13 V6 V13 V7 R302 V12 V11 V4 V8 V8 E R155 V5 E V5 V4 V2 V12 V10 V11 V3 E V9 V12 V4 V8 V1 V5 V6 V12 V9 V1 V5 V1 V2 V3 V10 V6 V7 V4 V2 V8 V6 V7 V2 V8 V9 V9 V4 V11 V13 V9 V5 V8 V4 R617 V2 V11 V5 V12 V1 R927 V1 V2 V1 V3 V7 E E E V7 V7 V2 V1 V9 V3 V11 V1 V1 V4 V9 V1 V9 V5 R100 V7 V7 V13 V4 V5 V3 V5 V4 R502 V12 V11 V8 V2 V9 V5 R613 V4 V13 R113 V2 V11 V5 E V7 V13 R1 R1 R1 V10 E E E E V12 V6 V13 V13 V11 V3 V8 V8 V4 V5 V7 V8 V5 V7 V3 V10 V4 V10 R410 R1 V5 E E V4 V1 V5 V8 V8 V12 V7 V11 V13 V6 V8 V9 V4 V4 V6 V8 V7 V11 V10 V11 R78 V6 E V8 V3 V4 V12 V10 V2 V5 V5 V7 R10 V12 V9 E V12 V10 V5 V6 E V2 V3 V4 V12 V1 V4 V7 V2 V4 V9 V11 V2 V1 V9 V4 R10 V9 V9 V11 V6 V13 V4 V5 V6 V4 V10 V7 V10 V2 V9 R2 R2 V5 E E E V13 V2 V7 V7 V8 V13 V1 V2 V11 E R19 V9 V3 E V11 V7 V1 V5 V9 V3 V5 E R54 V7 E R4 V11 E diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.ans b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.ans new file mode 100644 index 0000000000000000000000000000000000000000..e6b5d216cfee3e2ff8e7dfe873c04dc3fe24ea63 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.ans @@ -0,0 +1 @@ +1066327047874 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.in b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.in new file mode 100644 index 0000000000000000000000000000000000000000..820773165a7bf166d688bd7ac78837a367bfd45d --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/data/secret-harvard-037.in @@ -0,0 +1,2 @@ +13 4 +V4 V5 V3 V11 R354 V1 R627 V6 V12 V10 V1 V7 V6 R217 V11 V12 V1 V7 V13 V7 V2 V11 R677 V6 V7 V13 V8 V7 V7 V5 V5 V5 V4 V6 V2 V1 V7 V5 V2 V4 V1 V5 V2 V9 V1 V3 V8 V10 V5 V10 V7 R2 V2 E E V12 V6 V10 V4 V12 V12 V10 V11 V6 V7 V6 V10 V2 V11 V5 V11 V6 V9 V13 V5 V8 V6 V8 V9 V13 V7 V7 V11 V13 V2 V11 V7 V2 V12 R261 V2 E V7 V8 V9 V8 R44 V13 V2 V13 E V2 V5 V1 V11 V12 V4 V6 V5 V8 V7 V5 R1 R1 V1 E E E V7 V13 V9 V6 V8 V3 V8 V5 V8 V12 V7 V8 V13 V10 V7 V2 V13 R153 V12 E V7 V13 V9 V5 V5 V7 V7 V6 R28 V12 E V10 V1 V3 V13 V9 V2 V12 E V2 V12 V2 V8 V6 V4 V7 V12 V2 V13 V7 R411 V9 E V10 V9 V2 V6 V10 V7 V10 V2 V1 R51 R2 R1 V12 E E E V5 V2 V6 V10 V8 V5 V3 V9 V1 V2 V3 E V3 V2 R11 R1 V12 E E V9 R3 V10 E V7 diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/judge.json b/problems/PA-05/ICPC_WF/2013/E-harvard/judge.json new file mode 100644 index 0000000000000000000000000000000000000000..bc51495c82f4577158c27d4848e99343c04c79d1 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/judge.json @@ -0,0 +1,14 @@ +{ + "time_limit_s": 10.0, + "time_limit_source": "statement", + "memory_mb": 2048, + "memory_source": "default (the statement gives none)", + "mode": "exact", + "validator_flags": null, + "comparison": "Kattis default output validator: token-wise, whitespace- and case-insensitive", + "validator": null, + "tests_dir": "data", + "answer_suffix": ".ans", + "n_sample": 4, + "n_secret": 37 +} diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/meta.json b/problems/PA-05/ICPC_WF/2013/E-harvard/meta.json new file mode 100644 index 0000000000000000000000000000000000000000..ffdcc8c7f70e32fde6594cd53aff0d65f317d9c6 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/meta.json @@ -0,0 +1,9 @@ +{ + "year": "2013", + "letter": "E", + "title": "Harvard", + "slug": "harvard", + "source_pdf": "contest_problems.pdf", + "page_start": 9, + "page_end": 10 +} diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/page.html b/problems/PA-05/ICPC_WF/2013/E-harvard/page.html new file mode 100644 index 0000000000000000000000000000000000000000..b09a801f17d62ef7b69134418aff7bf4b0b39e66 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/page.html @@ -0,0 +1,211 @@ + E. Harvard | Ethan Pham
EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2013
ICPC 2013

E. Harvard

2013 World Finals St. Petersburg HOSTED BY ITMO Problem E Harvard Time Limit: 10 seconds The term “Harvard architecture” applies to a computer that has physically separate memories for instructions and data. The term originated with the Harvard Mark I com...

Updated May 21, 2026
Track ICPC
Year 2013
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

+Statement text + +Statement PDF +
Time limit 10 seconds

computer that has physically separate memories for instructions and data. The term originated with the Harvard Mark I computer, delivered by IBM in 1944, which used paper tape for instruc- tions and relays for data. Some modern microcontrollers use the Harvard architecture – but not paper tape and relays! Data memory is organized in banks, each con- taining the same number of data items. Each data-referencing instruction has a byte offset f Picture from Wikimedia Commons to a bank, and a bit a that is used to select the bank to be referenced. If a is 0, then bank 0 is referenced. If a is 1, then the value in a bank select register (BSR) identifies the bank to be used. Assume each instruction takes the same time to execute, and there is an instruction that can set the BSR’s value. For example, suppose there are 4 banks of 8 bytes each. To access location 5, either use a single instruction with a = 0 and f = 5, or set the BSR to 0 in one instruction and then use an instruction with a = 1 and f = 5. The first approach is faster since it does not require setting the BSR. Now suppose (with the same memory) the location to access is 20. Only one approach will work here: execute an instruction that sets the BSR to 2 (unless the BSR already has the value 2) and then use an instruction with a = 1 and f = 4. A program is a sequence of operations. Each operation is either

+
  • a variable reference, written as Vi, where i is a positive integer, or
+
  • a repetition, written as Rn <program> E, where n is a positive integer and <program> is an arbitrary program. This operation is equivalent to n sequential occurrences of <program>.
+

Your problem is to determine the minimum running time of programs. In particular, given the number and size of the memory banks and a program to be executed, find the minimum number of instructions (which reference memory location and possibly set the BSR) that must be executed to run the program. To do this you must identify a mapping of variables to memory banks that yields the smallest execution time, and report that execution time – that is, the number of memory references and BSR register settings required. The BSR’s value is initially undefined, and changes only when an instruction explicitly sets its value.

+
                                                              ICPC 2013
+2013 World Finals
+                                                        St. Petersburg
+                                                         HOSTED BY   ITMO

Input

The input consists of a single test case. A test case consists of two lines. The first line contains two integers b and s, where 1 ≤ b ≤ 13 is the number of memory banks and 1 ≤ s ≤ 13 is the number of variables that can be stored in each memory bank. The second line contains a non-empty program with at most 1 000 space-separated elements (each Rn, Vi, and E counts as one element). You may assume the following:

+
  • In a repetition Rn, the number of repetitions satisfies 1 ≤ n ≤ 106 .
+
  • In a loop operation Rn <program> E, the loop body <program> is not empty.
+
  • In a variable reference Vi, the variable index satisfies 1 ≤ i ≤ min(b · s, 13).
+
  • The total number of variable references performed by an execution of the program is at most 1012 .

Output

Display the minimum number of instructions that must be executed to complete the program.

Sample Tests

Sample 1
Sample Input 
 1 2
+ V1 V2 V1 V1 V2
Sample Output 
5
Sample 2
Sample Input 
 2 1
+ V1 V2 V1 V1 V2
Sample Output 
6
Sample 3
Sample Input 
 1 2
+ R10 V1 V2 V1 E
Sample Output 
30
Sample 4
Sample Input 
 4 1
+ V1 R2 V2 V4 R2 V1 E V3 E
Sample Output 
17
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

+

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

+

Algorithm

+

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

+

Correctness Proof

+

We prove that the algorithm returns the correct answer.

+

Lemma 1.

+

State the first key claim.

+

Proof.

+

Provide a concise proof.

+

Lemma 2.

+

State the next claim if needed.

+

Proof.

+

Provide a concise proof.

+

Theorem.

+

The algorithm outputs the correct answer for every valid input.

+

Proof.

+

Combine the lemmas and finish the argument.

+

Complexity Analysis

+

State the running time and memory usage in terms of the input size.

+

Implementation Notes

+

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#include <bits/stdc++.h>
+using namespace std;
+
+namespace {
+
+void solve() {
+    // Fill in the full solution logic for the problem here.
+}
+
+}  // namespace
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    solve();
+    return 0;
+}
+
Resources

Source Files and Assets

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TeX write-upcompetitive_programming/icpc/2013/E-harvard/solution.texC++ implementationcompetitive_programming/icpc/2013/E-harvard/solution.cppStatement textcompetitive_programming/icpc/2013/E-harvard/statement.txtStatement PDFcompetitive_programming/icpc/2013/E-harvard/statement.pdfMetadatacompetitive_programming/icpc/2013/E-harvard/meta.jsonYear packetcompetitive_programming/icpc/2013/contest_problems.pdf
Endnote

+Personal site by Ethan Pham (Pham Van Nghia). Project Euler, competitive programming, essays, + and code notes from Ho Chi Minh City, Vietnam. +

Project Euler, competitive programming, and notes on the things I keep studying.

\ No newline at end of file diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/solution.cpp b/problems/PA-05/ICPC_WF/2013/E-harvard/solution.cpp new file mode 100644 index 0000000000000000000000000000000000000000..3008b30386d8ff8a624de011a41777e8f2602603 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/solution.cpp @@ -0,0 +1,178 @@ +#include +using namespace std; +using int64 = long long; + +struct Node { + int var = -1; + int64 times = 1; // Number of executions of the loop body. + int64 back = 0; // Total number of boundaries between its repetitions. + vector children; +}; + +struct PartitionSearch { + int m, capacity, maxBanks; + vector weight; + int bankMask[13]{}; + int bankSize[13]{}; + int64 best; + + // Banks are numbered in the order in which their first variable appears. + // Thus every unlabeled partition is visited exactly once. + void dfs(int pos, int used, int64 saved) { + if (pos == m) { + best = max(best, saved); + return; + } + int bit = 1 << pos; + for (int k = 0; k < used; ++k) { + if (bankSize[k] == capacity) continue; + int old = bankMask[k]; + bankMask[k] = old | bit; + ++bankSize[k]; + dfs(pos + 1, used, saved + weight[old | bit] - weight[old]); + --bankSize[k]; + bankMask[k] = old; + } + if (used < maxBanks) { + bankMask[used] = bit; + bankSize[used] = 1; + dfs(pos + 1, used + 1, saved); + } + } + + int64 solve() { + best = 0; + dfs(0, 0, 0); + return best; + } +}; + +int main() { + ios::sync_with_stdio(false); + cin.tie(nullptr); + + int b, s; + if (!(cin >> b >> s)) return 0; + + vector nodes(1); // Root is a sequence executed once. + nodes.reserve(1001); + vector st{0}; + int present = 0; + int64 references = 0; + string token; + while (cin >> token) { + if (token[0] == 'E') { + st.pop_back(); + continue; + } + int parent = st.back(); + Node node; + if (token[0] == 'V') { + node.var = stoi(token.substr(1)) - 1; + present |= 1 << node.var; + references += nodes[parent].times; + } else { + int64 repeat = stoll(token.substr(1)); + node.times = nodes[parent].times * repeat; + node.back = nodes[parent].times * (repeat - 1); + } + int id = (int)nodes.size(); + nodes.push_back(move(node)); + nodes[parent].children.push_back(id); + if (token[0] == 'R') st.push_back(id); + } + + int n = 0; + int rename[13]; + for (int v = 0; v < 13; ++v) { + if (present & (1 << v)) rename[v] = n++; + } + for (Node &node : nodes) { + if (node.var >= 0) node.var = rename[node.var]; + } + + if (n <= s) { + cout << references << '\n'; + return 0; + } + if (n <= 2 * s) { + cout << references + 1 << '\n'; + return 0; + } + + PartitionSearch search; + search.m = n - s; + search.capacity = s; + search.maxBanks = min(b - 1, search.m); + search.weight.resize(1 << search.m); + + vector first(nodes.size()), last(nodes.size()); + int64 answer = LLONG_MAX; + + // Filling bank zero cannot increase the required number of BSR settings. + for (int zero = 0; zero < (1 << n); ++zero) { + if (__builtin_popcount((unsigned)zero) != s) continue; + int64 adjacency[13][13]{}; + auto addEdge = [&](int u, int v, int64 count) { + if (u >= 0 && v >= 0 && u != v) { + adjacency[u][v] += count; + adjacency[v][u] += count; + } + }; + + // Delete bank-zero references and count consecutive remaining pairs. + // Children occur after their parent in the node array. + for (int id = (int)nodes.size() - 1; id >= 0; --id) { + const Node &node = nodes[id]; + if (node.var >= 0) { + first[id] = last[id] = ((zero >> node.var) & 1) ? -1 : node.var; + } else { + int f = -1, l = -1; + for (int child : node.children) { + if (first[child] < 0) continue; + if (f < 0) f = first[child]; + else addEdge(l, first[child], node.times); + l = last[child]; + } + first[id] = f; + last[id] = l; + addEdge(l, f, node.back); + } + } + + vector remaining; + for (int v = 0; v < n; ++v) { + if (!(zero & (1 << v))) remaining.push_back(v); + } + int m = search.m; + int64 graph[13][13]{}; + int64 totalEdges = 0; + for (int i = 0; i < m; ++i) { + for (int j = i + 1; j < m; ++j) { + graph[i][j] = graph[j][i] = adjacency[remaining[i]][remaining[j]]; + totalEdges += graph[i][j]; + } + } + + // weight[mask] is the number of transitions saved by putting all + // variables in mask in the same nonzero bank. + search.weight[0] = 0; + for (int mask = 1; mask < (1 << m); ++mask) { + int v = __builtin_ctz((unsigned)mask); + int rest = mask & (mask - 1); + int64 value = search.weight[rest]; + for (int t = rest; t; t &= t - 1) { + int u = __builtin_ctz((unsigned)t); + value += graph[v][u]; + } + search.weight[mask] = value; + } + + int64 saved = search.solve(); + // One initial BSR setting, then one for each between-bank transition. + answer = min(answer, references + 1 + totalEdges - saved); + } + + cout << answer << '\n'; + return 0; +} diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/solution.tex b/problems/PA-05/ICPC_WF/2013/E-harvard/solution.tex new file mode 100644 index 0000000000000000000000000000000000000000..f4af36cf39c87c586f56c4f62b44fbc2b601340e --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/solution.tex @@ -0,0 +1,109 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{enumitem} + +\title{ICPC World Finals 2013\\E. Harvard} +\author{} +\date{} + +\begin{document} +\maketitle + +\section*{Problem Summary} + +Assign variables to $b$ banks of capacity $s$. Each reference costs one instruction. Bank $0$ is accessible directly without changing the bank select register (BSR); other banks require the BSR to select them. Setting this initially undefined register costs another instruction. Minimize the total instruction count. + +The program uses nested repetitions, with at most $1000$ tokens, $13$ variables, and $10^{12}$ executed references. + +\section*{Key Observations} + +Let $n$ count distinct referenced variables and $N$ count executed references. Since $N$ is fixed, only BSR settings need optimization. + +For a fixed assignment, delete bank-$0$ references when counting BSR settings. A nonempty remaining sequence needs one initial setting and one at every change of bank. Deleted references still contribute to $N$. + +Moving a variable into free space in bank $0$ cannot hurt, so when $n>s$, enumerate only size-$s$ sets for bank $0$. Nonzero banks are interchangeable: enumerate partitions of the remaining variables, not assignments to labeled banks. + +\section*{Algorithm} + +\begin{enumerate}[leftmargin=*] + \item Parse with a stack into an ordered tree of reference leaves and repetition nodes beneath a root sequence. For a loop repeated $r$ times whose parent body executes $p$ times, store global multiplicities + \[ + \texttt{times}=pr,\qquad \texttt{back}=p(r-1). + \] + The root has values $1$ and $0$. Each reference adds its parent's \texttt{times} to $N$. Compress the occurring variable indices. + + If $n\le s$, return $N$. If $ss$, some optimum fills bank $0$, and an assignment's BSR cost is one plus the number of bank changes in its filtered sequence. + +\paragraph{Proof.} +Move variables into unused bank-$0$ space until full. Direct access permits retaining all old BSR instructions, so cost cannot increase. + +The filtered sequence is nonempty. Its first reference requires a setting, and every subsequent change of bank requires another. Direct references impose no BSR requirement. Setting the BSR exactly at these points attains the bound. \qed + +\paragraph{Lemma 2.} +The tree traversal computes exactly the filtered sequence's adjacency weights. + +\paragraph{Proof.} +Deletion commutes with concatenation and repetition, so the endpoint rules hold. A consecutive pair is inside a child, between consecutive nonempty children in one body execution, or between successive body repetitions. The first kind is counted within the child. The other kinds occur globally \texttt{times} and \texttt{back} times, exactly as added at the node. This recursive decomposition counts every pair once. Equal-variable pairs never cause a bank change. \qed + +\paragraph{Lemma 3.} +For fixed $Z$, backtracking maximizes achievable within-bank weight. + +\paragraph{Proof.} +Order any feasible partition's groups by their smallest variable. This determines a unique search path: each variable joins its existing group or creates the next group. Conversely, every completed path obeys the capacity and bank-count limits. + +Each insertion changes only its group's weight by the increment used in the search. Hence every leaf's saved weight is $\sum_i W(S_i)$, and maximizing over leaves considers every feasible partition. \qed + +\paragraph{Theorem.} +The algorithm returns the minimum instruction count. + +\paragraph{Proof.} +At least $N$ instructions are necessary, and $n>s$ forces a BSR setting. The early-return placements attain these bounds. Otherwise, Lemma 1 guarantees enumeration of an optimal bank-$0$ set. By Lemma 2, subtracting within-bank weights from $T$ leaves exactly the bank changes, yielding the cost formula. Lemma 3 optimizes it for each set; minimizing over all sets gives the global optimum. \qed + +\section*{Complexity Analysis} + +Let $L$ be the token count, $m=n-s$, and $P_s(k)$ count partitions of $k$ items into groups of size at most $s$, with $P_s(0)=1$. For one bank-$0$ set, graph construction costs $O(L+n^2)$ and subset weights cost $O(m2^m)$. Ignoring the bank-count limit, there are at most $\sum_{k=0}^m P_s(k)$ search calls, each trying at most $m$ banks. Including parsing and scanning all masks, time is +\[ + O\left(L+2^n+\binom ns + \left(L+n^2+m2^m+m\sum_{k=0}^m P_s(k)\right)\right). +\] +For $n\le13$, fewer than $22.2$ million search calls occur over all bank-$0$ choices, even without the bank-count restriction. Memory usage is $O(L+n^2+2^m)$. Early returns use $O(L)$ time and space. + +\section*{Implementation Notes} + +\begin{itemize}[leftmargin=*] + \item Reverse node creation order is bottom-up, since children follow their parents. + \item Restore existing banks' masks and sizes after recursive branches. + \item Use 64-bit integers for multiplicities and counts; the answer is at most $2N\le2\cdot10^{12}$. +\end{itemize} + +\end{document} diff --git a/problems/PA-05/ICPC_WF/2013/E-harvard/statement.txt b/problems/PA-05/ICPC_WF/2013/E-harvard/statement.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e604b67980f93efd2a871b1d21b23c33137d1a5 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2013/E-harvard/statement.txt @@ -0,0 +1,85 @@ +ICPC 2013 + 2013 World Finals + St. Petersburg + HOSTED BY ITMO + + Problem E + Harvard + Time Limit: 10 seconds +The term “Harvard architecture” applies to a +computer that has physically separate memories +for instructions and data. The term originated +with the Harvard Mark I computer, delivered by +IBM in 1944, which used paper tape for instruc- +tions and relays for data. +Some modern microcontrollers use the Harvard +architecture – but not paper tape and relays! +Data memory is organized in banks, each con- +taining the same number of data items. Each +data-referencing instruction has a byte offset f +to a bank, and a bit a that is used to select the +bank to be referenced. If a is 0, then bank 0 is referenced. If a is 1, then the value in a bank select +register (BSR) identifies the bank to be used. Assume each instruction takes the same time to execute, +and there is an instruction that can set the BSR’s value. +[Image: Picture from Wikimedia Commons] +For example, suppose there are 4 banks of 8 bytes each. To access location 5, either use a single +instruction with a = 0 and f = 5, or set the BSR to 0 in one instruction and then use an instruction with +a = 1 and f = 5. The first approach is faster since it does not require setting the BSR. +Now suppose (with the same memory) the location to access is 20. Only one approach will work here: +execute an instruction that sets the BSR to 2 (unless the BSR already has the value 2) and then use an +instruction with a = 1 and f = 4. +A program is a sequence of operations. Each operation is either + + • a variable reference, written as Vi, where i is a positive integer, or + + • a repetition, written as Rn E, where n is a positive integer and is an + arbitrary program. This operation is equivalent to n sequential occurrences of . + +Your problem is to determine the minimum running time of programs. In particular, given the number +and size of the memory banks and a program to be executed, find the minimum number of instructions +(which reference memory location and possibly set the BSR) that must be executed to run the program. +To do this you must identify a mapping of variables to memory banks that yields the smallest execution +time, and report that execution time – that is, the number of memory references and BSR register settings +required. The BSR’s value is initially undefined, and changes only when an instruction explicitly sets its +value. + + ICPC 2013 + 2013 World Finals + St. Petersburg + HOSTED BY ITMO + +Input + +The input consists of a single test case. A test case consists of two lines. The first line contains two +integers b and s, where 1 ≤ b ≤ 13 is the number of memory banks and 1 ≤ s ≤ 13 is the number of +variables that can be stored in each memory bank. The second line contains a non-empty program with +at most 1 000 space-separated elements (each Rn, Vi, and E counts as one element). +You may assume the following: + + • In a repetition Rn, the number of repetitions satisfies 1 ≤ n ≤ 10^6. + + • In a loop operation Rn E, the loop body is not empty. + + • In a variable reference Vi, the variable index satisfies 1 ≤ i ≤ min(b · s, 13). + + • The total number of variable references performed by an execution of the program is at most 10^12. + +Output + +Display the minimum number of instructions that must be executed to complete the program. + + Sample Input 1 Sample Output 1 + 1 2 5 + V1 V2 V1 V1 V2 + + Sample Input 2 Sample Output 2 + 2 1 6 + V1 V2 V1 V1 V2 + + Sample Input 3 Sample Output 3 + 1 2 30 + R10 V1 V2 V1 E + + Sample Input 4 Sample Output 4 + 4 1 17 + V1 R2 V2 V4 R2 V1 E V3 E diff --git a/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/sample-2.in b/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/sample-2.in new 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0000000000000000000000000000000000000000..cd14b31844d5a3e60dbef8b4036af088fb6e1087 --- /dev/null +++ b/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-04-random-large.in @@ -0,0 +1,2 @@ +364 233 807 +565131267980025569 -563973445141848388 diff --git a/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-05-random-large.in b/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-05-random-large.in new file mode 100644 index 0000000000000000000000000000000000000000..b28b435beabcf1aaa7bc419c1d21b8ced3fc40bc --- /dev/null +++ b/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-05-random-large.in @@ -0,0 +1,2 @@ +832 25 89 +-249079765382362173 760948473357112058 diff --git a/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-07-random-large.ans b/problems/PA-05/ICPC_WF/2024/E-flipping-container/data/secret-07-random-large.ans new file mode 100644 index 0000000000000000000000000000000000000000..37ae5c69880349f0af62b6c086e0e6a555abef6a --- 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