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I'm designing an algorithm for a class that will determine if a directed graph is unique with respect to a vertex $v$ such that for any $u \ne v$ there is at most one path from $v$ to $u$. I've started by using BFS (breadth-first search) to find the shortest path from v to another vertex u, and then running BFS again t... |
I want to understand what the LEADING and TRAILING of non-terminal in an operator precedence grammar physically mean.
I am confused by the various definitions I have read on them.
I understand that the LEADING of a non-terminal is the first terminal which can be present in it's derivation.
On the other hand, the TRAI... |
Let's say $F$ is an oracle for a problem in $\mathbb{NP}$, but I cannot call this oracle with any input instance. Instead, whenever I call $F$ I get returned a random instance and solution. Thus, I know that $F$ is in fact capable of solving arbitrary $\mathbb{NP}$-hard problems, I just can't specify which one I wan... |
I see structural induction the usual way for proving an algorithm's termination property, but it's not that easy to prove by means of induction on a tree algorithm. Now I am struggling on proving that the pre-order tree traversal algorithm is terminable:
preorder(node)
if node == null then return
visit(node)
preo... |
Given a relation $R(A, B, C, D, E, F)$, with the following functional dependencies:
$\{A \rightarrow BC, CD \rightarrow E, B \rightarrow D, E \rightarrow A\}$.
The objective is to decompose $R$ into 3NF relations.
So far, I have determined that the following candidate keys are present in the given relation:
$AF$, $EF$,... |
My homework is implementing algorithms BFS, DFS, depth-limited and IDS for the map as a 2D grid with 8 directions of movement.
I read that the IDS algorithm is optimal, but in my case is not optimal paths.
Is the IDS algorithm optimal for graphs (e.g. map as a grid), or does optimality only apply to searching trees?
|
The notion of best-case running time is kind of ambiguous for me. According to wikipedia, the definition of best case running time is:
The term best-case performance is used in computer science to describe the way of an algorithm behaves under optimal conditions. For example, the best case for a simple linear search o... |
In question How to reduce MaxUNSAT to MaxSAT? I was asking, how to reduce the MaxUNSAT problem to MaxSAT. With help of the given answer I could give a polynomial reduction : $MaxUNSAT \leq decisionProblemMaxUNSAT \leq SAT \leq MaxSAT$.
Question: Is there a more direct reduction without much overhead from MaxUNSAT to Ma... |
I read in some sources that valid formulas are tautologies (valid under every evaluation). In the others, I read that these are formulas that have conclusions true when premises are true. Are these just equivalent definitions because ⊨ P → Q is equivalent to P ⊨ Q?
|
I'm having some trouble understanding the following:
When we look at satisfiability problems in conjunctive normal form, an underconstrained problem is one with relatively few clauses constraining the variables. For eg. here is a randomly generated 3-CNF sentence with five symbols and five clauses. (Each clause contain... |
In a lecture a professor mentioned that modern computers don't have as much computational power as a Turing machine because they don't have infinite memory, and since no computer can have infinite memory the Turing machine is therefore unattainable and simply represents the upper limit of computing. Is there a measure,... |
I was wondering, if there is an MD5 hash collision in the UNSIGNED MEDIUMINT Range (0 - 16777215). Sadly, I'm unable to run a script to check this myself, due to Memory (RAM) limitations. The answer is probably no; the next step would be - how many characters are even needed, until the first collision is found?
An MD5... |
How to prove that BFS directed-graph traversal algorithm terminates?
(I copy the pseudocode from here) Input: A graph G and a root v of G.
procedure BFS(G,v):
create a queue Q
enqueue v onto Q
mark v
while Q is not empty:
t ← Q.dequeue()
if t is what we are looking for:
... |
I'm looking for good resources regarding Support Vector Machines, or suggestions where to start learning SVM.
Already used references:
Stanford ML course by Andrew Ng is great place to star
A Tutorial on Support Vector Machines for Pattern Recognition, Burges, 1998
SVM tutorials
Neural Networks and Learning Machin... |
I want to prove that the complement of $\{0^n1^n \mid n \geq{} 0\}$ is not regular using closure properties.
I understand pumping lemma can be used to prove that $\{0^n1^n \mid n \geq{} 0\}$ is not a regular language. I also understand regular languages are closed under complement operation. However, does that also imp... |
I am trying to teach myself the principles of cryptograhpy, and want to solve the following question:
Let G be the algorithm that takes an input x = (x1, . . . , xn) from
{0, 1} n (so each xi ∈ {0, 1}) and outputs the string G(x1, . . . ,
xn) = (x1, . . . , xn, x1 ⊕ x2) in {0, 1} n+1. Rigorously prove that G
is not a ... |
Consider a square, ABCD. Intuitively it seemed to me that its chromatic polynomial is $\lambda(\lambda - 1)(\lambda - 1)(\lambda - 2)$ where there are $\lambda$ colours available..
That is there are $\lambda$ ways in which a colour for A can be picked, there are $\lambda - 1$ ways for colours for B and D to be picked(B... |
Not sure exactly how to define this but here's the basis of the question (very loosely)
All lists can be contained in tree's
all trees can be contained in a graph
all graphs can be contained in a multi graph,
and all multigraphs can be contained in a hypergraph (never actually heard of a hypergraph before searching f... |
Can anyone tell me what is the difference between the following regular expressions: $(0^*1^*)^*$ and $(0+1)^*$ ? To me they look like generating the same string.
|
In recent years the field of "collective intelligence" sometimes known as Web 2.0 has had a big impact on computer science, software engineering, and software development. Stackexchange software itself is a large scale evolving application or implementation and experiment along the lines of harnessing collective intell... |
Given a directed graph and two distinct vertices S and T, is there a polynomial-time algorithm which finds every vertex which is on at least one simple path from S to T?
It is not difficult to find all vertices that are both successors of S and predecessors of T but this is only a superset of the set above. For example... |
I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" want to prove that the average height $h_n$ of a tree with $n$ nodes (that is, the maximum number of nodes from the root... |
I've read the wikipedia page for rule 110 in cellular automata, and I more or less know how they work (a set of rules decides where to draw the next 1 or 0).
I've just read they're Turing complete, but I can't even fathom how would you 'program' in 'rule 110'?
|
Say there is a data table $D$ that we cannot see, with $M$ columns. We are given exact cross-tabulation frequencies for all ${M \choose 2}$ pairs of columns, that is how often each combination of two values occurs.
From the cross-tabulations, we can derive a set of possible rows $R$ of $D$ and maximum frequencies for e... |
I am learning about graph coloring. One of the exercise problems(grimaldi) led me to derive the chromatic polynomial for any cycle($C_n, n \ge 3$).
$P(C_n, \lambda) = (\lambda - 1)^n + (-1)^n(\lambda - 1)$, where $\lambda$ is the number of colours available
Is it possible to interpret the terms of above polynomial? I n... |
I am dealing with the problem of computing $ s = \lfloor sqrt(x)\rfloor$ with $x \in [0,30000^2]$. The common sqrtf(x) on C language is too slow for this case, however it always gives me the correct result. I've tried with the Newton's method but I get very small errors when the square root of a number is exact. This l... |
I am studying algorithms from CLRS book. I try to understand the difference between
probability of hiring the $i$th person out of $n$ and
probability of hiring the $i$th person out of $n$ persons based on ranks.
Using the normal probability, we know the answer for the first one is $\frac{1}{n-i}$ and the answer for ... |
What's the best way that anyone can do to have a good introduction to the theory of distributed system, any books or references, and topics should be covered first and requirements to start learning in this topic.
|
An instance of the SUBSET SUM problem (given $y$ and $A = \{x_1,...,x_n\}$ is there a non-empty subset of $A$ whose sum is $y$) can be represented on a one-tape Turing Machine with a list of comma separated numbers in binary format.
If $\Sigma = \{0,1,\#\}$ a reasonable format could be:
$( 1 \; (0|1)^* \; \#)^* \#$
Whe... |
I am learning algorithms. So, I came along with something very interesting.
The asymptotic bound of linear function $an+b$ is $O(n^2)$ for all $a>0$.
This is same as for $an^2 + bn + c$. But shouldn't it be different?
|
I am looking for a fast k-mismatch string matching algorithm. Given a pattern string P of length m, and a text string T of length n, I need a fast (linear time) algorithm to find all positions where P matches a substring of T with at most k mismatches. This is different from the k-differences problem (edit distance). A... |
What would the order of growth for this loop be:
int sum = 0;
for (int n = N; n > 0; n /= 2)
for(int i = 0; i < n; i++)
sum++;
The first loop seems to run for $\log N + 1$ times and the second loop runs $n$ times.
So is the correct answer $O(n \log n)$?
|
I would like to ask how to determine the number of states when designing a Turing machine from the description for a language? For example:
$\qquad \displaystyle L = \{wcw \mid w \in \{0,1\}^*\}.$
I mean how to know how many states are there in the set $Q$, with the information from the description of that language.
|
A surprising number of problems have fairly natural reductions to linear programming (LP). See Chapter 7 of [1] for examples such as network flows, bipartite matching, zero-sum games, shortest paths, a form of linear regression, and even circuit evaluation!
Since circuit evaluation reduces to linear programming, any p... |
I work on a recommender system framework which is implemented with a variant on Funk SVD (See his explanation of his algorithm here).
However the framework that we are trying to integrate doesn't support Funk SVD, only SGD (Stochastic Gradient Descent).
Since shouldn't these be compatible? In other words, I should be... |
I´m not exactly sure what kind of impact these two things have on kernel design. Are servers more flexible because of running in user space, for instance, and does adding a module require a new build of a kernel?
|
I'm working out of the 3rd edition CLRS Algorithms textbook and in Chapter 3 a discussion begins about asymptotic notation which starts with $\Theta$ notation. I understood the beginning definition of:
$$\Theta(g(n)) = \{ f(n)\,|\, \exists\, c_1, c_2 > 0, n_0 \in \mathbb{N}: 0 \leq c_1 g(n) \leq f(n) \leq c_2 g(n)\ \ \... |
What is a good textbook for a seminar about
virtual environments, virtual worlds, augmented reality and
similar topics?
The seminar is at the graduate studies level and is supposed to give introduction to virtual environments, technologies and applications as well as current research directions.
A creative commo... |
Planar 3SAT is NP-complete. A planar 3SAT instance is a 3SAT instance for which the graph built using the following rules is planar:
add a vertex for every $x_i$ and $\bar{x_i}$
add a vertex for every clause $C_j$
add an edge for every $(x_i,\bar{x_i})$ pair
add an edge from vertex $x_i$ (or $\bar{x_i}$) to each verte... |
Let us say we know that problem A is hard, then we reduce A to the unknown problem B to prove B is also hard.
As an example: we know 3-coloring is hard. Then we reduce 3-coloring to 4-coloring. By conflating one of the colors in the 3-coloring you have a 4-coloring, ergo 4-coloring is hard.
That's the how. But why is t... |
I took my theory of computation exams a few weeks ago, and this was one of the questions:
Assume language $L=\{(a^nb^m)^r \mid n,m,r\ge 0\}$
Is L regular? If yes provide a regular expression or an automaton for it.
After I briefly asked him the answer after the exam, it appears it really is regular (I believe he said... |
As the title says, I spent a couple of hours last weekend trying to wrap up my mind about the class of languages matched by Perl-compatible regular expressions, excluding any matching operator that allows to execute arbitrary code inside the pattern.
If you don't know what PCREs are, please read this and this.
The pro... |
I'm trying to prove the following lemma:
$c$ is a positive real number and $f, g$ are functions from natural numbers to non-negative real numbers. I'm trying to prove rigorously that:
$\Omega(cf(n))$ = $\Omega(f(n))$.
I know that it is obvious but I'm trying to construct a proof that is as complete as possible. My curr... |
Partition refinement is a technique in which you start with a finite set of objects and progressively split the set. Some problems, like DFA minimization, can be solved using partition refinement quite efficiently. I don't know of any other problems that are usually solved using partition refinement other than the ones... |
How can you show that the Language accepted by an NFA and the reverse NFA is the same?
For a language $L$, there is an $L^R=\{ w^R \mid w \in L\}$
Let's say that $w^R$ is the string obtained by reversing the string $w$.
I know that it involves using induction on the length of the input, but I would really appreciate so... |
I am trying to create an algorithm in linear time where if given a directed acyclic graph I can add edges to make it strongly connected components.
I believe I have an algorithm to identify sources and sinks in the input list of edges in form
1 2
2 5
3 1
etc
And I know that the minimum number of edges to be added in ... |
From Wikipedia:
Given two subsets A and B of N and a set of functions F from N to N
which is closed under composition, A is called reducible to B under F
if $$
\exists f \in F \mbox{ . } \forall x \in \mathbb{N} \mbox{ . } x \in A \Leftrightarrow f(x) \in B $$ We write $$
A \leq_{F} B $$ Let S be a subse... |
I have seen some questions related to functional programming on stackexchange sites which suggests it has significant popularity. I have some experience with it from many years ago in Lisp/Scheme and have seen commentary on the difference between functional and imperative/mutable approaches eg [4].
The wikipedia page ... |
I have been reading Introduction to Algorithms by Cormen et al. and I'm reading the statement of the Master theorem starting on page 73. In case 3 there is also a regularity condition that needs to be satisfied to use the theorem:
... 3. If
$\qquad \displaystyle f(n) = \Omega(n^{\log_b a + \varepsilon})$
for some cons... |
"Alan Turing proved in 1936 that a general algorithm to solve the
halting problem for all possible program-input pairs cannot exist"
Can I find a general algorithm to solve the halting problem for some possible program input pairs?
Can I find a programming language (or languages), where I for every kind of program... |
I have a homework assignment where I need to convert a grammar into LL(1). I've already removed the left recursion, but I'm having trouble doing left-factoring. All of the examples I've found are simple, and look something like this:
A -> aX | aY
becomes:
A -> aZ
Z -> X | Y
I understand that. However, my grammar looks ... |
I face a problem with computing a complexity.
I have this equality : $P(u) = (\sqrt{u}+1)P(\sqrt{u}) + \theta(\sqrt{u})$
And I want to prove that $P(u) = O(u)$
This is how I process :
I put $m = \lg\lg u \implies P(u) = P(2^{2^{m}}) = (2^{2^{m-1}}+1)P(2^{2^{m-1}}) + \theta(2^{2^{m-1}})$
Now, I consider $S(m)$ that is ... |
What problem does the halting problem solve in computing, whether theoretical or practical?
It is very easy to debug code which loops forever, just signal the debugger to break if the program is running for too long? What purpose / good is the halting problem? Why was Turing praised for it?
|
We got tower $T_1$ with $n$ odd disks (1,3,5,...) and tower $T_2$ with $n$ even disks (2,4,6,...).
Now we want to move all $2n$ disks to tower $T_3$.
If $T(p,q)$ is a recurrence relation of minimum number of moves we make to move $p$ disks from $T_1$ and $q$ disks from $T_2$ to $T_3$, we must find $T(n,n)$.
If anyone h... |
From Wikipedia:
The type of computational problem: The most commonly used problems are
decision problems. However, complexity classes can be defined based on
function problems, counting problems, optimization problems, promise
problems, etc.
I also saw the definitions of NP-complete, NP-hard, NP, ..., are defin... |
From Wikipedia
a computational problem is understood to be a task that is in principle amenable to being solved by a computer (i.e. the problem can
be stated by a set of mathematical instructions). Informally, a
computational problem consists of problem instances and solutions to
these problem instances. For exa... |
G{V, E} is directed, cyclic, weighted graph. What is the algorithm of finding all paths between any given two nodes?
Can you suggest any good reading?
|
If I understand correctly, the complexity of solving a computational problem is defined in terms of which instance $I$ of the problem, what size $n$ of the problem instance, and what algorithm $A$ for solving the problem instance.
Is the complexity of the problem at a given size $n$ of the problem instance defined as
$... |
You can modify any graph $G$ so that Dijkstra's finds the solution with the minimal number of edges thusly:
Multiply every edge weight with a number $a$, then add $1$ to the weight to penalize each additional edge in the solution, i.e.
$w'(u,v)=a*w(u,v)+1$
This does not work for all values of $a$; $a$ needs to be at l... |
I am recently studying about Compilers designing. I came to know about two types of grammar one is LL grammar and other is LR grammar.
We also know the facts that every LL grammar is LR that is LL grammar is a proper subset of LR grammar. First one is used in top-down parsing and the second one is used in bottom-up par... |
I am trying to understand different interactive proof systems, in particular AM and MA.
Is there a typical problem for the complexity class MA
as Graph-NonIsomorphism problem is for AM?
Is there a problem in MA that is not known to be in NP or BPP?
It seems to me that MA is a more natural proof system than AM.
Is th... |
This is part of a larger problem, which I believe I have reduced to this. Given a tree $T$ having positive edge weights, and $k$ leaves (nodes which have exactly one connected node), I need to delete some edges in the tree so that no two leaves in the original tree are connected (by a path) in the newly formed forest (... |
Foo and Bar are playing a game of strategy. At the start of the game, there are $N$ apples, placed in a row (in straight line). The apples are numbered from $1$ to $N$. Each apple has a particular price value.
The price of $i$th apple is $p_i$.
In this game, the players Foo and bar make an alternative move.
In each mov... |
In my Computing Theory course, a lot of our problems involve using induction on the length of the input string to prove statements about finite automata. I understand mathematical induction, however when strings come into play I get real tripped up. I'd really appreciate it if someone would go through the process of ... |
Representing numeric values using positional notation is one of the milestones in the history of arithmetic. Babylons used a base 60 system, Maya a base 20 system; base 10 system became "the standard" used by modern civilizations; digital computers use the Binary numeral system, ....
But if we look at nature, we found ... |
Let's say I have a graph $|G|$ with $|E|=O(V^2)$ edges. I want to run BFS on $G$ which has a running time of $O(V+E)$.
It feels natural to write that the running time on this graph would be $O(O(V^2)+V)$ and then simplify to $O(V^2)$.
Are there any pitfalls to using such a "remove-the-nested-O" shortcut (not just in th... |
I understand that using DFS "as is" will not find a shortest path in an unweighted graph.
But why is tweaking DFS to allow it to find shortest paths in unweighted graphs such a hopeless prospect? All texts on the subject simply state that it cannot be done. I'm unconvinced (without having tried it myself).
Do you kno... |
What is the runtime complexity of the following implementation of Bubblesort (for integers)?
#define SWAP(a,b) { int t; t=a; a=b; b=t; }
void bubble( int a[], int n )
/* Pre-condition: a contains n items to be sorted */
{
int i, j;
/* Make n passes through the array */
for(i=0;i<n-... |
I have a formula $ \neg((q \implies \neg q) \vee p \vee (\neg q \implies (r \wedge p))) $.
As it contains 3 subformulas between the $\vee$'s, how can i put it into a parse tree, as a parse tree contains 2 branches from each node.
|
Let $A$ be an adjacency matrix of a directed graph. What's the meaning of the $(i,j)-$entry of the matrix $((A^T)^{7} \cdot (A^{7}))$ ?
My initial interpretation is that $(i,j)$ of this matrix is zero whenever nodes $i$ and $j$ have no 7-length in-coming paths from a common source. Is that right? Any attention is appr... |
For a formal language $L \subseteq \Sigma^{*}$ I define the set Pref(L) to be:
$\text{pref}(L) = \{\alpha \in \Sigma^{*} : \exists \beta \in \Sigma^{*} \text{ such that } \alpha \beta \in L\}$
ie. the set of all (not necessarily proper) prefixes of words in $L$. I know that if $L$ is context-free then pref(L) is contex... |
Shannon's entropy [plog(1/p)] for an image is a probabilistic method for comparing two pixels or a group of pixels.Suppose an image with a matrix of 3x3 has pixel intensity values
1 1 2
2 3 3
4 4 5
and another image with 3x3 matrix has group of pixels having intensity values
5 5 6
6 7 7
8 8 9
Then shannon's entropy ... |
Dynamic programming with large number of subproblems. So I'm trying to solve this problem from Interview Street:
Grid Walking (Score 50 points)
You are situated in an $N$-dimensional grid at position $(x_1,x_2,\dots,x_N)$. The dimensions of the grid are $(D_1,D_2,\dots,D_N$). In one step, you can walk one step ahead... |
I need to find the easiest cost path between two vertices of a graph. Easiest here means the path with the smallest maximum-weigth edge.
In the above graph, the easiest path from 1 to 2 is:
1 > 3 > 4 > 2
Because the maximum edge weight is only 2. On the other hand, the shortest path 1 -> 2 has maximum weight 4.
So... |
Prove or disprove the following statements:
$T\left( n \right) = 2T\left( {\frac{n}{2}} \right) + f\left( n \right),f\left( n \right) = \theta \left( {{n^2}} \right) $ then $ {\rm{ }}T\left( n \right) = \theta \left( {f\left( n \right)} \right) $ for all $ {\rm{ n = }}{{\rm{2}}^k}$
$T\left( n \right) = 2T\left( {\frac... |
In most PL textbooks (even in language specifications) statments like if, while, for, break are categorized as flow of control statements.
However, it is not consistent; i.e. sometimes the same text uses control-flow statement (note the dash), or control flow statment, I read sentences that says control flows...
I und... |
Can I always increase the complexity of a problem via polynomial reduction? (in which case 'reduction' is really a misnomer) For example, if I have a classic P problem (say, finding the smallest element in an array, by iterating through and making comparisons with the smallest value found thus far), what would be a cor... |
Im trying to figure out how to describe fifty-six strings to test if a three state FA over the alphabet $\{a,b\}$ has a finite language.
The number fifty-six comes from a theorem that states if a machine has $N$ states and an alphabet has $m$ letters, then in total there are $m^N + m^{N + 1} + m^{N + 2} +\ldots+ m^{2N-... |
Are $\log_{10}(x)$ and $\log_{2}(x)$ in the same big-O class of functions? In other words, can one say that $\log_{10}(x)=O(\log x)$ and $\log_{2}(x)=O(\log x)$?
|
I want the conditional probability for each topic (being the word that we give as input). For example, the text being
have seen and reviewed your requirements you posted here. If you can
give me the fix criteria/category of your data mining then I can do
this job. If you want me to define and allot criteria and c... |
I'm currently doing some reading into AI and up to this point couldn't find a satisfying answer to this question: what's the difference between a rule based system and an artificial neural network?
From my understanding both are trying to do inference based on a variety of different inputs.
|
I'm in a programming course and one Problem given to us is to mark the order in which BFS visits nodes in a weighted graph.
My question is whether BFS adds the distance of the previous path while adding a new vertex, as that will definitely change the order in which the vertices are added?
|
Im trying to prove that the following language is not regular.
$$\text{Notprime} = \{a^n \text{where \(n\) isn't prime}\}
= \{\epsilon, a, aaaa, aaaaaa, aaaaaaaa, \ldots\}$$
Heres what I have:
"If Notprime were regular, then its complement would be regular also. However, the complement of Notprime is the language Pri... |
I got no responses on stackoverflow, so I'm asking here:
How useful is the LIS (Longest Increasing Subsequence) problem in tackling other CS problems? There are a few algorithms, using patience sorting, dynamic programming or with decision trees. How are these used in real life -- maybe to data streams or something?
... |
Possible Duplicate:
How to convert finite automata to regular expressions?
Im curious if anyone knows if its possible to write a program to generate a regular expression given a finite automation.
To make things less complicated I want to limit the number of states to about 4, assume the FA is in minimal form and ... |
This looks like a fundamental result but I can't find a resource online that gives an intuitive interpretation of this complexity. Any basic explanation is appreciated.
|
Below is the lambda expression which I am finding difficult to reduce i.e. I am not able to understand how to go about this problem.
$$(\lambda mn.(\lambda sz.ms(nsz)))(\lambda sz.sz)(\lambda sz.sz)$$
I am lost with this.
if anyone could lead me in the right direction that would be much appreciated
|
There are two context-sensitive languages, $L_1$ and $L_2$. Which of the following statements about them are decidable respectively undecidable?
$L_1 = \emptyset$
$L_1 = \Sigma^*$
$L_1 \cap L_2 = \emptyset$
$\overline{L_1}$ is also a context-sensitive language.
$L_1 = L_2$
|
This is a similar algorithm to one I used in a previous question, but I'm trying to illustrate a different problem here.
for (int i = 0; i < numbers.length - 1; i++) {
for (int j = i + 1; j < numbers.length; j++) {
if (numbers[i] + numbers[j] == 10) {
System.out.println(i+" and "+j+" add up to 1... |
Substring Diff
Given two strings of length $n$, $P = p_1\dots p_n$ and $Q = q_1 \dots q_n$, we define $M(i, j, L)$ as the number of
mismatches between $p_i \dots p_{i+L-1}$ and $q_j \dots q_{j+L-1}$. In set
notation, $M(i, j, L)$ refers to the size of the set $\{0 \leq x < L \mid p_{i + x} \neq q_{j + x}\}$.
Giv... |
Is there an algorithm to raise a matrix to the $n$th power in $O(\log n)$ time?
I have been searching online, but have been unsuccessful thus far.
|
The definitions I've found were highly technical and using terms I've never seen before.
Say, I have a certain irrational number e that I get get closer and closer to with a computer and I want to determine if there is a particular digit d appearing after a certain position. Or what about the halting problem (give a p... |
What differences and relationships are between randomized algorithms and nondeterministic algorithms?
From Wikipedia
A randomized algorithm is an algorithm which employs a degree of
randomness as part of its logic. The algorithm typically uses
uniformly random bits as an auxiliary input to guide its behavior, in
... |
I'm familiar with a few problems of each class and even though the definitions are based on sets and polynomial reducibility, I see a pattern with time complexity. NP problems appear to be $O(2^n)$ (minus the P problems of course), and NP-hard problems seem to be worse: $n^n$ or $n!$ (Chess, TSP). Is this a misleading ... |
Im having trouble figuring out how to determine if two finite automata are the same apart from renumbered states.
More specifically, heres an example:
It's easy to generate a regular expression by hand and see that both FA produce:
b + (ab*a(a+b)), though their states are renumbered, they are identical.
What I'm t... |
$$\begin{align*}
T[1] &= 1 \\
T[2] &= 2 \\
T[i] &= T[i-1] + T[i-3] + T[i-4] & \text{for \(i \gt 2\)} \\
\end{align*}$$
I have to calculate $T[N]$, but $N$ is too big ($\approx 10^9$), how can I optimize it?
|
I like to implement an arc-length Parameterization of a cubic bezier curve. So far I have implemented the method of calculating the arc length of the curve and now I'm stuck at calculating the times to divide the original curve into equal arc length segments.
What i have:
$m = 5$ The number of segments to create
$i = ... |
I have a forest, i.e., nodes with directed edges and no cycles (directed or undirected). I define the height of a vertex $v$ as 0 if it does not have any incoming edges, or the maximum number of edges to traverse in reverse to reach a vertex of height 0.
I also know that the average degree of a node is a small consta... |
In radix sort we first sort by least significant digit then we sort by second least significant digit and so on and end up with sorted list.
Now if we have list of $n$ numbers we need $\log n$ bits to distinguish between those number. So number of radix sort passes we make will be $\log n$. Each pass takes $O(n)$ time... |
I have a simple problem. I can't seem to even find the right search terms to get me pointed in the direction I need to be heading.
I'm writing a bunch of integers to disk. Lot's of them.
Starting with the integer 2, I add it to another integer, write the result to disk, and start the process again using the result as t... |
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