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My setup is something like this: I have a sequence of sets of integers $C_i (1\leq i\leq n)$, with $|C_i|$ relatively small - on the order of four or five items for all $i$. I want to choose a sequence $x_i (1\leq i\leq n)$ with each $x_i\in C_i$ such that the total variation (either $\ell_1$ or $\ell_2$, i.e. $\sum_{... |
I have a set of strings $S$ and I am using the edit-distance (Levenshtein) to measure the distance between all pairs.
Is there an algorithm for finding the string $x$ which minimizes the sum of the distances to all strings in $S$, that is
$\arg_x \min \sum_{s \in S} \text{edit-distance}(x,s)$
It seems like there should... |
Consider the Kripke structure:
$$
\begin{array}{ccccccc}
\to & (p, \neg q) & \to & (\neg p, \neg q) & \to & (\neg p, q) \\
& \circlearrowright & & \circlearrowright & & \circlearrowright & \\
\end{array}
$$
where $(p, \neg q)$ means “$p$ and not $q$” and $\circlearrowright$ is a self loop. We number the states $s_1... |
Suppose we are given a list of $n$ points, whose $x$ and $y$ coordinates are all non-negative. Suppose also that there are no duplicate points. We can only go from point $(x_i, y_i)$ to point $(x_j, y_j)$ if $x_i \le x_j$ and $y_i \le y_j$. The question is: given these $n$ points, what is the maximum number of points t... |
I'm trying to understand how the 2nd parity bit or byte is set in RAID 6. I'm reading a paper by H. Peter Anvin, and it goes into Galois field algebra, which is somewhat new to me. Anyway, a rep from HP was trying to explain RAID 6 to me and she thought it was merely two XOR operations, one for the 1st parity bit and o... |
I created a simple regular expression lexer and parser to take a regular expression and generate its parse tree. Creating a non-deterministic finite state automaton from this parse tree is relatively simple for basic regular expressions. However I can't seem to wrap my head around how to simulate backreferences, lookah... |
There are obvious analogs (pardon the pun) between Boolean algebra and algebra. They have similar laws, operators and properties. I can't figure out why Karnaugh Maps and sum of products, which are used to derive a Boolean function from a truth table, doesn't have an equivalent in algebra. Perhaps it does, but I haven... |
I am having trouble finding good resources that give a worst case $O(n \ln n)$ in place stable sorting algorithm. Does anyone know of any good resources?
Just a reminder, in place means it uses the array passed in and the sorting algorithm is only allowed to use constant extra space. Stable means that elements with t... |
I have a run-time implementation question regarding the 3-dimensional (unweighted 2-)approximation algorithm below:
How can I construct the maximum matching M_r in S_r in linear time in line 8?
$X, Y, Z $ are disjoint sets; a matching $M$ is a subset of $S$ s.t. no two triples in $M$ have the same coordinate at any dim... |
Duplicate:
NP-completeness of graph isomorphism through edge contractions with an edge validity condition
I know that graph contractability is $NP$-complete. To be specific given $G=(V_1,E_1)$ and $H=(V_2,E_2)$, can a graph isomorphic to H be obtained from G by a sequence of edge contractions ?
However my problem is... |
Basically, the problem I am solving is this. Initially, the array $A$ is empty. Then I am given data to fill the array and at any time I have to make a query to print the $|A|/3$-th largest element inserted so far.
I was solving the problem with segment trees, but I am not able to make a little modification to the quer... |
We are given a random number generator RandNum50 which generates a random integer uniformly in the range 1–50.
We may use only this random number generator to generate and print all integers from 1 to 100 in a random order. Every number must come exactly once, and the probability of any number occurring at any place mu... |
I'm having a very hard time trying to figure out how to solve this problem efficiently. Let me describe how it goes:
"A hard working mom bought several fruits with different nutritional values for her 3 kids, Amelia, Jessica and Bruno. Both girls are overweight, and they are very vicious and always leave poor Bruno wit... |
We take the sequence of integers from $1$ to $n$, and we push them onto a stack one by one in order. Between each push, we can choose to pop any number of items from the stack (from 0 to the current stack size).
Every time we pop a value from the stack, we will print it out.
For example, $1,2,3$ is printed out when we ... |
Suppose we are given an array of positive integers $P = [p_1, p_2, \dots, p_N]$ where each $p_i$ represents the price of a product on a different day $i = 1 \dots N$.
I would like to design an algorithm to find the maximum profit that you can given this array of prices. Profit is made by buying at a given date $i$ and... |
There's a lot of debate about what exactly the Church-Turing thesis is, but roughly it's the argument that "undecidable" should be considered equivalent to "undecidable by a universal turing machine."
I'm wondering if there's an analogous statement for time complexity, i.e. an argument that if some language is decided ... |
Given a sequence of natural numbers, you can add any natural number to any number in the sequence such that their xor becomes zero. My goal is to minimize the sum of added numbers.
Consider the following examples :
For $1, 3$ the answer is $2$; adding $2$ to $1$ we get $3 \oplus 3=0$.
For $10, 4, 5, 1$ the answer is ... |
I have a problem with the application of the Shannon expansion for to obtain the negation of a formula boolean, than will need for implement the negation operator on OBDD (Order Binary Decision Diagram) that is, show that:
$\qquad \displaystyle \neg f(x_1,\ldots,x_n) = (\neg x_1 \wedge \neg f|_{x_1=0}) \vee (x_1 \wedg... |
Given a cactus, we want to weight its edges in such a way that
For each vertex, the sum of the weights of edges incident to the vertex is no more than 1.
The sum of all edge weights is maximized.
Clearly the answer is no more than $\frac{n}{2}$ for $n$ vertices ($\sum d_i = 2D$ where $d_i$ is the sum for one vertex a... |
I've been studying the Spearman's rank correlation coefficient
$\qquad \displaystyle \rho = \frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_i (x_i-\bar{x})^2 \sum_i(y_i-\bar{y})^2}}$.
for two lists $x_1, \dots, x_n$ and $y_1, \dots, y_n$. What's the complexity of the algorithm?
Since the algorithm should just comput... |
Suppose that we have a random generator that outputs
numbers in the range $[0..R-1]$ with uniform distribution and we
need to generate random numbers in the range $[0..N-1]$
with uniform distribution.
Suppose that $N < R$ and $N$ does not evenly divide $R$;
in order to get a truly uniform distribution we can use the
re... |
I have read that the degree of nodes in a "knowledge" graph of people roughly follows a power law distribution, and more exactly can be approximated with a Pareto-Lognormal distribution.
Where can I find a kind of algorithm that will produce a random graph with this distribution?
See for example the paper Revisiting De... |
The lookup time in perfect hash-tables is $O(1)$ in the worst case. Does that simply mean that the average should be $\leq O(1)$?
|
Sometimes it is necessary to dig into Artificial Intelligence or specifically Machine Learning problems to make a research. Common googling (in my own experience) usually doesn't help much due to a lot of irrelevant or paid material among the results. Google Scholar in contrast is limited to scientific publications onl... |
Say instead of using a linked list as buckets for a hash table of size $m$, we use another hash table of size $p$ as buckets this time. What would be the average case for this problem?
I looked up perfect hashing and I got a very close algorithm, and it is $O(1)$. Can someone please clarify?
|
In Algorithmic Randomness and Complexity from Downey and Hirschfeldt, it is stated on page 129 that
$\qquad \displaystyle \sum_{K(\sigma)\downarrow} 2^{-K(\sigma)} \leq 1$,
where $K(\sigma)\downarrow$ means that $K$ halts on $\sigma$, $\sigma$ being a binary string. $K$ denotes the prefix-free Kolmogorov complexity.
... |
If a graph $G$ is connected and has no path with a length greater than $k$, prove that every two paths in $G$ of length $k$ have at least one vertex in common.
I think that that common vertex should be in the middle of both the paths. Because if this is not the case then we can have a path of length $>k$. Am I right?... |
Semi clustering algorithm is mentioned in the Google Pregel paper. The score of a semi cluster is calculated using the below formula
$\qquad \displaystyle S_c =\frac{I_c - f_BB_c}{\frac{1}{2}V_c(V_c - 1)}$
where
$I_c$ is sum of the weights of all the internal edges,
$B_c$ is the sum of the weights of all the boundary ... |
Is the language
$$L = \{a,b\}^* \setminus \{(a^nb^n)^n\mid n \geq1 \}$$
context-free? I believe that the answer is that it is not a CFL, but I can't prove it by Ogden's lemma or Pumping lemma.
|
I have a high interest in priority-queues (E.g., see my answers on: Does there exist a priority queue with $O(1)$ extracts?), and was wondering if there is a priority-queue or similar data-structure where you can sort by multiple values?
For example, if I wanted to sort by numval and sort by strval, and be able to get ... |
Given an array $A$ of $N$ integers, each element in the array can be increased by a fixed number $b$ with some probability $p[i]$, $0 \leq i < n$. I have to find the expected number of swaps that will take place to sort the array using bubble sort.
I've tried the following:
The probability for an element $A[i] > A[j]$... |
In "Computers and Intractability A Guide to the Theory of NP-Completeness" textbook pp 236, "Sequencing to minimize tardy tasks" is NP-complete.
To be specific the problem is as follows:
For each task $t \in T$, partial order $<$ on $T$, a length $l(t)$, and a deadline $d(t)$ and a positive integer $K \leq |T|$. Is th... |
Given Graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$. Can a graph isomorphic to $H$ be obtained from $G$ by a sequence of edge contractions ? We know this problem is NP-complete. What about if only a subset of edges are valid for contraction at each step of the sequence. For example when deciding the first edge for contraction... |
I have asked this question at math.stackexchange, but I have been redirected here.
My question is, if there is any historical relation between the fields of Dynamical systems (and related fields such as Optimal control) and (theoretical) Computer science. The reason for which I ask this question is, that several older ... |
Do there exist any Turing complete typed lambda calculi? If so, what are a few examples?
|
I wanted ask if you know an algorithm to find the witness for $EU(\phi_1,\phi_2)$ (CTL formula "Exist Until") using BDDs (Binary Decision Diagram). In pratice you should use the fixed point for calculating $EU(\phi_1,\phi_2)$, that is:
$\qquad \displaystyle EU(\phi_1,\phi_2)=\mu.Q (\phi_2 \vee (\phi_1 \wedge EX Q)) $
... |
I want to specify, what it means to give an algebra as input to an algorithm and didn't find very much literature about it. So first I want to ask if you can recommend a book or paper that deals with the topic of complexity analysis of algebras over fields and clearly define the decision problem.
After some digging I f... |
Is there a fundamental difference between top-down and bottom-up dynamic programming?
In particular, is there a problem which can be solved bottom-up but not top-down? Or is the bottom-up approach just an unwinding of the recurrence in the top-down approach?
|
Is it possible to always construct a hamiltonian path on a tournament graph $G=(V,E)$ by sorting (using any sorting algorithm) with the following total order:
$\qquad \displaystyle a \leq b \iff (a,b) \in E \lor \left(\exists\, c \in V. a \leq c \land c \leq b\right)$
For context, this came from an observation that the... |
There are $n$ people at a table. The $i$th person has to pay $p_i$ dollars.
Some people don't have the right bills to pay exactly $p_i$, so they come up with the following algorithm.
First, everyone puts some of their money on the table. Then each individual takes back the money they overpaid.
The bills have a fixe... |
I wanted to ask if someone can help me to construct a regular expression over the alphabet $\{a,b,x\}$ for the language $L$ which is constituted by all strings containing an odd number of $a$'s, and in which between each pair of consecutive $a$'s there is an even number of $b$'s (and an arbirtary number of $x$'s).
For ... |
Suppose that someone found a polynomial algorithm for a NP-complete decision problem. Would this mean that we can modify the algorithm a bit and use it for solving the problems that are in NP, but not in NP-complete? Or would this just shows the availability of a polynomial algorithm for each NP problem indirectly?
edi... |
I just learned that when we have a polynomial algorithm for NP-complete problems, it is possible to use that algorithm to solve all NP problems.
So, the question is how we then distinguish non-NP-complete NP problems from NP-complete problems? It seems that all these problems will have a polynomial algorithm to conver... |
The discrete logarithm is the same as finding $b$ in $a^b=c \bmod N$, given $a$, $c$, and $N$.
I wonder what complexity groups (e.g. for classical and quantum computers) this is in, and what approaches (i.e. algorithms) are the best for accomplishing this task.
The wikipedia link above doesn't really give very concrete... |
The problem of finding the largest subgraph of a graph that has a Hamiltonian path can be restated as finding the longest path of a graph. Is this NP-complete? Also, is finding the $k$-length path of a graph NP-complete? Is it still NP-complete if we require the path to visit a given vertex?
|
The question is exercise 1.9 from Arora-Barak's book Computational Complexity — A Modern Approach:
Define a RAM Turing machine to be a Turing machine that has random access memory. We formalize this as follows: The machine has an infinite array A that is initialized to all blanks. It accesses this array as follows. On... |
The language is:
$$
L = \{ (a^n) (b^m) \mid n + m = 3k, k \ge 0 \}
$$
My attempt at an answer:
$$
(a \cup b)^{3k}
$$
This will work if the a OR b can change for each instance in the string that is (3k) long. If not, what can I do to fix this?
|
I've been tasked with solving some recurrence relations, and I've been running into trouble with so called 'chip & conquer' relations.
Here are some example problems:
$$T(n) = T(n-5) + cn^2$$
and
$$T(n) = T(n-2) + \log{n}$$
I'm supposed to be giving an answer in $\Theta$ notation. How do I go around and solving relatio... |
I am trying to understand the approach for this problem:
"If all streets are one way, there is still a legal way to drive from
one intersection to another"
The question is to prove that it can be done in linear time. I am not looking for direct answers but the approach to this problem.
How can I think about this pr... |
Given an undirected graph, I define a structure called k-key as a path containing $k$ vertices which are connected to a simple cycle which contains $k$ vertices as well.
Here's the k-key problem: given an undirected graph $G$ and a number $k$, decide whether $G$ contains k $k$-key.
I want to show that the k-key problem... |
I'm working on an expression typing system and looking for insights on what algorithms may be available which solve my problem -- or a proof that its complexity is too high to be reasonable to implement. The problem is defined below.
I have a set of types which form a directed graph $T = (V,E)$ (assume no cycles). This... |
I've been reading a bit of the literature lately, and have found some rather interesting data-structures.
I have researched various different methods of getting update times down to $\mathcal{O}(1)$ worst-case update time [1-7].
Recently I begun looking into lock-free data-structures, to support efficient concurrent ac... |
Let $\Sigma$ be a given alphabet. Is there a way to code up Deterministic Finite state Automata (DFA) over $\Sigma$ as strings of $\Sigma$ in such a way that the corresponding subset of $\Sigma^*$ is a regular language?
For example for Turing machines, the set of codes of Turing machines over a fixed alphabet is decida... |
There is a time series of say $100$ data points. I wish to assign symbols of $0, 1, 2$ for each unique data point. The issue is I have tried but got stuck since no matter I specify the symbols, the program just outputs probability of $1$'s and $0$'s. The following are the questions:
How to find probability or correct ... |
Definition: Karp Reduction
A language $A$ is Karp reducible to a language $B$ if there is a polynomial-time computable function $f:\{0,1\}^*\rightarrow\{0,1\}^*$ such that for every $x$, $x\in A$ if and only if $f(x)\in B$.
Definition: Levin Reduction
A search problem $V_A$ is Levin reducible to a search problem $V_B$ ... |
Consider the set of graphs in which the maximum degree of the vertices is a constant number $\Delta$ independent of the number of vertices. Is the vertex coloring problem (that is, color the vertices with minimum number of colors such that no pair of adjacent nodes have the same color) on this set still NP-hard? Why?
|
if I have a grammar having a production that contains both left recursion and left factoring like
$\qquad \displaystyle F \to FBa \mid cDS \mid c$
which one has priority, left recursion or left factoring?
|
Let $\mathbb F\in\{\mathbb R, \mathbb C\}$ the field of real or complex numbers. Then [1, page 22 in the middle] claims that the following equation can easily be solved in deterministic polynomial time:
$$ \sum_{i=1}^n a_ix_i^2=b$$
with $a_i, b\in\mathbb F$. Some discussion suggests, that the algorithmical model assume... |
Suppose that someone found an algorithm A for a NP problem (that is not NP-complete) that uses an algorithm B for PSPACE-complete or #P-complete problem during execution. (Remaining part of the algorithm takes polynomial time.)
Then suppose there is also an algorithm C for a NP problem that uses the polynomial-consumin... |
I define a long CNF to contain at least $2^\frac{n}{2}$ clauses, where $n$ is the number of its variables. Let $\text{Long-SAT}=\{\phi: \phi$ is a satisfiable long CNF formula$\}$.
I'd like to know why $\text{Long-SAT} \in P$. First I thought it is $\text{NPC}$ since I can do a polynomial-time reduction from $\text{SA... |
I'm looking for a simple calculus that supports reasoning about reflection, namely, the introspection and manipulation of running programs.
Is there an untyped $\lambda$-calculus extension that enables one to convert $\lambda$-terms into a form that can be syntactically manipulated and then subsequently evaluated?
I ... |
Given a directed graph $G$ and two nodes $s,t$, decide whether there is some node $s'$ such that $s'$ is reachable from $s$ while $t$ is not reachable from $s'$.
I am wondering whether this problem is in NL.
|
Is $NP$ with oracle access to $NP$ larger than just $NP$?
As I understand $NP^
{NP}$ is just a turing machine that can make queries to an other $NP$ machine if so than $NP$ can simulate $NP^{NP}$? Is there something wrong with this argument?
|
If I have a Type 3 Grammar, it can be represented on a pushdown automaton (without doing any operation on the stack) so I can represent regular expressions by using context free languages. But can I know if a type 3 grammar is $LR(1)$, $LL(1)$, $SLR(1)$, etc. without constructing any parse tables?
|
Given a set of segments, I would like to compute the set of closed polygons inside the convex hull of the set of the end of those segments. The vertices of the polygons are the intersections of the segments. For example, if you draw the 6 lines restricted which equations are: $x=-1$, $x=0$, $x=1$, $y=-1$, $y=0$, $y=1$,... |
I was going through the discussion on the question How to define quantum Turing machines? and I feel that quantum TM and nondetermistic TM are one and the same. The answers to the other question do not touch on that. Are these two models one and the same?
If no,
What are the differences between quantum TM and NDTM?
I... |
I've been looking into Recurrent Neural Networks, but I don't understand what the architecture of a neural network would look like when the output length is not necessarily fixed.
It seems like most networks I've read descriptions of require the output length to be equal to the input length or at least a fixed size. B... |
Context
If you have to pay an amount of money at a store and have a limited collection of payment items (i.e. coins and banknotes) -- let's for simplicity assume there are only coins -- a trivial algorithm is to always greedily pick the biggest coin that is smaller or equal to the remaining value you have to pay. Anoth... |
I have a bivariate function like
$
f(x,y) = \frac{1}{x^3 \sqrt{\pi}}. e^{\frac{2-x}{x^2}} . y^3 . e^{3.y \over 3-y}
$
and I want to find its global maximum over a range of
$
x \in [0, 200] \text{, and } y \in [300,50000]
$
What kind of algorithms I can use to find the global maximum. I want to have keywords for search... |
A Turing machine that returns to a previously encountered state with its read/write head on the same cell of the exact same tape will be caught in a loop. Such a machine doesn't halt.
Can someone give an example of a never-halting machine that doesn't loop?
|
Mark lives in a tiny country populated by people who tend to over-think things. One day, the king of the country decides to redesign the country's currency to make giving change more efficient. The king wants to minimize the expected number of coins it takes to exactly pay any amount up to (but not including) the amoun... |
I'm trying to convert a context free grammar to a pushdown automaton (PDA); I'm not sure how I'm gonna get an answer or show you my progress as it's a diagram... Anyway this is the last problem I have on a homework that's due later today, so I'd appreciate some kind of help, even if it's just an explanation of the corr... |
Given a TSP instance $T$, decide whether changing the city coordinates by adding a vector of coordinates $v$ will change the optimal TSP objective by atleast $x$. The city coordinates are integers.
The problem is in PSPACE but even the verification problem seems to be NP-hard. Is that true?
If the verification problem ... |
Linear temporal logic and deterministic Büchi automata are incomparable: DBA cannot express $FGa$, and LTL cannot express "at least each odd letter is 'a'". But sometimes it is interesting to know whether the language of a DBA can be expressed in LTL.
I need an algorithm that decides whether a language of a given DBA i... |
I'd like to learn something about this optimization problem: For given non-negative whole numbers $a_{i,j,k}$,
find a function $f$ minimizing the expression
$$\max_k \sum_i a_{i,f(i),k}$$
An example using a different formulation might make it clearer:
You're given a set of sets of vectors like
{
{(3, 0, 0, 0, 0), (... |
I have this finite-state process with the corresponding labeled transition system:
The FSP is:
MAKER = (kochen -> fertig -> MAKER).
USER = (fertig -> essen -> USER).
||MAKER_USER = (USER||MAKER).
Why is essen possible after the second kochen? USER needs to have fertig in order to able to essen right?
|
I know it's possible to build a Finite State Transducer for converting numbers from base 2 to base 4 or 8 or other powers of 2 (translating from base N to base N^M is easy). However I've never seen a FST that can convert numbers from base 1 to base 2 or viceversa. Can a FST even do this? If so, can you please give some... |
Consider one specific useful function of our human brain: abstraction of object. Take the example of two pictures: if we are told the pictures are similar, we actually make conclusion about the aspects in which they are close to each other.
I'm considering whether machine can have the ability described. More accurately... |
IPv6 addresses in the form of 862A:7373:3386:BF1F:8D77:D3D2:220F:D7E0 are much harder to memorize or even transcribe than the 4 octets of IPv4.
There have been attempts to mitigate this, making IPv6 addresses somehow more memorable.
Is there an intentionally-weak hashing function which could be reversed to find that ... |
The only paper I can find about an implementation of a lossless video codec is an article by Michael Niedermayer which explains the FF1V compression pipeline.
I'm wondering if anyone else has found any articles similar to this. I've looked on IEEE and ACM and their articles mostly focus on one algorithm that's part of ... |
Does there exist any data structure that efficiently (linearithmic-or-similar in-order traversal, sublinear time insertion/search/removal) implements an ordered set (i.e. a set that allows for enumerating the items in insertion order)?
If so, what data structure is it/how does it do this?
|
Is there any formal definition about the average height of a binary tree?
I have a tutorial question about finding the average height of a binary tree using the following two methods:
The natural solution might be to take the average length of all possible
paths from the root to a leaf, that is
$\qquad \displaystyle \... |
I am interested in exploring the world of BDI agents (software agents that possess "beliefs, desires, intentions", essentially the agent has knowledge of the world, a set of motivations, and carries out certain plans).
I recently read A Canonical Agent Model for Healthcare Applications [1], which left me with a lot of ... |
What's the definition of Kolmogorov complexity for a decision problem? For example, how to define the length of the shortest program that solves the 3SAT problem? Is it the "smallest" Turing machine that recognizes the 3SAT langauge?
|
Suppose we have $N$ fixed points (set $S$ with $|S|=N$) on the plane and $m$ agents with fixed, known initial positions ($m<N$) outside $S$. We should transfer the agents so that in our final configuration they are all positioned to different points of $S$. How could we achieve it by minimizing the total distance cover... |
A context-free grammar is said to be self-embedding if and only
if contains a derivation of the form $\xi\stackrel*\Rightarrow u\xi v$, where $\xi$ is a non-terminal and $u$, $v$ are some non-empty words.
Please note that I am aware the undecidability of the conversion of context-free grammar to regular grammar. But gi... |
I'm not sure if this question belongs here and I apologize if not. What I am looking to do is to develop a programmatic way in which I can probabilistically determine whether a given string "belongs" in a bag of strings. For example, if I have bag of 10,000 US city names, and then I have the string "Philadelphia", I ... |
Text books everywhere assume that the Bounded Post Correspondence Problem is NP-complete (no more than $N$ indexes allowed with repetitions). However, nowhere is one shown a simple (as in, something that an undergrad can understand) polynomial time reduction from another NP-complete problem.
However every reduction I c... |
At my current project I had a network problem come up for which I could not find a solution. In a peer-to-peer network I needed to send an action to all peers, and each peer was to act on it only if it could verify that all other peers would also act on it.
That is, given a network of peers $P = { P_1, ..., P_n }$. We ... |
The book "Neural Networks and Learning Machines" by Simon Haykin has many computer experiments to which many exercises are related. But there seems to be no data for these experiments available online. Where can I find them?
|
In computer science, we have often have to solve recurrence relations, that is find a closed form for a recursively defined sequence of numbers. When considering runtimes, we are often interested mainly in the sequence's asymptotic growth.
Examples are
The runtime of a tail-recursive function stepping downwards to $0... |
According to the Wikipedia article, the L in $LR(k)$ means "left-to-right scan", and the "R" means "rightmost derivation." However, in Knuth's original paper on $LR(k)$ grammars, he defines $LR(k)$ (on page 610) as a language that is "translatable from left to right with bound $k$."
I am guessing that this new termino... |
I have recently implemented the Paull's algorithm for removing left-recursion from context-free grammars:
Assign an ordering $A_1, \dots, A_n$ to the nonterminals of the grammar.
for $i := 1$ to $n$ do begin
$\quad$ for $j:=1$ to $i-1$ do begin
$\quad\quad$ for each production of the form $A_i \to A_j\alpha$ do begin
... |
I have the grammar:
$\qquad \begin{align}
S &\to S = P \mid S \neq P \mid P \\
P &\to NUM
\end{align}$
This grammar suffers from left recursion. To eliminate left recursion, I got:
$\qquad \begin{align}
S &\to PS' \\
S' &\to\, = PS' \mid\, \neq PS' \mid \varepsilon \\
P &\to NUM
\end{align}$
However when constr... |
How would you prove that the following language is not context-free?
$$ L= \{a^n b^m |\, gcd(n,m)=1 \}$$
I suspect the solution uses the pumping lemma, but I'm not sure how to apply it.
|
I cannot understand decidability really well. I have been reading from books and internet, but I am little bit confused.
According to the book (as I understood), we can decide on decidability of a problem by constructing Turing Machines. Let us say that we want to decide whether an NFA is a decidable language. First, ... |
The problem is as follows:
Given a finite set of rectangles ($S\subset\mathbb{R}\times\mathbb{R}$), build a data structure that will support the following operations:
Check, receives a rectangle $r\in\mathbb{R}\times\mathbb{R}$, and returns true iff there is a rectangle $c\in S$ so that $r_x \leq c_x$ and $r_y \leq c_... |
Summary: According to Rice's theorem, everything is impossible. And yet, I do this supposedly impossible stuff all the time!
Of course, Rice's theorem doesn't simply say "everything is impossible". It says something rather more specific: "Every property of a computer program is non-computable."
(If you want to split h... |
I asked a (seed) question about sums of Landau terms before, trying to gauge the dangers of abusing asymptotics notation in arithmetics, with mixed success.
Now, over here our recurrence guru JeffE does essentially this:
$\qquad \displaystyle \sum_{i=1}^n \Theta\left(\frac{1}{i}\right) = \Theta(H_n)$
While the end resu... |
Borůvka's algorithm is one of the standard algorithms for calculating the minimum spanning tree for a graph $G = (V,E)$, with $|V| = n, |E| = m$.
The pseudo-code is:
MST T = empty tree
Begin with each vertex as a component
While number of components > 1
For each component c
let e = minimum edge out of compon... |
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