id int64 1 141k | title stringlengths 15 150 | body stringlengths 45 28.5k | tags stringlengths 1 102 | label int64 1 1 | text stringlengths 128 28.6k | source stringclasses 1
value |
|---|---|---|---|---|---|---|
109,734 | Substitution for Landau's O notation formula | <p> I found the following description when I was reading a paper on computational complexity theory.</p>

<blockquote>
 <p>This can be done ... in time 2<sup><em>n</em></sup>･poly(log<em>s</em>,<em>n</em>)+2<sup><em>O</em>(log<em>s</em>)<sup><em>c</em></sup></sup>. For <em>s</em>≤2<sup><em>n</em><sup><em>o... | complexity theory landau notation | 1 | Substitution for Landau's O notation formula -- (complexity theory landau notation)
<p> I found the following description when I was reading a paper on computational complexity theory.</p>

<blockquote>
 <p>This can be done ... in time 2<sup><em>n</em></sup>･poly(log<em>s</em>,<em>n</em>)+2<sup><em>O</em>(... | habedi/stack-exchange-dataset |
109,743 | Does real linear programming produce bipartite perfect matching using maxflow reduction? | <p>Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:</p>
<p><a href="https://i.stack.imgur.com/YxPxq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YxPxq.png" alt="enter image description here" /></a></p>
<p>We can formulate max... | linear programming network flow integer programming bipartite matching bipartite graph | 1 | Does real linear programming produce bipartite perfect matching using maxflow reduction? -- (linear programming network flow integer programming bipartite matching bipartite graph)
<p>Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:</p>
<p><a href="ht... | habedi/stack-exchange-dataset |
109,752 | What is the specific name for basic data types? | <p>What is the specific name for data types such as integer, string, real (the basic ones)?</p>
 | terminology abstract data types | 1 | What is the specific name for basic data types? -- (terminology abstract data types)
<p>What is the specific name for data types such as integer, string, real (the basic ones)?</p>
 | habedi/stack-exchange-dataset |
109,768 | Shortest path with a given condition | <p>The problem says to find the shortest way (the smallest amount of intermediary points), with given source and destination points, such that between two consecutive intermediary points there are two alternative routes. Also, all the
connections between the points on the map are given.</p>

<p>I understood... | shortest path | 1 | Shortest path with a given condition -- (shortest path)
<p>The problem says to find the shortest way (the smallest amount of intermediary points), with given source and destination points, such that between two consecutive intermediary points there are two alternative routes. Also, all the
connections between the p... | habedi/stack-exchange-dataset |
109,771 | Does $L_1L_2 = L_2L_1$ imply $L_1 = L_2$? | <p>Let <span class="math-container">$L_1, L_2 \subseteq \Sigma^*$</span> be two languages, where <span class="math-container">$\Sigma$</span> is some finite Alphabet.</p>

<p>Does <span class="math-container">$L_1L_2 = L_2L_1$</span> imply <span class="math-container">$L_1 = L_2$</span>?</p>

<p>What if... | formal languages regular languages | 1 | Does $L_1L_2 = L_2L_1$ imply $L_1 = L_2$? -- (formal languages regular languages)
<p>Let <span class="math-container">$L_1, L_2 \subseteq \Sigma^*$</span> be two languages, where <span class="math-container">$\Sigma$</span> is some finite Alphabet.</p>

<p>Does <span class="math-container">$L_1L_2 = L_2L_1$</sp... | habedi/stack-exchange-dataset |
109,775 | Multiple knapsack problem with equal profit and different weight | <p>I am doing a research about the load balancing problem in 5G system, but I am not sure if my problem is a NP-complete problem.</p>

<p>The problem is:</p>

<ul>
<li>given a set of n items and a set of m knapsack</li>
<li>capacity of knapsacks are equal</li>
<li>the weight of item j in kna... | knapsack problems | 1 | Multiple knapsack problem with equal profit and different weight -- (knapsack problems)
<p>I am doing a research about the load balancing problem in 5G system, but I am not sure if my problem is a NP-complete problem.</p>

<p>The problem is:</p>

<ul>
<li>given a set of n items and a set of m knapsa... | habedi/stack-exchange-dataset |
109,793 | Understanding proof of upper bound on complexity of recursive computation of graph chromatic polynomial | <p>This question is about section 2.3 of Wilf's ``Algorithms and Complexity''</p>

<p><a href="https://www.math.upenn.edu/~wilf/AlgoComp.pdf" rel="nofollow noreferrer">https://www.math.upenn.edu/~wilf/AlgoComp.pdf</a></p>

<p>in which he analyses the complexity of a recursive computation of the chromati... | graphs runtime analysis recursion | 1 | Understanding proof of upper bound on complexity of recursive computation of graph chromatic polynomial -- (graphs runtime analysis recursion)
<p>This question is about section 2.3 of Wilf's ``Algorithms and Complexity''</p>

<p><a href="https://www.math.upenn.edu/~wilf/AlgoComp.pdf" rel="nofollow noreferrer">h... | habedi/stack-exchange-dataset |
109,795 | In predicate logic, is the "environment" only needed when free variables are present? | <p>Suppose there exist no free variables in a given predicate logic formula. Is then a model alone sufficient to fully interpret the formula and make inferences? Don't we need an <em>environment</em> or variable lookup table (as often mentioned)?</p>
 | predicate logic | 1 | In predicate logic, is the "environment" only needed when free variables are present? -- (predicate logic)
<p>Suppose there exist no free variables in a given predicate logic formula. Is then a model alone sufficient to fully interpret the formula and make inferences? Don't we need an <em>environment</em> or variable l... | habedi/stack-exchange-dataset |
109,798 | Reduction from minimum dominating set to the set cover | <p>To solve the min dominating set problem of a graph G, we can reduce it to a set cover problem.</p>

<p>For example to find the MDS of the graph G:</p>

<h1><a href="https://i.stack.imgur.com/DRuVU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DRuVU.png" alt="G"></a></h1>

... | complexity theory graphs computability np complete | 1 | Reduction from minimum dominating set to the set cover -- (complexity theory graphs computability np complete)
<p>To solve the min dominating set problem of a graph G, we can reduce it to a set cover problem.</p>

<p>For example to find the MDS of the graph G:</p>

<h1><a href="https://i.stack.imgur.com... | habedi/stack-exchange-dataset |
109,801 | When Huffman coding is inefficient? | <p>I have a question regarding the redundancy of Huffman coding. I know that for a general prefix code we have the following inequality: </p>

<p><span class="math-container">$$ H(X) \le R \le H(X) + 1 $$</span> </p>

<p><span class="math-container">$R$</span> being the rate (average codeword length) an... | entropy huffman coding | 1 | When Huffman coding is inefficient? -- (entropy huffman coding)
<p>I have a question regarding the redundancy of Huffman coding. I know that for a general prefix code we have the following inequality: </p>

<p><span class="math-container">$$ H(X) \le R \le H(X) + 1 $$</span> </p>

<p><span class="math-c... | habedi/stack-exchange-dataset |
109,823 | When inserting an element in a priority queue and the heap size is already at max capacity, should you output an error OR increase the array size? | <p>I'm currently learning about how to implement priority queues using heaps, but I've hit a wall while trying to implement the insertion operation. Assume that the size of the array storing the elements of the heap is <strong><em>n</em></strong>, and the size of the heap is also <strong><em>n</em></strong>. Assume fu... | heaps priority queues | 1 | When inserting an element in a priority queue and the heap size is already at max capacity, should you output an error OR increase the array size? -- (heaps priority queues)
<p>I'm currently learning about how to implement priority queues using heaps, but I've hit a wall while trying to implement the insertion operati... | habedi/stack-exchange-dataset |
109,828 | What are the current state of art approximation algorithm for NP-Hard problems? | <p>I came cross some works try to use deep learning to approximate NP-Hard</p>

<p><a href="https://arxiv.org/pdf/1810.10659.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1810.10659.pdf</a></p>

<p>Though the paper seems to have very good results but based on the citations. I'm quit doubt if it r... | np complete np hard approximation | 1 | What are the current state of art approximation algorithm for NP-Hard problems? -- (np complete np hard approximation)
<p>I came cross some works try to use deep learning to approximate NP-Hard</p>

<p><a href="https://arxiv.org/pdf/1810.10659.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1810.10659.pdf<... | habedi/stack-exchange-dataset |
109,836 | Reduction from Exact Cover to Fixed Exact Cover | <p>I am trying to reduce Exact Cover to Fixed Exact Cover to show that Fixed Exact Cover is NP-Hard. </p>

<p><strong>Exact Cover</strong></p>

<p><strong>Input</strong></p>

<p>S = {x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>n</sub>} (set)</p>

<p>P = {P<sub>1</sub>, P<sub>2</sub>, ..., P... | reductions np hard | 1 | Reduction from Exact Cover to Fixed Exact Cover -- (reductions np hard)
<p>I am trying to reduce Exact Cover to Fixed Exact Cover to show that Fixed Exact Cover is NP-Hard. </p>

<p><strong>Exact Cover</strong></p>

<p><strong>Input</strong></p>

<p>S = {x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>... | habedi/stack-exchange-dataset |
109,850 | Is the language of all TMs *not* accepting a given string, Enumerable? | <p>Is the following language in <strong>RE</strong>?<br>
<span class="math-container">$$L = \{\langle M\rangle : M\text{ is a TM that does not accept }010\}$$</span> </p>

<p>I could use Rice's Theorem with the property <span class="math-container">$P = \{L : 010\text{ is not in }L\}$</span> to show it isn... | turing machines computability semi decidability | 1 | Is the language of all TMs *not* accepting a given string, Enumerable? -- (turing machines computability semi decidability)
<p>Is the following language in <strong>RE</strong>?<br>
<span class="math-container">$$L = \{\langle M\rangle : M\text{ is a TM that does not accept }010\}$$</span> </p>

<p>I could ... | habedi/stack-exchange-dataset |
109,867 | Minimum number of comparisons in comparison-based sorting algorithms | <p>I've seen that every comparison-based sorting algorithm must perform at least <span class="math-container">$\log_{2}(n!)=\Omega(nlog(n))$</span> comparisons on some input (<em>n</em> being the size of the input). </p>

<p>Why is the minimum number of comparisons <span class="math-container">$\log_{2}(n!)$</s... | sorting | 1 | Minimum number of comparisons in comparison-based sorting algorithms -- (sorting)
<p>I've seen that every comparison-based sorting algorithm must perform at least <span class="math-container">$\log_{2}(n!)=\Omega(nlog(n))$</span> comparisons on some input (<em>n</em> being the size of the input). </p>

<p>Why i... | habedi/stack-exchange-dataset |
109,872 | Regular expression notation clarification | <p>For the alphabet <span class="math-container">$Σ$</span>={a,b,c}</p>

<p>I was wondering how you would say:</p>

<blockquote>
 <p>T that has elements from Σ, so could be T=a, T=bc</p>
</blockquote>

<p>I was considering maybe <span class="math-container">$Σ^*$</span> or <span class="... | regular languages regular expressions | 1 | Regular expression notation clarification -- (regular languages regular expressions)
<p>For the alphabet <span class="math-container">$Σ$</span>={a,b,c}</p>

<p>I was wondering how you would say:</p>

<blockquote>
 <p>T that has elements from Σ, so could be T=a, T=bc</p>
</blockquote>

... | habedi/stack-exchange-dataset |
109,873 | A query regarding P vs NP | <p><strong>Problem Statement:</strong> Given a Problem <span class="math-container">$(A)$</span> such that: Any proposed solution <span class="math-container">$(S_0)$</span> is guaranteed to be polynomial size w.r.t. <span class="math-container">$A$</span>. Now, consider a fixed set of <span class="math-container">$K$<... | complexity theory | 1 | A query regarding P vs NP -- (complexity theory)
<p><strong>Problem Statement:</strong> Given a Problem <span class="math-container">$(A)$</span> such that: Any proposed solution <span class="math-container">$(S_0)$</span> is guaranteed to be polynomial size w.r.t. <span class="math-container">$A$</span>. Now, consider... | habedi/stack-exchange-dataset |
109,894 | Why does not valiant's reduction show NP=RP? | <p>Valiant converts <span class="math-container">$SAT$</span> formula to a <span class="math-container">$0/1$</span> matrix such that <span class="math-container">$Permanent$</span> of the matrix is <span class="math-container">$4^m\#SAT$</span>.</p>

<p>We know <span class="math-container">$Permanent$</span> c... | complexity theory counting | 1 | Why does not valiant's reduction show NP=RP? -- (complexity theory counting)
<p>Valiant converts <span class="math-container">$SAT$</span> formula to a <span class="math-container">$0/1$</span> matrix such that <span class="math-container">$Permanent$</span> of the matrix is <span class="math-container">$4^m\#SAT$</spa... | habedi/stack-exchange-dataset |
109,898 | Determine smallest possible parameter set for FPT | <p>I am reading a book about complexity analysis and cannot find a way to solve a problem in that book.
The problem is, that I do not understand how to determine the smallest possible parameters, given the runtime of an algorithm. Do I just need to insert some numbers and see if the parameters exceed n at some poin... | complexity theory parameterized complexity | 1 | Determine smallest possible parameter set for FPT -- (complexity theory parameterized complexity)
<p>I am reading a book about complexity analysis and cannot find a way to solve a problem in that book.
The problem is, that I do not understand how to determine the smallest possible parameters, given the runtime of a... | habedi/stack-exchange-dataset |
109,901 | Confused in calculating the shortest-path using Dijkstra's Algorithm | <p><a href="https://i.stack.imgur.com/I72ma.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/I72ma.png" alt="enter image description here"></a></p>

<p>I am required to find the shortest path using Dijkstra's Algorithm. After performing the calculations, am I getting the following values corre... | algorithms shortest path | 1 | Confused in calculating the shortest-path using Dijkstra's Algorithm -- (algorithms shortest path)
<p><a href="https://i.stack.imgur.com/I72ma.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/I72ma.png" alt="enter image description here"></a></p>

<p>I am required to find the shortest path usi... | habedi/stack-exchange-dataset |
109,905 | Give an NL-algorithm for complement of 2-SAT | <p><strong>Question1:</strong> What is the difference between 2SAT and the complement of 2SAT?</p>

<p><strong>Question2:</strong> It is known that NL is contained in P, but what we know about P over NL? can be said that an algorithm that runs in P, uses NL space?</p>

<p>Information That I gather</p>&#... | complexity theory 2 sat | 1 | Give an NL-algorithm for complement of 2-SAT -- (complexity theory 2 sat)
<p><strong>Question1:</strong> What is the difference between 2SAT and the complement of 2SAT?</p>

<p><strong>Question2:</strong> It is known that NL is contained in P, but what we know about P over NL? can be said that an algorithm that... | habedi/stack-exchange-dataset |
109,917 | Minimum depth of a leaf in a tree that corresponds to a comparison-based sorting algorithm | <p>The lower bound of comparisons in a comparison-based sorting algorithm is <span class="math-container">$\log_2 n!=Θ(n\log n)$</span>. Yet there can be less comparisons needed in an algorithm.
If you take a sorted array, it will take <span class="math-container">$n-1 = O(n)$</span> comparisons (with insertion sor... | sorting | 1 | Minimum depth of a leaf in a tree that corresponds to a comparison-based sorting algorithm -- (sorting)
<p>The lower bound of comparisons in a comparison-based sorting algorithm is <span class="math-container">$\log_2 n!=Θ(n\log n)$</span>. Yet there can be less comparisons needed in an algorithm.
If you take a sor... | habedi/stack-exchange-dataset |
109,933 | Complexity of "Fast Poisson Disk Sampling in Arbitrary Dimensions" | <p>I came across the paper <a href="https://www.cs.ubc.ca/~rbridson/docs/bridson-siggraph07-poissondisk.pdf" rel="nofollow noreferrer">Fast Poisson Disk Sampling in Arbitrary Dimensions</a> which gives an algorithm for generating Poisson disk points in <span class="math-container">$\mathbb{R}^n$</span>. 
It's claim... | time complexity graphics | 1 | Complexity of "Fast Poisson Disk Sampling in Arbitrary Dimensions" -- (time complexity graphics)
<p>I came across the paper <a href="https://www.cs.ubc.ca/~rbridson/docs/bridson-siggraph07-poissondisk.pdf" rel="nofollow noreferrer">Fast Poisson Disk Sampling in Arbitrary Dimensions</a> which gives an algorithm for gene... | habedi/stack-exchange-dataset |
109,935 | The Ising Model and Computational Complexity | <p>I've been told recently that one can use the Ising model can find solutions to certain NP-hard problems, such as Clique, although it doesn't do so in polynomial time.</p>

<p>Googling gets a few Arxiv results but where might I find an introduction to this interesting relationship?</p>
 | complexity theory reference request np hard physics | 1 | The Ising Model and Computational Complexity -- (complexity theory reference request np hard physics)
<p>I've been told recently that one can use the Ising model can find solutions to certain NP-hard problems, such as Clique, although it doesn't do so in polynomial time.</p>

<p>Googling gets a few Arxiv result... | habedi/stack-exchange-dataset |
109,936 | cheap multi agent path planning algorithms | <p>I am trying to make a tower defence game where some critters are trying to make their ways through a 2D grid-like obstacle course. The idea is these critters cannot walk into the obstacles or into each other during their path, and they would like to get to the other side as soon as possible.</p>

<p>I just n... | a star search | 1 | cheap multi agent path planning algorithms -- (a star search)
<p>I am trying to make a tower defence game where some critters are trying to make their ways through a 2D grid-like obstacle course. The idea is these critters cannot walk into the obstacles or into each other during their path, and they would like to get t... | habedi/stack-exchange-dataset |
109,939 | Floating point arithmetic | <p>I need to change x1 = 0.3 and x2 = -0.29 to a FP(floating point) number with one sign bit, a 4 bit mantissa, a 3 bit exponent. The results I got are:</p>

<p>x1: 0 001 0011</p>

<p>x2: 1 001 0010</p>

<p>I am also trying to calculate the number of significant digits of x1 and x2 and I have fo... | floating point | 1 | Floating point arithmetic -- (floating point)
<p>I need to change x1 = 0.3 and x2 = -0.29 to a FP(floating point) number with one sign bit, a 4 bit mantissa, a 3 bit exponent. The results I got are:</p>

<p>x1: 0 001 0011</p>

<p>x2: 1 001 0010</p>

<p>I am also trying to calculate the number of... | habedi/stack-exchange-dataset |
109,944 | NL - iterating all edges of a graph in log space | <p>Given a turing machine which has logrtmic space, and consists of an input tape and a working tape, Is it possible to iterate all egdes of an input graph? I know the answer is probably NO, because clique and other similer NP problems belongs to NP and not to NL. Unfortunately, I can't find a satisfying and clear exp... | complexity theory graphs | 1 | NL - iterating all edges of a graph in log space -- (complexity theory graphs)
<p>Given a turing machine which has logrtmic space, and consists of an input tape and a working tape, Is it possible to iterate all egdes of an input graph? I know the answer is probably NO, because clique and other similer NP problems belo... | habedi/stack-exchange-dataset |
109,945 | How to find highest normal form of a relation, given its functional dependencies | <p>Let's say that I have a relational model, defined as:</p>

<p><span class="math-container">$$X(A,B,C,D,E,F,G)$$</span> with the functional dependencies: 
<span class="math-container">$$f_1: \{A,B,C\} \to \{D,E,F,G\}$$</span>
<span class="math-container">$$f_2: \{A,B\} \to \{C\}$$</span></p>

... | data structures database theory databases | 1 | How to find highest normal form of a relation, given its functional dependencies -- (data structures database theory databases)
<p>Let's say that I have a relational model, defined as:</p>

<p><span class="math-container">$$X(A,B,C,D,E,F,G)$$</span> with the functional dependencies: 
<span class="math-conta... | habedi/stack-exchange-dataset |
109,948 | Existence of path under weight and value budgets | <p>Consider the following problem:</p>
<blockquote>
<p>Input: An undirected graph <span class="math-container">$G = (V, E)$</span>, each edge has a non-negative weight <span class="math-container">$w_i$</span> and a non-negative value <span class="math-container">$v_i$</span>. There are two vertices to represen... | graphs np hard | 1 | Existence of path under weight and value budgets -- (graphs np hard)
<p>Consider the following problem:</p>
<blockquote>
<p>Input: An undirected graph <span class="math-container">$G = (V, E)$</span>, each edge has a non-negative weight <span class="math-container">$w_i$</span> and a non-negative value <span cl... | habedi/stack-exchange-dataset |
109,951 | Algorithm for fewest moves in 1's, 10's, 100's, etc | <p>I'm building a math application (not homework) and I want to build a component that allows players to drag-and-drop 1's, 10's, 100's, etc to complete problems and <em>a bonus reward</em> if they do it in the fewest moves possible--<strong>How can I calculate this?</strong> e.g.</p>

<pre><code>50 + X = 96&#x... | algorithms | 1 | Algorithm for fewest moves in 1's, 10's, 100's, etc -- (algorithms)
<p>I'm building a math application (not homework) and I want to build a component that allows players to drag-and-drop 1's, 10's, 100's, etc to complete problems and <em>a bonus reward</em> if they do it in the fewest moves possible--<strong>How can I ... | habedi/stack-exchange-dataset |
109,952 | NP-completeness of Induced disjoint paths | <p>In graph theory, it is well-known to be NP-complete the problem of given a set of <span class="math-container">$k$</span> pairs of source-sink, deciding whether there exists <span class="math-container">$k$</span> vertex-disjoint paths connecting these pairs.</p>

<p>Our problem is a variant in which the req... | graphs np complete np | 1 | NP-completeness of Induced disjoint paths -- (graphs np complete np)
<p>In graph theory, it is well-known to be NP-complete the problem of given a set of <span class="math-container">$k$</span> pairs of source-sink, deciding whether there exists <span class="math-container">$k$</span> vertex-disjoint paths connecting t... | habedi/stack-exchange-dataset |
109,960 | Number of nil-links in a binary tree | <p>In its section <a href="https://en.wikipedia.org/wiki/Binary_tree#Properties_of_binary_trees" rel="nofollow noreferrer">Properties of binary trees</a> Wikipedia states:</p>

<blockquote>
 <p>The maximum possible number of null links (i.e., absent children of the nodes) in a <strong>complete</strong> bin... | binary trees | 1 | Number of nil-links in a binary tree -- (binary trees)
<p>In its section <a href="https://en.wikipedia.org/wiki/Binary_tree#Properties_of_binary_trees" rel="nofollow noreferrer">Properties of binary trees</a> Wikipedia states:</p>

<blockquote>
 <p>The maximum possible number of null links (i.e., absent ch... | habedi/stack-exchange-dataset |
109,963 | Intuition behind eigenvalues of an adjacency matrix | <p>I am currently working to understand the use of the <a href="https://en.wikipedia.org/wiki/Cheeger_bound" rel="noreferrer">Cheeger bound</a> and of Cheeger's inequality, and their use for spectral partitioning, conductance, expansion, etc, but I still struggle to have a start of an intuition regarding the second eig... | graphs adjacency matrix | 1 | Intuition behind eigenvalues of an adjacency matrix -- (graphs adjacency matrix)
<p>I am currently working to understand the use of the <a href="https://en.wikipedia.org/wiki/Cheeger_bound" rel="noreferrer">Cheeger bound</a> and of Cheeger's inequality, and their use for spectral partitioning, conductance, expansion, e... | habedi/stack-exchange-dataset |
109,969 | What is a constructor in java? | <p>I have checked many websites but did not get the right answer. what is the meaning of constructor in java? </p>
 | java | 1 | What is a constructor in java? -- (java)
<p>I have checked many websites but did not get the right answer. what is the meaning of constructor in java? </p>
 | habedi/stack-exchange-dataset |
109,972 | Showing Cycle is NL-complete? | <p>Consider the following decision problem : </p>

<blockquote>
 <p>Cycle:
 Given a directed graph G, does G contains a directed cycle?</p>
</blockquote>

<p>It is very clear why Cycle belongs to NL. 
My question is - how to show Cycle is also NL-hard?
it seems almost obvious to s... | complexity theory graphs | 1 | Showing Cycle is NL-complete? -- (complexity theory graphs)
<p>Consider the following decision problem : </p>

<blockquote>
 <p>Cycle:
 Given a directed graph G, does G contains a directed cycle?</p>
</blockquote>

<p>It is very clear why Cycle belongs to NL. 
My question is - how to ... | habedi/stack-exchange-dataset |
109,974 | How to solve the optimization of bin packing using the decision version | <p>Let us say the optimization version of the bin packing problem asks you to give a packing using the fewest bins possible and the decision version asks if it is possible to pack the bins into <span class="math-container">$k$</span> bins.</p>

<blockquote>
 <p>How can you reduce the optimization version t... | optimization np complete reductions | 1 | How to solve the optimization of bin packing using the decision version -- (optimization np complete reductions)
<p>Let us say the optimization version of the bin packing problem asks you to give a packing using the fewest bins possible and the decision version asks if it is possible to pack the bins into <span class="... | habedi/stack-exchange-dataset |
109,975 | Find candidate keys given functional dependencies | <p>In my database class we practiced how to find the candidate keys, given the functional dependencies. However they were easy examples, based on a single attribute being functionally dependent on a single attribute but in our book it gives the following example of more complex functional dependencies. Let's say that I... | data structures database theory databases | 1 | Find candidate keys given functional dependencies -- (data structures database theory databases)
<p>In my database class we practiced how to find the candidate keys, given the functional dependencies. However they were easy examples, based on a single attribute being functionally dependent on a single attribute but in ... | habedi/stack-exchange-dataset |
109,981 | Confused between bandwidth and frequency | <p>I am new to Computer Science and started learning about computer networking very recently. I know that <em>bandwidth</em> means how many bits can be sent through the channel in 1 second. However, recently I came across another definition of bandwidth which says it is the range of frequency where the signal is receiv... | computer networks | 1 | Confused between bandwidth and frequency -- (computer networks)
<p>I am new to Computer Science and started learning about computer networking very recently. I know that <em>bandwidth</em> means how many bits can be sent through the channel in 1 second. However, recently I came across another definition of bandwidth wh... | habedi/stack-exchange-dataset |
109,991 | Log-space reduction - How to calculate overall space needed? | <p>Assume A, B are two decision problems. Given a reduction from A to B that uses log^c1(n) space and an algorithm that solves B in log^c2(n) space, Can we determine some lower bound to the space needed by an algorithm which solves A using this reduction?</p>
 | complexity theory | 1 | Log-space reduction - How to calculate overall space needed? -- (complexity theory)
<p>Assume A, B are two decision problems. Given a reduction from A to B that uses log^c1(n) space and an algorithm that solves B in log^c2(n) space, Can we determine some lower bound to the space needed by an algorithm which solves A us... | habedi/stack-exchange-dataset |
110,004 | List of major Open Problems in Computational Complexity and their Likelihood? | <p>I remember reading an article/paper (or perhaps a talk, most probably by Scott Arranson) where he lists the major open problems and their likelihood of being true or false in a table/graph. This is listed along with the 'surprise factor' of each result if its true/false.</p>

<p>I am unable to locate the art... | complexity theory reference request | 1 | List of major Open Problems in Computational Complexity and their Likelihood? -- (complexity theory reference request)
<p>I remember reading an article/paper (or perhaps a talk, most probably by Scott Arranson) where he lists the major open problems and their likelihood of being true or false in a table/graph. This is ... | habedi/stack-exchange-dataset |
110,005 | Cannibals missionaries problem - solving usings graphs | <p>I am trying to solve the cannibals - missionaries problem; we have the number of cannibals, the number of missionaries and the position of the boat. We are trying to transfer all of them to the other side, however there can't be more cannibals than missionaries on either side. The capacity of the boat is limited by ... | graphs graph traversal | 1 | Cannibals missionaries problem - solving usings graphs -- (graphs graph traversal)
<p>I am trying to solve the cannibals - missionaries problem; we have the number of cannibals, the number of missionaries and the position of the boat. We are trying to transfer all of them to the other side, however there can't be more ... | habedi/stack-exchange-dataset |
110,010 | Difference between greedy and work conserving scheduler for DAG | <p>For both schedulers I have found the definition, that no processor stays idle, if there is more work it can do.</p>

<p>However, I found two different upper bounds on the computation time of <span class="math-container">$T$</span>. For the following equations <span class="math-container">$T_{\infty}$</span> ... | graphs parallel computing greedy algorithms scheduling | 1 | Difference between greedy and work conserving scheduler for DAG -- (graphs parallel computing greedy algorithms scheduling)
<p>For both schedulers I have found the definition, that no processor stays idle, if there is more work it can do.</p>

<p>However, I found two different upper bounds on the computation ti... | habedi/stack-exchange-dataset |
110,031 | How does Rogozhin's (2, 18) universal turing machine work? | <p>I am trying to understand <a href="https://www.sciencedirect.com/science/article/pii/S0304397596000771" rel="nofollow noreferrer">Rogozhin's (2, 18) universal turing machine</a> by stepping
through a simple <a href="https://en.wikipedia.org/wiki/Tag_system" rel="nofollow noreferrer">2-tag</a> encoding that I bel... | turing machines | 1 | How does Rogozhin's (2, 18) universal turing machine work? -- (turing machines)
<p>I am trying to understand <a href="https://www.sciencedirect.com/science/article/pii/S0304397596000771" rel="nofollow noreferrer">Rogozhin's (2, 18) universal turing machine</a> by stepping
through a simple <a href="https://en.wikipe... | habedi/stack-exchange-dataset |
110,042 | Pebble game lower bound? | <p>This paper says pebble games have super linear lower bound for every fixed <span class="math-container">$k$</span> <a href="https://dl.acm.org/citation.cfm?doid=62.322433" rel="nofollow noreferrer">https://dl.acm.org/citation.cfm?doid=62.322433</a>.</p>

<p>Why is it not considered proof of constructive exam... | complexity theory turing machines time complexity np lower bounds | 1 | Pebble game lower bound? -- (complexity theory turing machines time complexity np lower bounds)
<p>This paper says pebble games have super linear lower bound for every fixed <span class="math-container">$k$</span> <a href="https://dl.acm.org/citation.cfm?doid=62.322433" rel="nofollow noreferrer">https://dl.acm.org/cita... | habedi/stack-exchange-dataset |
110,044 | Prove that the upper bound in the Noiseless-coding theorem is strict | <p>Given a probability distribution <span class="math-container">$p$</span> across an alphabet, we define redundancy as:</p>

<p>Expected Length of codewords - entropy of p = <span class="math-container">$\ E(S) - h(p)$</span></p>

<p>Prove that for each <span class="math-container">$\epsilon$</span> wi... | information theory coding theory encoding scheme | 1 | Prove that the upper bound in the Noiseless-coding theorem is strict -- (information theory coding theory encoding scheme)
<p>Given a probability distribution <span class="math-container">$p$</span> across an alphabet, we define redundancy as:</p>

<p>Expected Length of codewords - entropy of p = <span class="m... | habedi/stack-exchange-dataset |
110,066 | Suffix array and counting distinct substrings of specific length | <p>I am trying to use the suffix array, and the LCP array to count all distinct substrings of a specified length. </p>

<p>I started with the algorithm for counting ALL distinct substrings. I solved it after this explanation: <a href="https://www.geeksforgeeks.org/count-distinct-substrings-string-using-suffix-... | algorithms strings substrings | 1 | Suffix array and counting distinct substrings of specific length -- (algorithms strings substrings)
<p>I am trying to use the suffix array, and the LCP array to count all distinct substrings of a specified length. </p>

<p>I started with the algorithm for counting ALL distinct substrings. I solved it after thi... | habedi/stack-exchange-dataset |
110,071 | If a problem C is NP hard and there is an existing reduction from/to A,B,D, are they NP hard as well? | <p>Lets say there is an reduction in polynomial time from problem A to B, from problem B to C and from problem C to D. Now lets say C is NP hard. Does this mean A,B,D are NP hard as well?</p>
 | reductions np hard | 1 | If a problem C is NP hard and there is an existing reduction from/to A,B,D, are they NP hard as well? -- (reductions np hard)
<p>Lets say there is an reduction in polynomial time from problem A to B, from problem B to C and from problem C to D. Now lets say C is NP hard. Does this mean A,B,D are NP hard as well?</p>&#x... | habedi/stack-exchange-dataset |
110,076 | Sampling in large graph using simple random walk | <p>I'm studying sampling techniques in online social networks. The assumption is we don't have full access to the network(i.e, <strong>we don’t know the size of the network</strong>). However crawling is supported, i.e, starting with any arbitrary node we can access its neighbors. Hence we go for random walk(Markov Ch... | sampling social networks random walks | 1 | Sampling in large graph using simple random walk -- (sampling social networks random walks)
<p>I'm studying sampling techniques in online social networks. The assumption is we don't have full access to the network(i.e, <strong>we don’t know the size of the network</strong>). However crawling is supported, i.e, startin... | habedi/stack-exchange-dataset |
110,078 | Find typing derivation of STLC term with reference types | <p>The problem is to find the typing derivation of a term of the call-by-value STLC extended with reference types. The evaluation and typing rules for this language is given in Types and Programming Languages by Pierce on page 166. The term is
<span class="math-container">$$
t = (\lambda a:\mathrm{Ref}\mathrm{B... | lambda calculus type theory typing | 1 | Find typing derivation of STLC term with reference types -- (lambda calculus type theory typing)
<p>The problem is to find the typing derivation of a term of the call-by-value STLC extended with reference types. The evaluation and typing rules for this language is given in Types and Programming Languages by Pierce on p... | habedi/stack-exchange-dataset |
110,102 | Is λx. a valid Lamda Calculus abstraction? | <p>For demonstration purposes I was wondering about some very easy to grasp LC abstractions and I came to the idea of a function that simply eats its argument, and nothing more. </p>

<p>If you apply λx. (Yes no lambda term after the point) to an argument the abstraction reduces to nothing.</p>

<p>Alth... | lambda calculus | 1 | Is λx. a valid Lamda Calculus abstraction? -- (lambda calculus)
<p>For demonstration purposes I was wondering about some very easy to grasp LC abstractions and I came to the idea of a function that simply eats its argument, and nothing more. </p>

<p>If you apply λx. (Yes no lambda term after the point) to an a... | habedi/stack-exchange-dataset |
110,107 | Number of possible heaps on $\{1,...,2^h-1\}$ | <blockquote>
<p>Let <span class="math-container">$C_h$</span> be the number of possible heaps for the set of keys <span class="math-container">$\{1,...,2^h-1\}$</span>. Determine a recurrence relation for <span class="math-container">$C_h$</span> via the substitution method and prove it.</p>
<p><strong>Definiti... | combinatorics recurrence relation heaps enumeration check my answer | 1 | Number of possible heaps on $\{1,...,2^h-1\}$ -- (combinatorics recurrence relation heaps enumeration check my answer)
<blockquote>
<p>Let <span class="math-container">$C_h$</span> be the number of possible heaps for the set of keys <span class="math-container">$\{1,...,2^h-1\}$</span>. Determine a recurrence relat... | habedi/stack-exchange-dataset |
110,108 | Average codeword length in a Huffman tree is $\Omega(\log n)$ | <blockquote>
 <p>Prove that the average codeword length in a Huffman tree is <span class="math-container">$\Omega(\log n)$</span>, where <span class="math-container">$n$</span> is the number of characters.</p>
</blockquote>

<p>My try:</p>

<p>I think that the worst case is when the tree is ful... | algorithms binary trees huffman coding | 1 | Average codeword length in a Huffman tree is $\Omega(\log n)$ -- (algorithms binary trees huffman coding)
<blockquote>
 <p>Prove that the average codeword length in a Huffman tree is <span class="math-container">$\Omega(\log n)$</span>, where <span class="math-container">$n$</span> is the number of characters.</p>... | habedi/stack-exchange-dataset |
110,116 | Finding row wise sum of transpose of hv-convex binary matrix | <p>I'm stuck on a problem involving the Gale-Ryser Theorem. The problem's input gives me the row-wise sum of an hv-convex binary matrix(n*m).</p>

<pre><code>e.g. I get {4,3,2,2,1} in the input. It's the row wise sum of the following matrix:

 1 1 1 1
 1 1 1 0
 1 1 0 0
 1 1 0 0&#... | graphs combinatorics | 1 | Finding row wise sum of transpose of hv-convex binary matrix -- (graphs combinatorics)
<p>I'm stuck on a problem involving the Gale-Ryser Theorem. The problem's input gives me the row-wise sum of an hv-convex binary matrix(n*m).</p>

<pre><code>e.g. I get {4,3,2,2,1} in the input. It's the row wise sum of the f... | habedi/stack-exchange-dataset |
110,124 | How to reduce $\{w \mid |T(M_w)| \geq 42\}$ to the halting problem? | <p>For a string <span class="math-container">$w$</span>, <span class="math-container">$M_w$</span> denotes the Turing machine whose encoding is <span class="math-container">$w$</span>. </p>

<p>I want to reduce the language <span class="math-container">$L=\{w \mid |T(M_w)| \geq 42\}$</span> to <span class="math... | formal languages turing machines reductions proof techniques | 1 | How to reduce $\{w \mid |T(M_w)| \geq 42\}$ to the halting problem? -- (formal languages turing machines reductions proof techniques)
<p>For a string <span class="math-container">$w$</span>, <span class="math-container">$M_w$</span> denotes the Turing machine whose encoding is <span class="math-container">$w$</span>. <... | habedi/stack-exchange-dataset |
110,127 | Minimum cost to match $n$ people with $m$ shops | <p>We are given coordinates of <span class="math-container">$n$</span> people and <span class="math-container">$m$</span> shops. We should find a matching such that each person is matched with exactly one shop, and one shop is matched with at most one person.The total cost of one matching is sum of all matchings, and f... | bipartite matching | 1 | Minimum cost to match $n$ people with $m$ shops -- (bipartite matching)
<p>We are given coordinates of <span class="math-container">$n$</span> people and <span class="math-container">$m$</span> shops. We should find a matching such that each person is matched with exactly one shop, and one shop is matched with at most ... | habedi/stack-exchange-dataset |
110,128 | Maximum weight vertex-disjoint paths | <p>I have a complete (every vertex is connected by an edge to every other vertex) undirected positively weighted graph. I want to find vertex-disjoint paths in the graph whose total weight is as large as possible. </p>

<p>To me this sound like a mix of the longest path problem (NP-complete) and minimum spanni... | algorithms graphs weighted graphs | 1 | Maximum weight vertex-disjoint paths -- (algorithms graphs weighted graphs)
<p>I have a complete (every vertex is connected by an edge to every other vertex) undirected positively weighted graph. I want to find vertex-disjoint paths in the graph whose total weight is as large as possible. </p>

<p>To me this s... | habedi/stack-exchange-dataset |
110,138 | Conditions for maximum period of quadratic congruential method (PRNG) | <p><span class="math-container">$X_{n} = (d^2X_{n-1} + aX_{n-1} + c) \operatorname{mod} m$</span></p>

<p>Knuth lists out the necessity and sufficiency of 4 conditions (Exercise 8 in page 49 of "The art of computer programming Vol.II"):</p>

<blockquote>
 <ol>
 <li><span class="math-container"... | algorithms algorithm analysis modular arithmetic | 1 | Conditions for maximum period of quadratic congruential method (PRNG) -- (algorithms algorithm analysis modular arithmetic)
<p><span class="math-container">$X_{n} = (d^2X_{n-1} + aX_{n-1} + c) \operatorname{mod} m$</span></p>

<p>Knuth lists out the necessity and sufficiency of 4 conditions (Exercise 8 in page ... | habedi/stack-exchange-dataset |
110,140 | What are the sufficient conditions for weighted binary code to be self compliementary? | <p>I was going through binary codes and found out that if the weighted binary code is self complimentary then it's weights addition will be equal to 9. Is the converse true? If not then what are the sufficient conditions for making a weighted binary code self complimentary. My research was in vain. Please help and clea... | error correcting codes binary | 1 | What are the sufficient conditions for weighted binary code to be self compliementary? -- (error correcting codes binary)
<p>I was going through binary codes and found out that if the weighted binary code is self complimentary then it's weights addition will be equal to 9. Is the converse true? If not then what are the... | habedi/stack-exchange-dataset |
110,144 | CFG for all strings of a’s and b’s that contain a different number of a’s and b’s | <p>I am trying to write CFG for all strings on {a,b} that contains different numbers of a’s and b’s? After two hours of brainstorming, I came up with this: </p>

<p>S→A|B<br>
A→aE|aA|EA<br>
B→bE|bB|EB<br>
E→aEbE|bEaE|Λ </p>

<p>I tried running few strings on this, it worked, But then I run ... | automata context free strings | 1 | CFG for all strings of a’s and b’s that contain a different number of a’s and b’s -- (automata context free strings)
<p>I am trying to write CFG for all strings on {a,b} that contains different numbers of a’s and b’s? After two hours of brainstorming, I came up with this: </p>

<p>S→A|B<br>
A→aE|aA|EA<br>&... | habedi/stack-exchange-dataset |
110,145 | $PSPACE$ not equal $DSPACE(2^n) $ | <p>It seems pretty obvious that <span class="math-container">$PSPACE$</span> is not equal to <span class="math-container">$DSPACE(2^n) $</span>. Can this be shown using the space hierarchy theorem? Is that the most simple and straight-forward way? </p>
 | complexity theory | 1 | $PSPACE$ not equal $DSPACE(2^n) $ -- (complexity theory)
<p>It seems pretty obvious that <span class="math-container">$PSPACE$</span> is not equal to <span class="math-container">$DSPACE(2^n) $</span>. Can this be shown using the space hierarchy theorem? Is that the most simple and straight-forward way? </p>
 | habedi/stack-exchange-dataset |
110,151 | Description for languages that can be solved in time(n)? | <p>How can one describe all languages that are in
<span class="math-container">$\mathrm{TIME}(n)$</span>? </p>

<p>It can't be all the regular languages only, as for example <span class="math-container">$L = \{a^n b^nw \mid w \in \Sigma^* \land n \geq 1\}$</span> is not regular but in <span class="math-cont... | formal languages time complexity | 1 | Description for languages that can be solved in time(n)? -- (formal languages time complexity)
<p>How can one describe all languages that are in
<span class="math-container">$\mathrm{TIME}(n)$</span>? </p>

<p>It can't be all the regular languages only, as for example <span class="math-container">$L = \{a^n... | habedi/stack-exchange-dataset |
110,156 | How to reduce EQU to UNI? | <p>Let <span class="math-container">$$\texttt{EQU}=\{u\#v \mid T(M_u)=T(M_v)\} \\
\texttt{UNI}=\{w \mid T(M_w)= \Sigma^*\}$$</span> </p>

<p>How can you prove <span class="math-container">$\texttt{EQU} \leq \texttt{UNI}$</span>? </p>

<p>The idea I have so far is, to simulate the TM that decides whe... | formal languages turing machines reductions | 1 | How to reduce EQU to UNI? -- (formal languages turing machines reductions)
<p>Let <span class="math-container">$$\texttt{EQU}=\{u\#v \mid T(M_u)=T(M_v)\} \\
\texttt{UNI}=\{w \mid T(M_w)= \Sigma^*\}$$</span> </p>

<p>How can you prove <span class="math-container">$\texttt{EQU} \leq \texttt{UNI}$</span>? </p>... | habedi/stack-exchange-dataset |
110,172 | Context free grammar for a palindrome of even length + cc | <p>Suppose the language is <span class="math-container">$\{AA^R: A \in \{a,b\}^*\}$</span>.</p>

<p>I know that I can make a context free grammar for it:</p>

<p><span class="math-container">$$S\to aSa \mid bSb\mid\epsilon$$</span></p>

<p>Now if the language was <span class="math-container">$\{... | context free | 1 | Context free grammar for a palindrome of even length + cc -- (context free)
<p>Suppose the language is <span class="math-container">$\{AA^R: A \in \{a,b\}^*\}$</span>.</p>

<p>I know that I can make a context free grammar for it:</p>

<p><span class="math-container">$$S\to aSa \mid bSb\mid\epsilon$$</sp... | habedi/stack-exchange-dataset |
110,177 | How to construct a random error-correcting code (its generator matrix) according to the code parameters? | <p>I need to construct a random code which corrects T errors, has R check bits and has N maximum bits in the transmitting word.</p>

<p>I have researched the topic and found a few theorems about the bounds (The Varshamov-Gilbert bound, for instance). Neither of them suggests an algorithm of how to construct the... | coding theory error correcting codes | 1 | How to construct a random error-correcting code (its generator matrix) according to the code parameters? -- (coding theory error correcting codes)
<p>I need to construct a random code which corrects T errors, has R check bits and has N maximum bits in the transmitting word.</p>

<p>I have researched the topic a... | habedi/stack-exchange-dataset |
110,184 | Given $L$ and $D$ find $X, \text { such that } X * 10^L + D \equiv 0 \mod M$ | <p>Given <span class="math-container">$L$</span> and <span class="math-container">$D$</span> find <span class="math-container">$X, \text { such that } X * 10^L + D \equiv 0 \mod M$</span>. Integer <span class="math-container">$M$</span> is given and it is the same for all calculations however we need to solve for <span... | modular arithmetic | 1 | Given $L$ and $D$ find $X, \text { such that } X * 10^L + D \equiv 0 \mod M$ -- (modular arithmetic)
<p>Given <span class="math-container">$L$</span> and <span class="math-container">$D$</span> find <span class="math-container">$X, \text { such that } X * 10^L + D \equiv 0 \mod M$</span>. Integer <span class="math-cont... | habedi/stack-exchange-dataset |
110,191 | What are examples of applications of the tree decomposition of a graph? | <p>I am looking for <em>specific</em> applications of the <a href="https://en.wikipedia.org/wiki/Tree_decomposition" rel="nofollow noreferrer">tree decomposition</a> (of a graph), because I want to motivate its existence. What are examples of problems that are more easily solvable using the junction tree of a graph (ra... | graphs trees applications | 1 | What are examples of applications of the tree decomposition of a graph? -- (graphs trees applications)
<p>I am looking for <em>specific</em> applications of the <a href="https://en.wikipedia.org/wiki/Tree_decomposition" rel="nofollow noreferrer">tree decomposition</a> (of a graph), because I want to motivate its existe... | habedi/stack-exchange-dataset |
110,193 | Construction of a Deterministic Tree Automaton (DTA) | <p>Let <span class="math-container">$L \subseteq \Sigma^*$</span> be a regular language. Let <span class="math-container">$\Sigma' = \Sigma_0 \cup \Sigma_2$</span> where <span class="math-container">$\Sigma_0 =\Sigma$</span> and <span class="math-container">$\Sigma_2=\{*\}$</span>.</p>

<p>We define <span class... | regular languages finite automata trees tree automata | 1 | Construction of a Deterministic Tree Automaton (DTA) -- (regular languages finite automata trees tree automata)
<p>Let <span class="math-container">$L \subseteq \Sigma^*$</span> be a regular language. Let <span class="math-container">$\Sigma' = \Sigma_0 \cup \Sigma_2$</span> where <span class="math-container">$\Sigma_0... | habedi/stack-exchange-dataset |
110,202 | How to come up with a language that is recognizable but not co-recognizable? | <p>Forming a language that is recognizable but not co-recognizable. I'm having trouble coming up with a language with these properties. A recognizable language is a language <span class="math-container">$A \subseteq \Sigma^*$</span> iff <span class="math-container">$A = L(M)$</span> for some Turing machine <span class=... | turing machines church turing thesis | 1 | How to come up with a language that is recognizable but not co-recognizable? -- (turing machines church turing thesis)
<p>Forming a language that is recognizable but not co-recognizable. I'm having trouble coming up with a language with these properties. A recognizable language is a language <span class="math-container... | habedi/stack-exchange-dataset |
110,203 | Intersection of a recognizable language and a decidable language is decidable? | <p>I'm having trouble with proving that "Intersection of a recognizable language anda decidable language is decidable. </p>

<p>I assume this is true although I have no idea how to proof it. Can somebody point me in the right direction?</p>
 | computability | 1 | Intersection of a recognizable language and a decidable language is decidable? -- (computability)
<p>I'm having trouble with proving that "Intersection of a recognizable language anda decidable language is decidable. </p>

<p>I assume this is true although I have no idea how to proof it. Can somebody point me ... | habedi/stack-exchange-dataset |
110,209 | Graph ordering with smallest max vertex "discrepancy" | <p>Consider an undirected graph <span class="math-container">$G=(V,E)$</span> and a bijective function <span class="math-container">$f:V \rightarrow [|V|]$</span> which orders the vertices by mapping them onto the first <span class="math-container">$|V|$</span> natural numbers. </p>

<p>Define the cost of an or... | graphs time complexity | 1 | Graph ordering with smallest max vertex "discrepancy" -- (graphs time complexity)
<p>Consider an undirected graph <span class="math-container">$G=(V,E)$</span> and a bijective function <span class="math-container">$f:V \rightarrow [|V|]$</span> which orders the vertices by mapping them onto the first <span class="math-... | habedi/stack-exchange-dataset |
110,217 | Regularity of infinite concatenation | <p>It is well-known that an infinite union of regular languages is not necessarily regular, since <em>every</em> language can be written as a union of singletons. What about infinite concatenations?</p>

<p>Let <span class="math-container">$\{ L_z : z \in \mathbb{Z} \}$</span> be a family of languages, cofinite... | formal languages regular languages | 1 | Regularity of infinite concatenation -- (formal languages regular languages)
<p>It is well-known that an infinite union of regular languages is not necessarily regular, since <em>every</em> language can be written as a union of singletons. What about infinite concatenations?</p>

<p>Let <span class="math-contai... | habedi/stack-exchange-dataset |
110,219 | combinatoric grouping optimization problem based on time interval overlap, weight constraint, and distance minimization | <p>Let each element be an individual. Consider that an individual is defined such that each individual has a time range, weight, and location.</p>

<p>The goal is to group together individuals whose time ranges overlap while ensuring that, within the group, the sum of the weights of the individuals do not excee... | algorithms optimization combinatorics | 1 | combinatoric grouping optimization problem based on time interval overlap, weight constraint, and distance minimization -- (algorithms optimization combinatorics)
<p>Let each element be an individual. Consider that an individual is defined such that each individual has a time range, weight, and location.</p>

<... | habedi/stack-exchange-dataset |
110,220 | Lower Bound for Time Complexity of Pairing Problem | <p>Given an array X and array Y both of length n, the pairing algorithm will return the elements of the arrays matched so that the smallest element in X will be matched with the smallest element of Y, the second smallest in X matched with second smallest in Y and so on. i.e the algorithm will yield the pairs: <span cla... | time complexity sorting lower bounds selection problem | 1 | Lower Bound for Time Complexity of Pairing Problem -- (time complexity sorting lower bounds selection problem)
<p>Given an array X and array Y both of length n, the pairing algorithm will return the elements of the arrays matched so that the smallest element in X will be matched with the smallest element of Y, the seco... | habedi/stack-exchange-dataset |
110,229 | Computing the closest point inside an intersection of half-spaces | <p>Is there an efficient way to compute, given a point, the closest point to it that's on the inside of an intersection of half-spaces? </p>

<p>For example, given the half-spaces <code>a</code> & <code>b</code>, the closest point to <code>p</code> that's "inside" the half-spaces is <code>x</code>. <code>+<... | algorithms | 1 | Computing the closest point inside an intersection of half-spaces -- (algorithms)
<p>Is there an efficient way to compute, given a point, the closest point to it that's on the inside of an intersection of half-spaces? </p>

<p>For example, given the half-spaces <code>a</code> & <code>b</code>, the closest p... | habedi/stack-exchange-dataset |
110,234 | solving the recurrence t(n)=t(n-2)+d*(n^2)/2 with iteration method | <p>How can I solve <span class="math-container">$$T(n)=T(n-2)+\frac {d}{2}n^2$$</span>
I couldnt find <span class="math-container">$d$</span> (dont know if I have to) and after 3 iterations I got to <span class="math-container">$k= \frac{n-1}{2}$</span>
but had trouble to continue.</p>
 | recurrence relation | 1 | solving the recurrence t(n)=t(n-2)+d*(n^2)/2 with iteration method -- (recurrence relation)
<p>How can I solve <span class="math-container">$$T(n)=T(n-2)+\frac {d}{2}n^2$$</span>
I couldnt find <span class="math-container">$d$</span> (dont know if I have to) and after 3 iterations I got to <span class="math-contain... | habedi/stack-exchange-dataset |
110,238 | Why is the usual simulated annealing algorithm using a Boltzmann probability? | <p>I've coded a personal version for the simulated annealing problem, and I was wondering why, (other than by analogy to the physics), the probability is
<span class="math-container">$$ p = e^{-\dfrac{\Delta \text{length}}{T}}$$</span> (assuming we're applying to the TSP problem, so the parameter is here the length... | algorithms optimization heuristics simulated annealing | 1 | Why is the usual simulated annealing algorithm using a Boltzmann probability? -- (algorithms optimization heuristics simulated annealing)
<p>I've coded a personal version for the simulated annealing problem, and I was wondering why, (other than by analogy to the physics), the probability is
<span class="math-contai... | habedi/stack-exchange-dataset |
110,240 | What are elementary operations in time complexity definition? | <p>Wikipedia gives us the following defintion of time complexity: </p>

<blockquote>
 <p>"In computer science, the time complexity is the computational
 complexity that describes the amount of time it takes to run an
 algorithm. Time complexity is commonly estimated by counting the
 number ... | time complexity | 1 | What are elementary operations in time complexity definition? -- (time complexity)
<p>Wikipedia gives us the following defintion of time complexity: </p>

<blockquote>
 <p>"In computer science, the time complexity is the computational
 complexity that describes the amount of time it takes to run an&#... | habedi/stack-exchange-dataset |
110,247 | Prove propositional formula is a tautology | <p>I need to show this formula is a theorem of propositional calculus. I tried assuming antecedent and proving consequent but didn't work for this proof. Do I need to show it is equivalent to true? How should I start? It's not possible to use truth tables.</p>

<p><span class="math-container">$(p \rightarrow q)... | logic | 1 | Prove propositional formula is a tautology -- (logic)
<p>I need to show this formula is a theorem of propositional calculus. I tried assuming antecedent and proving consequent but didn't work for this proof. Do I need to show it is equivalent to true? How should I start? It's not possible to use truth tables.</p>
&... | habedi/stack-exchange-dataset |
110,267 | Examples for Partial Combinatory Algebras | <p>I am currently working on my Bachelor thesis about Turing Categories (see <em>Introduction to Turing Categories</em> [<a href="https://www.sciencedirect.com/science/article/pii/S0168007208000948?via%3Dihub" rel="nofollow noreferrer">1</a>]). In this context I got some questions regarding Partial Combinatory Algebras... | complexity theory computability reference request logic | 1 | Examples for Partial Combinatory Algebras -- (complexity theory computability reference request logic)
<p>I am currently working on my Bachelor thesis about Turing Categories (see <em>Introduction to Turing Categories</em> [<a href="https://www.sciencedirect.com/science/article/pii/S0168007208000948?via%3Dihub" rel="no... | habedi/stack-exchange-dataset |
110,275 | Is Unification "an Implementation of Existential Quantification"? | <p>I read a comment (I've forgotten the source), "Unification is an <em>implementation</em> of existential quantification." (Emphasis mine.)</p>

<p>If true, this point of view clears up many things.</p>

<p>For instance why omitting the "occurs check" in unification can lead to unsound inference: it w... | unification logic programming | 1 | Is Unification "an Implementation of Existential Quantification"? -- (unification logic programming)
<p>I read a comment (I've forgotten the source), "Unification is an <em>implementation</em> of existential quantification." (Emphasis mine.)</p>

<p>If true, this point of view clears up many things.</p>
&#x... | habedi/stack-exchange-dataset |
110,287 | In O(1) get the coordinate i,j of a diagonally ordered matrix | <p>Say you have a matrix like this:</p>

<pre><code>[][]int{
 {0, 2, 5, 9, 14},
 {1, 4, 8, 13, 18},
 {3, 7, 12, 17, 21},
 {6, 11, 16, 20, 23},
 {10, 15, 19, 22, 24},
}
</code></pre>

<p>As you can see, it is diagonally ordered</p>

<p>Question:
In O... | matrix | 1 | In O(1) get the coordinate i,j of a diagonally ordered matrix -- (matrix)
<p>Say you have a matrix like this:</p>

<pre><code>[][]int{
 {0, 2, 5, 9, 14},
 {1, 4, 8, 13, 18},
 {3, 7, 12, 17, 21},
 {6, 11, 16, 20, 23},
 {10, 15, 19, 22, 24},
}
</code></pre>

<p>A... | habedi/stack-exchange-dataset |
110,293 | Are Context Sensitive Languages Turing Complete? | <p>Related questions:</p>

<ol>
<li><a href="https://cs.stackexchange.com/questions/33666/can-regular-languages-be-turing-complete">Can regular languages be Turing complete?</a></li>
<li><a href="https://cs.stackexchange.com/questions/74095/why-are-linearly-bounded-turing-machines-more-powerful-than-fin... | formal languages turing machines turing completeness context sensitive | 1 | Are Context Sensitive Languages Turing Complete? -- (formal languages turing machines turing completeness context sensitive)
<p>Related questions:</p>

<ol>
<li><a href="https://cs.stackexchange.com/questions/33666/can-regular-languages-be-turing-complete">Can regular languages be Turing complete?</a></li>&... | habedi/stack-exchange-dataset |
110,296 | Changing a complete undecidable turing machine language | <p><span class="math-container">$L = \{\langle M\rangle\mid \text{M is a TM and }L(M) = \{101\}\}$</span> meaning M accepts only the string <span class="math-container">$101$</span>. Which is neither co-recognizable / recognizable. Can be proven easily by <span class="math-container">$HALT \leq_m L (\text{ or } \bar{L}... | turing machines | 1 | Changing a complete undecidable turing machine language -- (turing machines)
<p><span class="math-container">$L = \{\langle M\rangle\mid \text{M is a TM and }L(M) = \{101\}\}$</span> meaning M accepts only the string <span class="math-container">$101$</span>. Which is neither co-recognizable / recognizable. Can be prov... | habedi/stack-exchange-dataset |
110,301 | Best known (state of science) time complexity of an array access problem | <p>Consider an Array <span class="math-container">$A$</span> with <span class="math-container">$n$</span> values and the following operations:</p>

<ol>
<li>get(i): Returns the value of <span class="math-container">$A[i]$</span></li>
<li>insert (x): Insert the element x into the any free place in A (not... | data structures amortized analysis | 1 | Best known (state of science) time complexity of an array access problem -- (data structures amortized analysis)
<p>Consider an Array <span class="math-container">$A$</span> with <span class="math-container">$n$</span> values and the following operations:</p>

<ol>
<li>get(i): Returns the value of <span cla... | habedi/stack-exchange-dataset |
110,305 | Definition(s)/characterization(s) of intractability | <p>I've been searching around for definitions of <strong>intractability</strong>, and I realized that despite being widely used, the concept of intractability (when talking about computing problems of course) is <strong>not formally defined</strong>. </p>

<p>I'm currently writing a report in which I character... | complexity theory time complexity | 1 | Definition(s)/characterization(s) of intractability -- (complexity theory time complexity)
<p>I've been searching around for definitions of <strong>intractability</strong>, and I realized that despite being widely used, the concept of intractability (when talking about computing problems of course) is <strong>not forma... | habedi/stack-exchange-dataset |
110,321 | Geometric median of two disjoint sets of points lies on line between their respective medians | <p>I was working on a problem about geometric medians and I had an idea for a divide and conquer solution, but it would only work if a set of points, when split into two disjoint sets, and those sets respective geometric medians calculated and a line passed between, it holds that the geometric median of the overall set... | algorithms computational geometry linear algebra divide and conquer | 1 | Geometric median of two disjoint sets of points lies on line between their respective medians -- (algorithms computational geometry linear algebra divide and conquer)
<p>I was working on a problem about geometric medians and I had an idea for a divide and conquer solution, but it would only work if a set of points, whe... | habedi/stack-exchange-dataset |
110,333 | $stCON$ with path of length $≥ n/2$ | <p>The following problem seems very similar to the <span class="math-container">$stCON$</span> decision problem: </p>

<p>{<span class="math-container">$G, s,t | G = (V, E)$</span> such as <span class="math-container">$V$</span> is a graph, <span class="math-container">$s,t ∈ V$</span>,
there exists in <s... | complexity theory graphs | 1 | $stCON$ with path of length $≥ n/2$ -- (complexity theory graphs)
<p>The following problem seems very similar to the <span class="math-container">$stCON$</span> decision problem: </p>

<p>{<span class="math-container">$G, s,t | G = (V, E)$</span> such as <span class="math-container">$V$</span> is a graph, <sp... | habedi/stack-exchange-dataset |
110,334 | is $x^{100000000000}$ a "polynomial time"? | <p>per this <a href="https://www.mathsisfun.com/sets/np-complete.html" rel="nofollow noreferrer">post</a> </p>

<blockquote>
 <p><span class="math-container">$t = x^2$</span> means the problem is solvable in "Polynomial" time.</p>
</blockquote>

<p>per this <a href="https://math.stackexchange.c... | polynomial time | 1 | is $x^{100000000000}$ a "polynomial time"? -- (polynomial time)
<p>per this <a href="https://www.mathsisfun.com/sets/np-complete.html" rel="nofollow noreferrer">post</a> </p>

<blockquote>
 <p><span class="math-container">$t = x^2$</span> means the problem is solvable in "Polynomial" time.</p>
</blockq... | habedi/stack-exchange-dataset |
110,335 | Current value of flow in a network | <p>Confused about a question regarding flow networks.</p>

<p><strong>Question is:</strong> <em>Given the network below, what is the current value of flow in this network?</em>
<a href="https://i.stack.imgur.com/9adxi.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9adxi.jpg" alt="enter i... | graphs network flow | 1 | Current value of flow in a network -- (graphs network flow)
<p>Confused about a question regarding flow networks.</p>

<p><strong>Question is:</strong> <em>Given the network below, what is the current value of flow in this network?</em>
<a href="https://i.stack.imgur.com/9adxi.jpg" rel="nofollow noreferrer"... | habedi/stack-exchange-dataset |
110,339 | Finding the maximum disjoint weight in a weighted node graph | <p>I have a graph of nodes that reflect resource allocation. Each node has a weight to reflect this. A well formed graph is disjoint, so there will be no edges, and the weight of the graph is just the sum of the nodes. Any graph that contains edges needs to be reduced by removing one of the two nodes on an edge, until ... | algorithms graphs algorithm analysis weighted graphs | 1 | Finding the maximum disjoint weight in a weighted node graph -- (algorithms graphs algorithm analysis weighted graphs)
<p>I have a graph of nodes that reflect resource allocation. Each node has a weight to reflect this. A well formed graph is disjoint, so there will be no edges, and the weight of the graph is just the ... | habedi/stack-exchange-dataset |
110,344 | Parallelization of priority queue-based algorithms | <p>There's a number of algorithms that operate by maintaining and consuming a priority queue of "events". I'm thinking primarily of geometric algorithms, particularly sweep-line algorithms like Bentley-Ottmann and Fortune's algorithm, but also things like Dijkstra's algorithm and Prim's algorithm. </p>

<p>Ther... | reference request computational geometry parallel computing priority queues | 1 | Parallelization of priority queue-based algorithms -- (reference request computational geometry parallel computing priority queues)
<p>There's a number of algorithms that operate by maintaining and consuming a priority queue of "events". I'm thinking primarily of geometric algorithms, particularly sweep-line algorithms... | habedi/stack-exchange-dataset |
110,345 | Why isn't the Generalized Super Mario Bros. obviously in NP? | <p>It is shown in <a href="https://arxiv.org/pdf/1203.1895.pdf" rel="nofollow noreferrer">the paper: "Classic Nintendo Games are (Computationally) Hard"</a> by Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta that the Generalized Super Mario Bros. (SMB, for short) video game is NP-hard. </p>

<bl... | complexity theory np complete np computer games | 1 | Why isn't the Generalized Super Mario Bros. obviously in NP? -- (complexity theory np complete np computer games)
<p>It is shown in <a href="https://arxiv.org/pdf/1203.1895.pdf" rel="nofollow noreferrer">the paper: "Classic Nintendo Games are (Computationally) Hard"</a> by Greg Aloupis, Erik D. Demaine, Alan Guo, and G... | habedi/stack-exchange-dataset |
110,356 | How to verify Hamilton-Path in log-space? | <p>Given an undirected graph <span class="math-container">$G$</span> and an undirected path <span class="math-container">$p$</span>, Is it possible to verify <span class="math-container">$p$</span> is a Hamilton path in graph <span class="math-container">$G$</span> using logarithmic space? How is it possible to verify ... | complexity theory graphs | 1 | How to verify Hamilton-Path in log-space? -- (complexity theory graphs)
<p>Given an undirected graph <span class="math-container">$G$</span> and an undirected path <span class="math-container">$p$</span>, Is it possible to verify <span class="math-container">$p$</span> is a Hamilton path in graph <span class="math-cont... | habedi/stack-exchange-dataset |
110,363 | Find axis-aligned rectangle with maximum area in a point set in the Euclidean plane | <p>On the plane <span class="math-container">$n$</span> points <span class="math-container">$(x_i, y_i)$</span> are marked. Select 4 points so that they define a rectangle with the greatest area and sides parallel to the axes.</p>

<p>Time limit for python is 10 seconds, for other programming languages - 2 seco... | algorithms | 1 | Find axis-aligned rectangle with maximum area in a point set in the Euclidean plane -- (algorithms)
<p>On the plane <span class="math-container">$n$</span> points <span class="math-container">$(x_i, y_i)$</span> are marked. Select 4 points so that they define a rectangle with the greatest area and sides parallel to the... | habedi/stack-exchange-dataset |
110,366 | Recurrence with Minimum | <p>I need to solve the following recurrece:</p>

<p><span class="math-container">$T(n,m)=\begin{cases}
1, & m\leq 2(n-1)!\\
\min\limits_{a,b\geq 1\\a\cdot b\leq (n-1)!}{T(n-1,a)+T(n-1,b)+T(n,m-ab)}, & \text{else}
\end{cases}$</span></p>

<p>Note: in the first line beneath the <span c... | recurrence relation factorial | 1 | Recurrence with Minimum -- (recurrence relation factorial)
<p>I need to solve the following recurrece:</p>

<p><span class="math-container">$T(n,m)=\begin{cases}
1, & m\leq 2(n-1)!\\
\min\limits_{a,b\geq 1\\a\cdot b\leq (n-1)!}{T(n-1,a)+T(n-1,b)+T(n,m-ab)}, & \text{else}
\end{cases}$</span><... | habedi/stack-exchange-dataset |
110,370 | Decision problem - vertex with path to all other vertcies | <p>Consider the following decision problem: Given a directed graph <span class="math-container">$G$</span>, is there a vertex <span class="math-container">$v$</span> that has path to all other vertcies. 
I am able to place this problem in NL, similarly to the strongly-connected graph problem. But is this problem NL... | complexity theory graphs | 1 | Decision problem - vertex with path to all other vertcies -- (complexity theory graphs)
<p>Consider the following decision problem: Given a directed graph <span class="math-container">$G$</span>, is there a vertex <span class="math-container">$v$</span> that has path to all other vertcies. 
I am able to place this ... | habedi/stack-exchange-dataset |
110,375 | Is Lambda Calculus purely syntactic? | <p>I've been reading for a few weeks about the Lambda Calculus, but I have not yet seen anything that is materially distinct from existing mathematical functions, and I want to know whether it is just a matter of notation, or whether there are any new properties or rules created by the lambda calculus axioms that don't... | lambda calculus functional programming | 1 | Is Lambda Calculus purely syntactic? -- (lambda calculus functional programming)
<p>I've been reading for a few weeks about the Lambda Calculus, but I have not yet seen anything that is materially distinct from existing mathematical functions, and I want to know whether it is just a matter of notation, or whether there... | habedi/stack-exchange-dataset |
110,382 | Dove tailing questions about accepting language | <p><span class="math-container">$L = \{M | \text { M is a TM and there exist some string w that contains five 1's such that M halts}\}$</span></p>

<p>Where <span class="math-container">$\Sigma =\{0,1\}$</span></p>

<p>let <span class="math-container">$w_1, w_2, \cdots, \in \Sigma^*$</span> be an effect... | turing machines | 1 | Dove tailing questions about accepting language -- (turing machines)
<p><span class="math-container">$L = \{M | \text { M is a TM and there exist some string w that contains five 1's such that M halts}\}$</span></p>

<p>Where <span class="math-container">$\Sigma =\{0,1\}$</span></p>

<p>let <span class=... | habedi/stack-exchange-dataset |
110,394 | Solving recurrence relation with minimum and factorial | <p>I need to solve the following recurrence relation, where <span class="math-container">$T(n,m)$</span> is defined over <span class="math-container">$\Bbb N_+\times\Bbb N_+$</span>.</p>

<p><span class="math-container">$T(n,m)=\begin{cases}
1, & n=1\text{ or }m\leq 2(n-1)!\\
\min\limits_{a,b,c\geq ... | recurrence relation factorial | 1 | Solving recurrence relation with minimum and factorial -- (recurrence relation factorial)
<p>I need to solve the following recurrence relation, where <span class="math-container">$T(n,m)$</span> is defined over <span class="math-container">$\Bbb N_+\times\Bbb N_+$</span>.</p>

<p><span class="math-container">$T... | habedi/stack-exchange-dataset |
110,396 | Exercise 11 of section 3.2.2(The art of computer programming) | <p>I'm asking about its part a.</p>
<blockquote>
<p>a) If <span class="math-container">$f(z),\space a(z),\space b(z)$</span> are polynomials with integer coefficients, let
us write <span class="math-container">$a(z)\equiv b(z) (\operatorname{mod} f(z)\space and\space m)$</span>
if <span class="math-cont... | algorithms modular arithmetic | 1 | Exercise 11 of section 3.2.2(The art of computer programming) -- (algorithms modular arithmetic)
<p>I'm asking about its part a.</p>
<blockquote>
<p>a) If <span class="math-container">$f(z),\space a(z),\space b(z)$</span> are polynomials with integer coefficients, let
us write <span class="math-container">$... | habedi/stack-exchange-dataset |
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