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
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96,505 | quicksort recurrence relation | <p>In Concrete Mathematics
Textbook by Donald Knuth and Oren Patashnik ,
ch.2 Sum ,sec2.2<br>
He wrote:</p>

<blockquote>
 <p>The average number of comparison steps made by quicksort when it's applied to $n$ items in random order satisfies the recurrence :
 $$C_0=0;$$
 $$C_n = n + 1... | algorithm analysis runtime analysis recurrence relation quicksort | 1 | quicksort recurrence relation -- (algorithm analysis runtime analysis recurrence relation quicksort)
<p>In Concrete Mathematics
Textbook by Donald Knuth and Oren Patashnik ,
ch.2 Sum ,sec2.2<br>
He wrote:</p>

<blockquote>
 <p>The average number of comparison steps made by quicksort when it's... | habedi/stack-exchange-dataset |
96,547 | QTM & Halting problem | <p>"Can QTM (Quantum Turing machine) solve halting problem"
Why not have an immediate answer "No QTM Can't do this", we know that Turing proved it impossible when DTM , i meant , " Why we cant use the same proof which turing use it"
Let H be a program in a quantum turing machine that will solve the issue of hal... | turing machines quantum computing | 1 | QTM & Halting problem -- (turing machines quantum computing)
<p>"Can QTM (Quantum Turing machine) solve halting problem"
Why not have an immediate answer "No QTM Can't do this", we know that Turing proved it impossible when DTM , i meant , " Why we cant use the same proof which turing use it"
Let H be a program... | habedi/stack-exchange-dataset |
96,550 | What's the difference between user registers and kernel registers? | <p>I am reading <a href="http://pages.cs.wisc.edu/%7Eremzi/OSTEP/" rel="noreferrer">Operating Systems: Three Easy Pieces</a>, but I can't fully understand the following paragraph. It's from the <a href="http://pages.cs.wisc.edu/%7Eremzi/OSTEP/cpu-mechanisms.pdf" rel="noreferrer">6th chapter</a> of the book. I think it'... | operating systems | 1 | What's the difference between user registers and kernel registers? -- (operating systems)
<p>I am reading <a href="http://pages.cs.wisc.edu/%7Eremzi/OSTEP/" rel="noreferrer">Operating Systems: Three Easy Pieces</a>, but I can't fully understand the following paragraph. It's from the <a href="http://pages.cs.wisc.edu/%7... | habedi/stack-exchange-dataset |
96,551 | How to prove using pumping lemma that language generated by a(b*)c(d*)e is regular? | <p>I am studying pumping lemma from Introduction to theory of computation by Michael Sipser. I wanted to check if the language generated by regular expression</p>
<pre><code> a(b*)c(d*)e
</code></pre>
<p>can be proved to be regular using pumping lemma. The string generated by this expression are of the form... | regular languages finite automata regular expressions pumping lemma | 1 | How to prove using pumping lemma that language generated by a(b*)c(d*)e is regular? -- (regular languages finite automata regular expressions pumping lemma)
<p>I am studying pumping lemma from Introduction to theory of computation by Michael Sipser. I wanted to check if the language generated by regular expression</p>&... | habedi/stack-exchange-dataset |
96,557 | Finite list induction principle and the tail eliminator | <p>In Dybjer's <em>Inductive Families</em> the author present a method to derive an eliminator/induction principle for every inductive family of types.</p>

<p>In particular for the type of finite lists, namely
$$List' \colon (A \colon set) (n \colon N) set$$
we get the eliminator 
$$\begin{align*}&... | dependent types inductive datatypes | 1 | Finite list induction principle and the tail eliminator -- (dependent types inductive datatypes)
<p>In Dybjer's <em>Inductive Families</em> the author present a method to derive an eliminator/induction principle for every inductive family of types.</p>

<p>In particular for the type of finite lists, namely
... | habedi/stack-exchange-dataset |
96,563 | Subgraph isomorphism on star multi-graphs with labelled edges | <p>My approach to the problem has been to reformulate it into something more recognizable, but I don't know the best way to solve the reformulated problems either. I list the original problem, an example, and two reformulations below. The second reformulation is equivalent to an unbalanced assignment problem with a bin... | algorithms graphs search algorithms constraint satisfaction | 1 | Subgraph isomorphism on star multi-graphs with labelled edges -- (algorithms graphs search algorithms constraint satisfaction)
<p>My approach to the problem has been to reformulate it into something more recognizable, but I don't know the best way to solve the reformulated problems either. I list the original problem, ... | habedi/stack-exchange-dataset |
96,565 | Possibility to prove that a function in big-theta is in big-O and little-o | <p>I understand that many similar questions of this sort have been asked, but in this case, the solution does not appear to be provable - or is it?</p>

<p>Question: If $f(n) = \Theta (g(n))$ and $g(n) = o(h(n))$ then $f(n) = O(h(n))$</p>

<p>Current solution:</p>

<p>$f(n) = \Theta (g(n))$ impl... | algorithms algorithm analysis | 1 | Possibility to prove that a function in big-theta is in big-O and little-o -- (algorithms algorithm analysis)
<p>I understand that many similar questions of this sort have been asked, but in this case, the solution does not appear to be provable - or is it?</p>

<p>Question: If $f(n) = \Theta (g(n))$ and $g(n) ... | habedi/stack-exchange-dataset |
96,569 | Looking for a problem provably not in P | <p>My basic position is that everything is in P. Then comes the time hierachy theorem and EXP. That's easy: simulate and then diagonalize. After that comes EXP-completeness; that's difficult to swallow. Papadimitriou basically says that the proof is very similar to that of Cook-Levin theorem and do it yourself. Also, I... | reductions complexity classes | 1 | Looking for a problem provably not in P -- (reductions complexity classes)
<p>My basic position is that everything is in P. Then comes the time hierachy theorem and EXP. That's easy: simulate and then diagonalize. After that comes EXP-completeness; that's difficult to swallow. Papadimitriou basically says that the proo... | habedi/stack-exchange-dataset |
96,570 | Is DFA and Regular Expression equivalent? | <p>The language of a DFA can be the empty set (by defining no final states), but can a Regular Expression do that? </p>

<p>If Regular Expression cannot do that, does it mean that DFA and Regular Expression are not equivalent (in at least some cases)?</p>
 | formal languages regular languages finite automata regular expressions | 1 | Is DFA and Regular Expression equivalent? -- (formal languages regular languages finite automata regular expressions)
<p>The language of a DFA can be the empty set (by defining no final states), but can a Regular Expression do that? </p>

<p>If Regular Expression cannot do that, does it mean that DFA and Regula... | habedi/stack-exchange-dataset |
96,575 | How to create constraints for Mixed integer linear problem? | <p>i am a beginner to <code>Discrete optimization</code> domain. I am working on the real world problem, i.e., Scheduling of hybrid appliances. I have hybrid appliances which can use gas and electricity or electricity and hot water or hot water and gas for their operation. Each appliance has some set of tasks, each tas... | optimization linear programming integer programming constraint programming | 1 | How to create constraints for Mixed integer linear problem? -- (optimization linear programming integer programming constraint programming)
<p>i am a beginner to <code>Discrete optimization</code> domain. I am working on the real world problem, i.e., Scheduling of hybrid appliances. I have hybrid appliances which can u... | habedi/stack-exchange-dataset |
96,579 | M is a Determinstic TM with one tape, c2 is c1 reachable, if it's reachable within finite positive time | <p>$M=(Q,\sum,\Gamma, \delta, q_{0}, q_{accept}, q_{reject})$ is a TM with one tape.
let $c_{1}, c_{2}$ be two configurations of $M$.</p>

<p>A configuration is defined like this: </p>

<p>$uqv$ where $(q\in Q; u,v\in \Gamma^{*} )$</p>

<p>we say that $c_{1}$ is reachable from $c_{2}$ in $M... | complexity theory turing machines computability | 1 | M is a Determinstic TM with one tape, c2 is c1 reachable, if it's reachable within finite positive time -- (complexity theory turing machines computability)
<p>$M=(Q,\sum,\Gamma, \delta, q_{0}, q_{accept}, q_{reject})$ is a TM with one tape.
let $c_{1}, c_{2}$ be two configurations of $M$.</p>

<p>A configu... | habedi/stack-exchange-dataset |
96,583 | Determine whether two collections of items can be paired | <p>Given collections <code>I</code> (items) and <code>S</code> (slots), where <code>I >= S</code>.
And a pairing function that determines whether a slot can accept an item.</p>

<p>Is there an efficient way to find whether a "complete" pairing of items to slots exist, where "complete" means every slot is... | algorithms matching bipartite matching | 1 | Determine whether two collections of items can be paired -- (algorithms matching bipartite matching)
<p>Given collections <code>I</code> (items) and <code>S</code> (slots), where <code>I >= S</code>.
And a pairing function that determines whether a slot can accept an item.</p>

<p>Is there an efficient w... | habedi/stack-exchange-dataset |
96,596 | What is the use case of multi-type-parameters generic interface? | <p>Previously, I ask a <a href="https://cs.stackexchange.com/questions/95280/what-is-the-use-case-for-multi-type-parameter-generics">similar question</a>, but the answer given only demonstrates the usage of multiple-type-parameters(MTP) <strong>data structure</strong>, but not MTP <strong>generic interface</strong>. </... | type theory | 1 | What is the use case of multi-type-parameters generic interface? -- (type theory)
<p>Previously, I ask a <a href="https://cs.stackexchange.com/questions/95280/what-is-the-use-case-for-multi-type-parameter-generics">similar question</a>, but the answer given only demonstrates the usage of multiple-type-parameters(MTP) <... | habedi/stack-exchange-dataset |
96,606 | Chomsky Hierarchy and P vs NP | <p>I have read multiple questions here that involve this kind of subject but I haven't found any definite answer. In what class do regular languages belong? (P or NP or some regular are P and other NP), context-free languages? (same question) ,context-sensitive? and general languages? . I personally believe all regular... | complexity theory p vs np chomsky hierarchy | 1 | Chomsky Hierarchy and P vs NP -- (complexity theory p vs np chomsky hierarchy)
<p>I have read multiple questions here that involve this kind of subject but I haven't found any definite answer. In what class do regular languages belong? (P or NP or some regular are P and other NP), context-free languages? (same question... | habedi/stack-exchange-dataset |
96,616 | Proof that Turing machines and computers have same power | <p>How do we prove that any logical circuit can be simulated by a Turing machine?</p>

<p>For example, we take a logical circuit $L$ that is made of gates <em>and</em>, <em>or</em>, and <em>not</em>. This circuit determines a problem, for example, whether the input is an even number. How can we prove that if a ... | turing machines computability | 1 | Proof that Turing machines and computers have same power -- (turing machines computability)
<p>How do we prove that any logical circuit can be simulated by a Turing machine?</p>

<p>For example, we take a logical circuit $L$ that is made of gates <em>and</em>, <em>or</em>, and <em>not</em>. This circuit determi... | habedi/stack-exchange-dataset |
96,620 | Cost of computational representation in PAC-learning definition | <p>I'm currently reading Foundations of Machine Learning by M. Mohri, A. Rostamizadeh, A. Talwalkar and according to their definition a concept class $C$ is said to be PAC-learnable if</p>

<p>$$Pr_{S \sim D^m}[R(h_s)\leq\varepsilon] \geq 1-\delta$$</p>

<p>holds for any sample size $m \geq poly(1/\vare... | machine learning | 1 | Cost of computational representation in PAC-learning definition -- (machine learning)
<p>I'm currently reading Foundations of Machine Learning by M. Mohri, A. Rostamizadeh, A. Talwalkar and according to their definition a concept class $C$ is said to be PAC-learnable if</p>

<p>$$Pr_{S \sim D^m}[R(h_s)\leq\vare... | habedi/stack-exchange-dataset |
96,623 | Ordered set cover problem: is it NP-hard? | <p>Given a set of elements $U=\{1,2,\ldots,n\}$ and a collection of $m$ sets $\{S_1,S_2,\ldots,S_m\}$ whose union equals $U$. Each element $e$ of a set $S_i$ has a weight $w_i(e)$. The weight $w(S_i)$ of the set $S_i$ is defined as $w(S_i):=\max_{e\in S_i}w_i(e)$. </p>

<p>The problem is to choose an <strong>or... | algorithms np hard | 1 | Ordered set cover problem: is it NP-hard? -- (algorithms np hard)
<p>Given a set of elements $U=\{1,2,\ldots,n\}$ and a collection of $m$ sets $\{S_1,S_2,\ldots,S_m\}$ whose union equals $U$. Each element $e$ of a set $S_i$ has a weight $w_i(e)$. The weight $w(S_i)$ of the set $S_i$ is defined as $w(S_i):=\max_{e\in S_... | habedi/stack-exchange-dataset |
96,624 | How to solve Tower of Hanoi iteratively? | <p>It may seem obvious to many but i am having a hard time figuring out the iterative solution to the Tower of Hanoi problem. There are some solutions on the Internet but without explanations. So can anybody give a sound explanation so that it becomes more intuitive and easy to reason.</p>
 | algorithms | 1 | How to solve Tower of Hanoi iteratively? -- (algorithms)
<p>It may seem obvious to many but i am having a hard time figuring out the iterative solution to the Tower of Hanoi problem. There are some solutions on the Internet but without explanations. So can anybody give a sound explanation so that it becomes more intuit... | habedi/stack-exchange-dataset |
96,626 | What's the Big O runtime of a DFS word search through a matrix? | <p>The problem is to try and find a word in a 2D matrix of characters:</p>

<p>Given a 2D board and a word, find if the word exists in the grid.
The word can be constructed from letters of sequentially adjacent cell, where "adjacent" cells are those horizontally or vertically neighboring. The same letter ce... | graphs algorithm analysis runtime analysis recursion | 1 | What's the Big O runtime of a DFS word search through a matrix? -- (graphs algorithm analysis runtime analysis recursion)
<p>The problem is to try and find a word in a 2D matrix of characters:</p>

<p>Given a 2D board and a word, find if the word exists in the grid.
The word can be constructed from letters ... | habedi/stack-exchange-dataset |
96,635 | Tarjan's SCC : example showing necessity of lowlink definition and calculation rule? | <p>Several questions (<a href="https://cs.stackexchange.com/questions/11148/tarjans-strongly-connected-component-algorithm?s=5%7C26.1152">1</a>, <a href="https://cs.stackexchange.com/questions/62744/why-the-update-low-value-in-tarjans-for-the-ancestors-is-not-its-low-value-ins?s=4%7C30.5808">2</a>) have been asked abou... | graphs graph traversal connected | 1 | Tarjan's SCC : example showing necessity of lowlink definition and calculation rule? -- (graphs graph traversal connected)
<p>Several questions (<a href="https://cs.stackexchange.com/questions/11148/tarjans-strongly-connected-component-algorithm?s=5%7C26.1152">1</a>, <a href="https://cs.stackexchange.com/questions/6274... | habedi/stack-exchange-dataset |
96,650 | Nine-input sorting network of optimal depth with minimum number of comparators | <p>In the research on sorting networks, two parameters are typically of interest: The number of comparators required and the depth required.
They have practical implications for throughput and latency. Results in latter category are usually easier to achieve than corresponding results in the former. Already in 198... | algorithms sorting | 1 | Nine-input sorting network of optimal depth with minimum number of comparators -- (algorithms sorting)
<p>In the research on sorting networks, two parameters are typically of interest: The number of comparators required and the depth required.
They have practical implications for throughput and latency. Results in ... | habedi/stack-exchange-dataset |
96,660 | A directed graph colored path problem? | <p><strong>Given:</strong> A rooted, directed, a-cyclic graph $G$. Let $r_0$ be the root node and $t_0$ be another target node. Each node in $G$ is assigned a non unique id/color ($ID_i),\ 1<i<N$ for some integer N.</p>

<p><strong>Problem:</strong> Is there a directed path from $r_0$ to $t_0$, such that ... | algorithms complexity theory graphs np | 1 | A directed graph colored path problem? -- (algorithms complexity theory graphs np)
<p><strong>Given:</strong> A rooted, directed, a-cyclic graph $G$. Let $r_0$ be the root node and $t_0$ be another target node. Each node in $G$ is assigned a non unique id/color ($ID_i),\ 1<i<N$ for some integer N.</p>

<p... | habedi/stack-exchange-dataset |
96,661 | Why does n! have the least time? | <p><a href="https://i.stack.imgur.com/DZiN6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DZiN6.png" alt="Table with time intervals along the top, from one second to one century, and various functions of n down the side"></a></p>

<p>I was reading the book Introduction to Algorithms, Chapte... | algorithms time complexity | 1 | Why does n! have the least time? -- (algorithms time complexity)
<p><a href="https://i.stack.imgur.com/DZiN6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DZiN6.png" alt="Table with time intervals along the top, from one second to one century, and various functions of n down the side"></a></p>
... | habedi/stack-exchange-dataset |
96,663 | Cook completeness of a variant of Vertex Cover | <p>Is this variant of Vertex Cover Cook-complete for $\mathrm{NP}$?</p>

<p><strong>Input</strong>: An undirected graph $G(V, E)$ together with a vertex cover $C\subseteq V$</p>

<p><strong>Output</strong>: <strong>YES</strong> if there exists a vertex cover $C'\subseteq V$ such that $|C'|=|C|-1$, <stro... | np complete reductions np hard np decision problem | 1 | Cook completeness of a variant of Vertex Cover -- (np complete reductions np hard np decision problem)
<p>Is this variant of Vertex Cover Cook-complete for $\mathrm{NP}$?</p>

<p><strong>Input</strong>: An undirected graph $G(V, E)$ together with a vertex cover $C\subseteq V$</p>

<p><strong>Output</str... | habedi/stack-exchange-dataset |
96,672 | What is the "spatial information" in convolutional neural network | <p>deep learning research papers always claim that deeper layers of CNN have good "semantic information" but poor "spatial information". What is the spatial information exactly. Is that some activations in deeper layers?</p>
 | machine learning | 1 | What is the "spatial information" in convolutional neural network -- (machine learning)
<p>deep learning research papers always claim that deeper layers of CNN have good "semantic information" but poor "spatial information". What is the spatial information exactly. Is that some activations in deeper layers?</p>
 | habedi/stack-exchange-dataset |
96,678 | Best way to translate while loops to functions for software verification | <p>I have a verification engine where while loops are translated into functional code, like this:</p>

<pre><code>def f() = ... while(c) b ...

def f() = ... fwhile() ... 
def fwhile() = if(c){ fwhile() } 
</code></pre>

<p>We want for the users to be able to annot... | functional programming software verification imperative programming | 1 | Best way to translate while loops to functions for software verification -- (functional programming software verification imperative programming)
<p>I have a verification engine where while loops are translated into functional code, like this:</p>

<pre><code>def f() = ... while(c) b ...

def f() = ... ... | habedi/stack-exchange-dataset |
96,679 | Cardinality of the set of algorithms | <p>Someone in a discussion brought up that (he reckons) there can be at least continuum number of strategies to approach a specific problem. The specific problem was trading strategies (not algorithms but strategies) but I think thats beside the point for my question.</p>

<p>This got me thinking about the card... | algorithms turing machines combinatorics | 1 | Cardinality of the set of algorithms -- (algorithms turing machines combinatorics)
<p>Someone in a discussion brought up that (he reckons) there can be at least continuum number of strategies to approach a specific problem. The specific problem was trading strategies (not algorithms but strategies) but I think thats be... | habedi/stack-exchange-dataset |
96,700 | Is this reduction from 3D-MATCHING to PATH SELECTION invalid? | <p>I'm a bit confused about some proof that PATH-SELECTION-PROBLEM is NP-complete (Problem 9, chapter 8 in "Algorithm Design" by Tardos and Kleinberg) that I found in some solution manual here: </p>

<p><a href="https://github.com/weimin/CS-180/blob/master/Algorithm%20Design%20(Kleinberg%20Tardos%202005)%20-%... | np complete reductions | 1 | Is this reduction from 3D-MATCHING to PATH SELECTION invalid? -- (np complete reductions)
<p>I'm a bit confused about some proof that PATH-SELECTION-PROBLEM is NP-complete (Problem 9, chapter 8 in "Algorithm Design" by Tardos and Kleinberg) that I found in some solution manual here: </p>

<p><a href="https://... | habedi/stack-exchange-dataset |
96,705 | Why isn't Busy Beaver Infinite for n < trivial k? | <p>I am clearly not understanding the Busy Beaver function. Why wouldn't it be infinite for n > k, 
while(true)
{
 print("1");
}</p>
 | complexity theory | 1 | Why isn't Busy Beaver Infinite for n < trivial k? -- (complexity theory)
<p>I am clearly not understanding the Busy Beaver function. Why wouldn't it be infinite for n > k, 
while(true)
{
 print("1");
}</p>
 | habedi/stack-exchange-dataset |
96,706 | Algorithm for generating all unlabeled trees with n nodes? | <p>How can one generate all unlabeled trees with $\le n$ nodes?</p>

<p>That is, generate and store the <a href="https://en.wikipedia.org/wiki/Adjacency_matrix" rel="nofollow noreferrer">adjacency matrices</a> of those graphs? (not just <a href="http://oeis.org/A000055" rel="nofollow noreferrer">count them</a>)... | algorithms graphs trees | 1 | Algorithm for generating all unlabeled trees with n nodes? -- (algorithms graphs trees)
<p>How can one generate all unlabeled trees with $\le n$ nodes?</p>

<p>That is, generate and store the <a href="https://en.wikipedia.org/wiki/Adjacency_matrix" rel="nofollow noreferrer">adjacency matrices</a> of those graph... | habedi/stack-exchange-dataset |
96,718 | Does indirect diagonalization a relativize technique? | <p>My main question is can with R.kanon , Fortnow ,... <a href="http://www.cs.cmu.edu/~ryanw/ccc05.pdf" rel="nofollow noreferrer">technique</a> that shows lower bounds for SAT seperate P and NP ?</p>

<p>Baker-Gill-Solovay showed that $P?=NP$ could not be solved with relativization. Does <a href="http://www.cs.... | complexity theory proof techniques np p vs np | 1 | Does indirect diagonalization a relativize technique? -- (complexity theory proof techniques np p vs np)
<p>My main question is can with R.kanon , Fortnow ,... <a href="http://www.cs.cmu.edu/~ryanw/ccc05.pdf" rel="nofollow noreferrer">technique</a> that shows lower bounds for SAT seperate P and NP ?</p>

<p>Bak... | habedi/stack-exchange-dataset |
96,719 | Exponential Space Complexity equality | <p>Consider $$ \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE}(2^{c (\log{n})^2}) \quad \overset{?}{=} \quad \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE} ( n^{c \log{n}})$$</p>

<p>My lecture notes say that this is an equality, can someone point me to the theorem (or explain) why this is ? </p>

<p>I've obser... | complexity theory space complexity | 1 | Exponential Space Complexity equality -- (complexity theory space complexity)
<p>Consider $$ \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE}(2^{c (\log{n})^2}) \quad \overset{?}{=} \quad \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE} ( n^{c \log{n}})$$</p>

<p>My lecture notes say that this is an equality, can someone ... | habedi/stack-exchange-dataset |
96,724 | How many unique counting functions there are for strings of N bits? | <p>Let a counting function <code>inc</code> for inputs of <code>N</code> bits be defined as:</p>

<pre><code>inc(b0 | b1 ... | bN) = F0(b0) | F1(b0,b1) ... | FN(b0,b1,...bN)
</code></pre>

<p>Where <code>FN</code> is a function from <code>N</code> bits to a bit, and such that the period of <code>inc... | algorithms number formats binary | 1 | How many unique counting functions there are for strings of N bits? -- (algorithms number formats binary)
<p>Let a counting function <code>inc</code> for inputs of <code>N</code> bits be defined as:</p>

<pre><code>inc(b0 | b1 ... | bN) = F0(b0) | F1(b0,b1) ... | FN(b0,b1,...bN)
</code></pre>

<p>Wh... | habedi/stack-exchange-dataset |
96,729 | Is there a solution for this maze problem in polynomial time? | <p>Suppose you have a maze represented by a graph where each vertex represents a room and edges represent paths between rooms and each edge has a weight denoting the time it takes to go that way. Now here comes the tricky part: suppose each room needs a <strong>set</strong> of keys for you to enter, and inside each roo... | graphs time complexity shortest path traveling salesman | 1 | Is there a solution for this maze problem in polynomial time? -- (graphs time complexity shortest path traveling salesman)
<p>Suppose you have a maze represented by a graph where each vertex represents a room and edges represent paths between rooms and each edge has a weight denoting the time it takes to go that way. N... | habedi/stack-exchange-dataset |
96,733 | How easy is it for computers to compute in a 4th Dimension (or more)? | <p>I know its very hard to show a human what the 4th dimension would look like and the mind can't even comprehend more dimensions, but what about a computer? </p>

<p>How many "dimensions" can a computer handle?</p>

<p>Or maybe a better way to ask, does computing power exponentially increase when extra... | complexity theory | 1 | How easy is it for computers to compute in a 4th Dimension (or more)? -- (complexity theory)
<p>I know its very hard to show a human what the 4th dimension would look like and the mind can't even comprehend more dimensions, but what about a computer? </p>

<p>How many "dimensions" can a computer handle?</p>
... | habedi/stack-exchange-dataset |
96,737 | Collection of meta-reductions in theory of $\mathrm{NP}$-completeness | <p>I want to start a wiki post about meta-result of meta-reductions in the theory of $\mathrm{NP}$-completeness.</p>

<p>This can be regarded as a reference request post. Any links are appreciated.</p>

<p>At least, this can be helpful in drastically mitigating the frequently asked questions related to ... | complexity theory np complete reductions np hard np | 1 | Collection of meta-reductions in theory of $\mathrm{NP}$-completeness -- (complexity theory np complete reductions np hard np)
<p>I want to start a wiki post about meta-result of meta-reductions in the theory of $\mathrm{NP}$-completeness.</p>

<p>This can be regarded as a reference request post. Any links are ... | habedi/stack-exchange-dataset |
96,746 | Finding paths with minimum intersections | <p>Given a graph and a set of origin-destination <span class="math-container">$\langle o_i,d_i\rangle$</span> pairs. The goal is computing a set of paths such that every pair has a path and number of common vertices between paths is minimized.</p>

<p>If you have any relevant references to share I'd like to rea... | algorithms graphs optimization | 1 | Finding paths with minimum intersections -- (algorithms graphs optimization)
<p>Given a graph and a set of origin-destination <span class="math-container">$\langle o_i,d_i\rangle$</span> pairs. The goal is computing a set of paths such that every pair has a path and number of common vertices between paths is minimized.... | habedi/stack-exchange-dataset |
96,758 | Prove that different definitions of big-Oh with n>=1 or n>N are equivalent | <p>I am coming across two slightly different definitions of big-oh and need to prove that they are equivalent to each other:</p>

<p>Definition 1: f(n) = O(g(n)) if there exists constants c and N such that f(n) ≤ c g(n) for all n > N.</p>

<p>Definition 2: f(n) = O(g(n)) if there exists a constant c suc... | algorithms asymptotics | 1 | Prove that different definitions of big-Oh with n>=1 or n>N are equivalent -- (algorithms asymptotics)
<p>I am coming across two slightly different definitions of big-oh and need to prove that they are equivalent to each other:</p>

<p>Definition 1: f(n) = O(g(n)) if there exists constants c and N such that f(n... | habedi/stack-exchange-dataset |
96,767 | Is there a fast algorithm for computing the rolling mode of an array of integers? | <p>I was wondering if there exists an efficient algorithm for calculating the "rolling mode of an array of integers.</p>

<p>By rolling mode I mean that we have an array of integers of size $n$ and a sliding window of size $k$ and we we want to compute the mode for each window in the array.<br>
 There is an... | algorithms algorithm analysis data structures statistics streaming algorithm | 1 | Is there a fast algorithm for computing the rolling mode of an array of integers? -- (algorithms algorithm analysis data structures statistics streaming algorithm)
<p>I was wondering if there exists an efficient algorithm for calculating the "rolling mode of an array of integers.</p>

<p>By rolling mode I mean ... | habedi/stack-exchange-dataset |
96,778 | Does deep learning infer P = NP? | <p>The question comes from the following scenario,
assume we have the traveler problem which is NP (the one where a traveler wants to visit all countries with the lowest cost(by summing up all flights))</p>

<p>So basically Neural Network can not just predict, but generate results from things they've learne... | complexity theory np complete machine learning np hard | 1 | Does deep learning infer P = NP? -- (complexity theory np complete machine learning np hard)
<p>The question comes from the following scenario,
assume we have the traveler problem which is NP (the one where a traveler wants to visit all countries with the lowest cost(by summing up all flights))</p>

<p>So b... | habedi/stack-exchange-dataset |
96,794 | Oracle machine solving halting problem for other oracle machines | <p>Could someone give me a simple explanation why an oracle machine that can solve the halting problem for standard Turing machines, is however unable to solve the halting problem for other such oracle machines?</p>

<p>I have tried searching for the answer online but could not find any that really answered the... | complexity theory turing machines oracle machines | 1 | Oracle machine solving halting problem for other oracle machines -- (complexity theory turing machines oracle machines)
<p>Could someone give me a simple explanation why an oracle machine that can solve the halting problem for standard Turing machines, is however unable to solve the halting problem for other such oracl... | habedi/stack-exchange-dataset |
96,801 | Determining if a list contains two numbers whose difference is less than k | <blockquote>
<p>Given a list with <span class="math-container">$n$</span> positive elements and positive number <span class="math-container">$k$</span>, determine whether there are two numbers whose difference is less than <span class="math-container">$k$</span>.</p>
<p>The average time complexity should be <sp... | algorithms | 1 | Determining if a list contains two numbers whose difference is less than k -- (algorithms)
<blockquote>
<p>Given a list with <span class="math-container">$n$</span> positive elements and positive number <span class="math-container">$k$</span>, determine whether there are two numbers whose difference is less than <s... | habedi/stack-exchange-dataset |
96,820 | What is the real advantage of Google's new Fuchsia operating system kernel? | <p>Google's new in-production OS which it desires to replace not only Android but many other devices utilizes a completely new kernel:</p>

<blockquote>
 <p>The code differs from Android and Chrome OS due to its being based on the "Zircon" kernel (formerly "Magenta") rather than on the Linux kernel.</p>&#x... | operating systems os kernel | 1 | What is the real advantage of Google's new Fuchsia operating system kernel? -- (operating systems os kernel)
<p>Google's new in-production OS which it desires to replace not only Android but many other devices utilizes a completely new kernel:</p>

<blockquote>
 <p>The code differs from Android and Chrome ... | habedi/stack-exchange-dataset |
96,831 | Constant-depth threshold circuit for $\mathrm{PP}$ | <p>Is it proven that $\mathrm{PP}$ has uniform constant-depth threshold circuit?</p>
 | complexity theory | 1 | Constant-depth threshold circuit for $\mathrm{PP}$ -- (complexity theory)
<p>Is it proven that $\mathrm{PP}$ has uniform constant-depth threshold circuit?</p>
 | habedi/stack-exchange-dataset |
96,840 | Solving problems that DTM can't solve | <p>Let L be a problem that DTM can't solve. Can we prove that there is an abstract machine that can solve this problem?</p>

<p>Here, L is not Halting problem or Hilbert's tenth problem (because we proved that algorithms for these don't exist).</p>

<p>In other words, is there an abstract machine more p... | computability undecidability | 1 | Solving problems that DTM can't solve -- (computability undecidability)
<p>Let L be a problem that DTM can't solve. Can we prove that there is an abstract machine that can solve this problem?</p>

<p>Here, L is not Halting problem or Hilbert's tenth problem (because we proved that algorithms for these don't exi... | habedi/stack-exchange-dataset |
96,841 | Minimum commands to reach an value | <p>Sorry if I am doing it wrong in some way.</p>

<p>Some days ago I have passed trough an technical test in which I had 3 tasks to complete. The tasks were a problem and I have to write an algorithm to solve it.</p>

<p>One of the 3 tasks I could not complete. Really, I have absolutly no idea how to st... | algorithms | 1 | Minimum commands to reach an value -- (algorithms)
<p>Sorry if I am doing it wrong in some way.</p>

<p>Some days ago I have passed trough an technical test in which I had 3 tasks to complete. The tasks were a problem and I have to write an algorithm to solve it.</p>

<p>One of the 3 tasks I could not c... | habedi/stack-exchange-dataset |
96,845 | Need help converting machine code | <p>I would appreciate some help with this question.</p>

<p>The 32 bit numbers below represent a MIPS instruction. identify the different fields and state which instruction it is. give the assembler source code that can be converted into this instruction.</p>

<p>001000|10001|10010|0000000000001111</p>&... | binary mips | 1 | Need help converting machine code -- (binary mips)
<p>I would appreciate some help with this question.</p>

<p>The 32 bit numbers below represent a MIPS instruction. identify the different fields and state which instruction it is. give the assembler source code that can be converted into this instruction.</p>&#... | habedi/stack-exchange-dataset |
96,849 | Is it possible to give a guarantee about the accuracy of the local time? | <p>Assuming there is no attack/bug and I use NTP: Are there any formal guarantees about the accuracy of the local time? Can there be any over the network?</p>

<p>Even if the communication is going directly between the time server (stratum-1) and the client, I have no idea if the devices / protocols on OSI laye... | computer networks cryptography communication protocols | 1 | Is it possible to give a guarantee about the accuracy of the local time? -- (computer networks cryptography communication protocols)
<p>Assuming there is no attack/bug and I use NTP: Are there any formal guarantees about the accuracy of the local time? Can there be any over the network?</p>

<p>Even if the comm... | habedi/stack-exchange-dataset |
96,852 | How is the formula for calculation in row/column major obtained? | <p>In my book, this formula is given for calculation of address of row major order's element $[I,J]$: </p>

<p>Address of $[I,J]$th element in row major order $= B + W[n(I-L_r)+ [J-L_c]]$</p>

<p>where B denotes base address, W denotes element size in bytes, n is the number of columns; $L_r$ is the fir... | arrays java memory allocation | 1 | How is the formula for calculation in row/column major obtained? -- (arrays java memory allocation)
<p>In my book, this formula is given for calculation of address of row major order's element $[I,J]$: </p>

<p>Address of $[I,J]$th element in row major order $= B + W[n(I-L_r)+ [J-L_c]]$</p>

<p>where B... | habedi/stack-exchange-dataset |
96,856 | Proving parametricity for Gallina functions | <p>I have the following definitions</p>

<pre><code>Definition nat'' {X : Type} := (X -> X) -> X -> X.
Definition nat' := forall (X : Type), @nat'' X.
</code></pre>

<p>And when I wanted to prove a certain property I ended up in the following situation:</p>

<pre><code> m : nat... | functional programming semantics coq | 1 | Proving parametricity for Gallina functions -- (functional programming semantics coq)
<p>I have the following definitions</p>

<pre><code>Definition nat'' {X : Type} := (X -> X) -> X -> X.
Definition nat' := forall (X : Type), @nat'' X.
</code></pre>

<p>And when I wanted to prove a cer... | habedi/stack-exchange-dataset |
96,859 | Weakest reduction for 3-$\mathrm{SAT}$ | <p>Having read all these posts</p>

<p><a href="https://cs.stackexchange.com/questions/96831/constant-depth-threshold-circuit-for-mathrmpp/96832#96832">Constant-depth threshold circuit for $\mathrm{PP}$</a></p>

<p><a href="https://cs.stackexchange.com/questions/96453/is-there-any-interesting-consequenc... | reductions | 1 | Weakest reduction for 3-$\mathrm{SAT}$ -- (reductions)
<p>Having read all these posts</p>

<p><a href="https://cs.stackexchange.com/questions/96831/constant-depth-threshold-circuit-for-mathrmpp/96832#96832">Constant-depth threshold circuit for $\mathrm{PP}$</a></p>

<p><a href="https://cs.stackexchange.... | habedi/stack-exchange-dataset |
96,861 | Is it possible to simulate a fair coin with a finite number of tossing of a biased one? | <p>It is a classic problem to simulate a fair coin with a biased one.</p>
<p>According to <a href="https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin" rel="noreferrer">Fair Coin (wiki)</a>,</p>
<blockquote>
<p>John von Neumann gave the following procedure:</p>
<ol>
<li>Toss the ... | proof techniques probability theory randomized algorithms randomness probabilistic algorithms | 1 | Is it possible to simulate a fair coin with a finite number of tossing of a biased one? -- (proof techniques probability theory randomized algorithms randomness probabilistic algorithms)
<p>It is a classic problem to simulate a fair coin with a biased one.</p>
<p>According to <a href="https://en.wikipedia.org/wiki/... | habedi/stack-exchange-dataset |
96,862 | Question about mapping reducibility | <p>I am working on an assignment where one of the sub questions is:</p>

<blockquote>
 <p>Let $A$ and $B$ be languages. Suppose $A$ is context free and $A ≤_m B$, which means that there is a computable function $f\colon \Sigma^* \to \Sigma^*$ such that for all $w \in \Sigma^*$, $w \in A$ iff $f(w) \in B$. ... | automata regular languages context free reductions | 1 | Question about mapping reducibility -- (automata regular languages context free reductions)
<p>I am working on an assignment where one of the sub questions is:</p>

<blockquote>
 <p>Let $A$ and $B$ be languages. Suppose $A$ is context free and $A ≤_m B$, which means that there is a computable function $f\c... | habedi/stack-exchange-dataset |
96,866 | Priority of symbols in a notation for projections | <p>We define an initial function called projection as $$I^k_i(n_1,\ldots,n_k) = n_i, \quad i \leq 1 \leq k, \quad k \in \mathbb{N}.$$</p>

<p>Suppose now that we want to define it in some programming language where we have no access to superscripting nor subscripting. Should we call it <code>Iki</code> or <code... | computability | 1 | Priority of symbols in a notation for projections -- (computability)
<p>We define an initial function called projection as $$I^k_i(n_1,\ldots,n_k) = n_i, \quad i \leq 1 \leq k, \quad k \in \mathbb{N}.$$</p>

<p>Suppose now that we want to define it in some programming language where we have no access to supersc... | habedi/stack-exchange-dataset |
96,881 | Show that RP is closed under concatenation | <p>I'm trying to prove the following problem:</p>

<blockquote>
 <p>Show that $RP$ is closed under concatenation</p>
</blockquote>

<p>Now, let's say that the two languages are $L_{1}$ and $L_{2}$ (both in $RP$). Then I accept a word iff the TM of $L_{1}$ accepted the first part of the word (wi... | complexity theory complexity classes closure properties randomized algorithms | 1 | Show that RP is closed under concatenation -- (complexity theory complexity classes closure properties randomized algorithms)
<p>I'm trying to prove the following problem:</p>

<blockquote>
 <p>Show that $RP$ is closed under concatenation</p>
</blockquote>

<p>Now, let's say that the two langua... | habedi/stack-exchange-dataset |
96,886 | Hardness of $2$ edge-disjoint spanning trees decomposition | <p>The question is clear from the title. What is the complexity of the following decision problem:</p>
<blockquote>
<p><strong>Input</strong>: An undirected graph <span class="math-container">$G(V, E)$</span></p>
<p><strong>Output</strong>: <span class="math-container">$\mathrm{YES}$</span> if <span class="... | complexity theory np complete np hard np complexity classes | 1 | Hardness of $2$ edge-disjoint spanning trees decomposition -- (complexity theory np complete np hard np complexity classes)
<p>The question is clear from the title. What is the complexity of the following decision problem:</p>
<blockquote>
<p><strong>Input</strong>: An undirected graph <span class="math-contain... | habedi/stack-exchange-dataset |
96,900 | Proving special case of SAT is in P | <p>Let <em>SAT-100</em> be the following problem:</p>

<p><strong>Input:</strong> Any boolean logic formula</p>

<p><strong>Output:</strong> True if there exists a combination of exactly 100 input variables that satisfy the formula.</p>

<p>This is the description of a problem that is apparently... | algorithms satisfiability polynomial time | 1 | Proving special case of SAT is in P -- (algorithms satisfiability polynomial time)
<p>Let <em>SAT-100</em> be the following problem:</p>

<p><strong>Input:</strong> Any boolean logic formula</p>

<p><strong>Output:</strong> True if there exists a combination of exactly 100 input variables that satisfy t... | habedi/stack-exchange-dataset |
96,907 | How to prove even with structural induction, for expression | <p>E::= zero|two|expression + expression|expression*expression, E element of expression how do i prove E to be even.</p>

<p>I have no clue on how to go about tackling this.</p>
 | induction | 1 | How to prove even with structural induction, for expression -- (induction)
<p>E::= zero|two|expression + expression|expression*expression, E element of expression how do i prove E to be even.</p>

<p>I have no clue on how to go about tackling this.</p>
 | habedi/stack-exchange-dataset |
96,909 | P=NP giving a deterministic algorithm for SAT | <p>I'm trying to prove the following problem:</p>
<blockquote>
<p>Prove that if <span class="math-container">$P=NP$</span> then there is a polynomial time algorithm for the following problem:</p>
<p><strong>INPUT:</strong> A boolean formula <span class="math-container">$\phi$</span>. <br>
<strong>OUTPUT... | complexity theory time complexity satisfiability complexity classes | 1 | P=NP giving a deterministic algorithm for SAT -- (complexity theory time complexity satisfiability complexity classes)
<p>I'm trying to prove the following problem:</p>
<blockquote>
<p>Prove that if <span class="math-container">$P=NP$</span> then there is a polynomial time algorithm for the following problem:</... | habedi/stack-exchange-dataset |
96,922 | Encoding (binary) trees using lambda calculus | <p>I'm new to lambda calculus, and I read all kinds of interesting stuff about encoding data types as functions. Church booleans, numbers and lists.
<a href="https://en.wikipedia.org/wiki/Church_encoding" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Church_encoding</a></p>

<p>Is there a way to e... | trees lambda calculus | 1 | Encoding (binary) trees using lambda calculus -- (trees lambda calculus)
<p>I'm new to lambda calculus, and I read all kinds of interesting stuff about encoding data types as functions. Church booleans, numbers and lists.
<a href="https://en.wikipedia.org/wiki/Church_encoding" rel="nofollow noreferrer">https://en.w... | habedi/stack-exchange-dataset |
96,924 | Proving Postorder Traversal's Time Complexity | <p>I am looking at the following algorithm for performing a <strong>Postorder Traversal of a binary tree</strong></p>

<pre><code>POSTORDER(root):
if root != nullthen
 POSTORDER(root.left);
 POSTORDER(root.right);
 visit root;
end if;
</code></pre>

<p>I am supposed to b... | algorithms algorithm analysis binary trees graph traversal | 1 | Proving Postorder Traversal's Time Complexity -- (algorithms algorithm analysis binary trees graph traversal)
<p>I am looking at the following algorithm for performing a <strong>Postorder Traversal of a binary tree</strong></p>

<pre><code>POSTORDER(root):
if root != nullthen
 POSTORDER(root.left);&#... | habedi/stack-exchange-dataset |
96,932 | Relations between different time complexities of an algorithm | <p>If I have an algorithm A that, in the worst case, has a lower bound of $Ω(n\log n)$ and an upper bound of $O(n²)$, how can I go about determining possible time complexities for best and average cases?</p>

<p>For instance, only one of the following is possible:</p>

<p>a. Best case $\Theta(n)$</p>&#x... | algorithm analysis time complexity | 1 | Relations between different time complexities of an algorithm -- (algorithm analysis time complexity)
<p>If I have an algorithm A that, in the worst case, has a lower bound of $Ω(n\log n)$ and an upper bound of $O(n²)$, how can I go about determining possible time complexities for best and average cases?</p>

<... | habedi/stack-exchange-dataset |
96,939 | Prove that it is undecidable whether a given LBA accepts a regular set | <p>I know for an LBA the emptiness problem is undecidable. However I am not clear on how to reduce the halting problem of Turing machines to this as LBAs are strictly computationally less powerful than Turing machines. Or should I approach with reductions using computational histories of a Turing machine on an input.</... | complexity theory turing machines undecidability linear bounded automata | 1 | Prove that it is undecidable whether a given LBA accepts a regular set -- (complexity theory turing machines undecidability linear bounded automata)
<p>I know for an LBA the emptiness problem is undecidable. However I am not clear on how to reduce the halting problem of Turing machines to this as LBAs are strictly comp... | habedi/stack-exchange-dataset |
96,940 | How can average queuing delay exceed 1 if the $La/R \leq 1$? | <p>Consider a router with a packet arrival rate of $a$ where each packet has $L$ bits, and a transmission rate (in bits/s) of $R$. My textbook describes the traffic intensity as $\frac{La}{R}$ and states that as $\frac{La}{R} \to 1$, the average queueing delay approaches $\infty$. I'm just struggling to understand why ... | computer networks | 1 | How can average queuing delay exceed 1 if the $La/R \leq 1$? -- (computer networks)
<p>Consider a router with a packet arrival rate of $a$ where each packet has $L$ bits, and a transmission rate (in bits/s) of $R$. My textbook describes the traffic intensity as $\frac{La}{R}$ and states that as $\frac{La}{R} \to 1$, th... | habedi/stack-exchange-dataset |
96,945 | Language equivalent states in a deterministic parity automaton | <p>Given a deterministic parity automaton $\mathcal{A}$ with state set $Q$ and a state $q \in Q$, we denote with $\mathcal{A}_q$ the same automaton with initial state $q$. Two states $p$ and $q$ are language equivalent if $L(\mathcal{A}_p) = L(\mathcal{A}_q)$.</p>

<p>What is an efficient algorithm to compute t... | algorithms formal languages automata omega automata | 1 | Language equivalent states in a deterministic parity automaton -- (algorithms formal languages automata omega automata)
<p>Given a deterministic parity automaton $\mathcal{A}$ with state set $Q$ and a state $q \in Q$, we denote with $\mathcal{A}_q$ the same automaton with initial state $q$. Two states $p$ and $q$ are l... | habedi/stack-exchange-dataset |
96,946 | sha256 computability class | <p>Is it correct to say that since the SHA256 function domain is finite (<a href="https://stackoverflow.com/questions/17388177/is-there-a-limit-for-sha256-input">as reported here)</a> we can build a DFA that calculates this function (i.e. trivially a giant lookup table)?</p>

<p>Furthermore, if we eliminate the... | computability | 1 | sha256 computability class -- (computability)
<p>Is it correct to say that since the SHA256 function domain is finite (<a href="https://stackoverflow.com/questions/17388177/is-there-a-limit-for-sha256-input">as reported here)</a> we can build a DFA that calculates this function (i.e. trivially a giant lookup table)?</p... | habedi/stack-exchange-dataset |
96,947 | Why not polynomial-space reductions for $PSPACE$-hardness? | <p>A language $L'$ is $PSPACE$-hard if for every $L \in PSPACE$ we have $L \le_p L'$.</p>

<p>Here $L \le_p L'$ means that $L$ is polynomial-<em>time</em> reducible to $L'$.</p>

<p>Why does we use time reductions instead of space reductions in this situation?</p>
 | complexity theory time complexity space complexity | 1 | Why not polynomial-space reductions for $PSPACE$-hardness? -- (complexity theory time complexity space complexity)
<p>A language $L'$ is $PSPACE$-hard if for every $L \in PSPACE$ we have $L \le_p L'$.</p>

<p>Here $L \le_p L'$ means that $L$ is polynomial-<em>time</em> reducible to $L'$.</p>

<p>Why doe... | habedi/stack-exchange-dataset |
96,953 | Where can we put primitive recursive functions in Chomsky hierarchy? | <p>I am currently studying recursion theory, but I cannot really understand where to put the Primitive Recursive functions in the Chomsky hierarchy.</p>

<p>In my understanding, Primitive Recursive functions contain all finite languages but are not contained in Regular expressions.
For instance, I think tha... | chomsky hierarchy | 1 | Where can we put primitive recursive functions in Chomsky hierarchy? -- (chomsky hierarchy)
<p>I am currently studying recursion theory, but I cannot really understand where to put the Primitive Recursive functions in the Chomsky hierarchy.</p>

<p>In my understanding, Primitive Recursive functions contain all ... | habedi/stack-exchange-dataset |
96,956 | Does $\mathrm{SUBEXP}\subset \mathrm{P}/\mathrm{poly}$ imply anything? | <p>The assumption $\mathrm{SUBEXP}\subset \mathrm{P}/\mathrm{poly}$ seems to yield nothing interesting at all. Is that true?</p>
 | complexity theory | 1 | Does $\mathrm{SUBEXP}\subset \mathrm{P}/\mathrm{poly}$ imply anything? -- (complexity theory)
<p>The assumption $\mathrm{SUBEXP}\subset \mathrm{P}/\mathrm{poly}$ seems to yield nothing interesting at all. Is that true?</p>
 | habedi/stack-exchange-dataset |
96,962 | Mathematical proof for why gradient descent algorithm always converges | <p>I am currently learning machine learning and I stumbled across gradient descent. I understand why the algorithm always converges to the global/local minimum when the learning rate is small enough in my head, but is there a mathematical proof? </p>

<p>Thank you!</p>
 | machine learning | 1 | Mathematical proof for why gradient descent algorithm always converges -- (machine learning)
<p>I am currently learning machine learning and I stumbled across gradient descent. I understand why the algorithm always converges to the global/local minimum when the learning rate is small enough in my head, but is there a m... | habedi/stack-exchange-dataset |
96,972 | Median of medians: bound on pivot position | <p>If I understand correctly (from reading Wikipedia), median-of-medians pivot selection makes quickselect $O(n)$ because the pivot is guaranteed to be in between the 30th and 70th percentiles and so at least 30% of elements will be removed in each selection pass.</p>

<p>But I have tried to prove this percenta... | algorithms complexity theory algorithm analysis sorting selection problem | 1 | Median of medians: bound on pivot position -- (algorithms complexity theory algorithm analysis sorting selection problem)
<p>If I understand correctly (from reading Wikipedia), median-of-medians pivot selection makes quickselect $O(n)$ because the pivot is guaranteed to be in between the 30th and 70th percentiles and s... | habedi/stack-exchange-dataset |
96,984 | Find the lexicographically smallest order of N numbers such that the total of N numbers <= Threshold value | <p>GIven a number N, Threshold T and an array A. Find the lexicographically smallest order of N numbers from A such that the total of these N numbers is <= T.</p>

<p>This question is a simplification of <a href="https://www.interviewbit.com/problems/tushars-birthday-bombs/" rel="nofollow noreferrer">this dy... | dynamic programming greedy algorithms knapsack problems subset sum coin change | 1 | Find the lexicographically smallest order of N numbers such that the total of N numbers <= Threshold value -- (dynamic programming greedy algorithms knapsack problems subset sum coin change)
<p>GIven a number N, Threshold T and an array A. Find the lexicographically smallest order of N numbers from A such that the tota... | habedi/stack-exchange-dataset |
96,985 | $\mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{P}/\mathrm{poly} \implies \mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{MA}$ | <p>Is anyone able to give a concise proof for the implication stated in the title? This is gonna be in stark contrast to this <a href="https://cs.stackexchange.com/questions/96956/does-mathrmsubexp-subset-mathrmp-mathrmpoly-imply-anything">question</a>.</p>

<hr>

<p>For definition of $\mathrm{strict}$-... | complexity theory complexity classes | 1 | $\mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{P}/\mathrm{poly} \implies \mathrm{strict}$-$\mathrm{SUBEXP} \subset \mathrm{MA}$ -- (complexity theory complexity classes)
<p>Is anyone able to give a concise proof for the implication stated in the title? This is gonna be in stark contrast to this <a href="https://cs.... | habedi/stack-exchange-dataset |
97,008 | Reducing lambda expression to normal form | <p>Can someone explain the steps to reduce
 $$ (\lambda n. \lambda m. \lambda f. \lambda x.\ n\ (m\ f)\ x)\ (\lambda f. \lambda x.\ f\ (f\ x))\ (\lambda f. \lambda x.\ f\ x) $$
to $\lambda y. \lambda z.\ y\ (y\ z)$?</p>
 | lambda calculus functional programming | 1 | Reducing lambda expression to normal form -- (lambda calculus functional programming)
<p>Can someone explain the steps to reduce
 $$ (\lambda n. \lambda m. \lambda f. \lambda x.\ n\ (m\ f)\ x)\ (\lambda f. \lambda x.\ f\ (f\ x))\ (\lambda f. \lambda x.\ f\ x) $$
to $\lambda y. \lambda z.\ y\ (y\ z)$?</p>
 | habedi/stack-exchange-dataset |
97,018 | Finite state automaton for the reverse of the language, and multiple starting states | <p>I am studying about Finite State Automaton, and I found that the when reversing a language (i.e., transforming $L$ to $L^R$), I have to add new start state. Why is that?</p>

<p>Also, can a Finite State Automaton have more than one starting state?</p>
 | automata finite automata closure properties | 1 | Finite state automaton for the reverse of the language, and multiple starting states -- (automata finite automata closure properties)
<p>I am studying about Finite State Automaton, and I found that the when reversing a language (i.e., transforming $L$ to $L^R$), I have to add new start state. Why is that?</p>

... | habedi/stack-exchange-dataset |
97,020 | Recursive DP vs Graph Traversal solutions to path-based problems | <p>I am studying some algorithms interview questions and I am seeing many path-based questions like "if a robot is at the top left of a grid and can only move down or to the right, how many paths can take it to the bottom left of the grid". Or similarly sometimes they ask if the particular destination can be reached a... | algorithms dynamic programming shortest path | 1 | Recursive DP vs Graph Traversal solutions to path-based problems -- (algorithms dynamic programming shortest path)
<p>I am studying some algorithms interview questions and I am seeing many path-based questions like "if a robot is at the top left of a grid and can only move down or to the right, how many paths can take ... | habedi/stack-exchange-dataset |
97,025 | Difference between concatenation of two languages and Intersection of two languages | <p>Please also explain in terms of closure of these languages and plot a difference using specific example.</p>

<p>I am interpreted like this and now i am confused.</p>

<p>S1 and S2 are two languages.</p>

<p>"S --> S1. S2 " ( S1 followed by S2 hence concatenation )</p>

<p>and</p>
... | automata | 1 | Difference between concatenation of two languages and Intersection of two languages -- (automata)
<p>Please also explain in terms of closure of these languages and plot a difference using specific example.</p>

<p>I am interpreted like this and now i am confused.</p>

<p>S1 and S2 are two languages.</p>... | habedi/stack-exchange-dataset |
97,027 | what are the conditions for two NFAs to be equivalent? | <p>It's a pretty basic question, but I couldn't find an explicit answer in the book which I'm using, also when searching for it - I didn't find a reliable reference.</p>

<p>Correct me if I'm wrong but for two DFA's, $A , A'$ -given that $L(A) = L(A')$ doesn't mean that the DFA's are equivalent, so what are th... | automata finite automata | 1 | what are the conditions for two NFAs to be equivalent? -- (automata finite automata)
<p>It's a pretty basic question, but I couldn't find an explicit answer in the book which I'm using, also when searching for it - I didn't find a reliable reference.</p>

<p>Correct me if I'm wrong but for two DFA's, $A , A'$ -... | habedi/stack-exchange-dataset |
97,031 | Differences between Church and Scott encoding | <p>I'm kind of new to lambda calculus and I found this Wikipedia article
<a href="https://en.wikipedia.org/wiki/Mogensen%E2%80%93Scott_encoding" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Mogensen%E2%80%93Scott_encoding</a></p>

<p>The section <strong>Comparison to the Church encoding</strong> ... | lambda calculus | 1 | Differences between Church and Scott encoding -- (lambda calculus)
<p>I'm kind of new to lambda calculus and I found this Wikipedia article
<a href="https://en.wikipedia.org/wiki/Mogensen%E2%80%93Scott_encoding" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Mogensen%E2%80%93Scott_encoding</a></p>

... | habedi/stack-exchange-dataset |
97,050 | Algorithm to solve constraint satisfaction problems | <p>I have group of people. Each person can be described by few characteristics:</p>

<pre><code>age, occupation, city, favorite_color
</code></pre>

<p>I would like to generate 60 random people and combine them into pairs.
With a constrain that no person in a pair can share any characteristic, f... | constraint satisfaction constraint programming | 1 | Algorithm to solve constraint satisfaction problems -- (constraint satisfaction constraint programming)
<p>I have group of people. Each person can be described by few characteristics:</p>

<pre><code>age, occupation, city, favorite_color
</code></pre>

<p>I would like to generate 60 random people an... | habedi/stack-exchange-dataset |
97,055 | Karp hardness of searching for a matching cut | <p>Follow-up question in the series:</p>
<p><a href="https://cs.stackexchange.com/questions/97124/karp-hardness-of-searching-for-a-matching-erosion">Karp hardness of searching for a matching erosion</a></p>
<p><a href="https://cs.stackexchange.com/questions/97167/karp-hardness-of-searching-for-a-matching-split"... | complexity theory graphs np complete np hard np | 1 | Karp hardness of searching for a matching cut -- (complexity theory graphs np complete np hard np)
<p>Follow-up question in the series:</p>
<p><a href="https://cs.stackexchange.com/questions/97124/karp-hardness-of-searching-for-a-matching-erosion">Karp hardness of searching for a matching erosion</a></p>
<p><a ... | habedi/stack-exchange-dataset |
97,061 | Requirements for a TM to halt | <p>I'm just beginning self-studying a course on computability and automata; my textbook, describing a generic Turing Machine, lists two following halt conditions:</p>

<ol>
<li>A TM halts, if for a particular state $q \in Q$, $\delta(q, a) = (q, a, HALT)$ for all $a$ in the alphabet and this $q$ is called a... | turing machines | 1 | Requirements for a TM to halt -- (turing machines)
<p>I'm just beginning self-studying a course on computability and automata; my textbook, describing a generic Turing Machine, lists two following halt conditions:</p>

<ol>
<li>A TM halts, if for a particular state $q \in Q$, $\delta(q, a) = (q, a, HALT)$ f... | habedi/stack-exchange-dataset |
97,065 | $\mathrm{BPEXP} = \mathrm{BPP} \iff \mathrm{BPEE} = \mathrm{BPE}$ | <p>Concerning about a wide variety of complexity classes, I have come up with the above conjecture.</p>

<p>Please, establish the claim in the title formally.</p>
 | complexity theory | 1 | $\mathrm{BPEXP} = \mathrm{BPP} \iff \mathrm{BPEE} = \mathrm{BPE}$ -- (complexity theory)
<p>Concerning about a wide variety of complexity classes, I have come up with the above conjecture.</p>

<p>Please, establish the claim in the title formally.</p>
 | habedi/stack-exchange-dataset |
97,068 | $\mathrm{ZPEXP} = \mathrm{BPP} \iff \mathrm{ZPEE} = \mathrm{BPE}$ | <p>Please, establish the above claim formally. It seems that the structure of complexity classes has so much bizarre features everywhere.</p>

<p>For the $\Longrightarrow$ direction, a padding technique will work out. For the other direction, the time bounds are pushed down, so hardly any current technique woul... | complexity theory | 1 | $\mathrm{ZPEXP} = \mathrm{BPP} \iff \mathrm{ZPEE} = \mathrm{BPE}$ -- (complexity theory)
<p>Please, establish the above claim formally. It seems that the structure of complexity classes has so much bizarre features everywhere.</p>

<p>For the $\Longrightarrow$ direction, a padding technique will work out. For t... | habedi/stack-exchange-dataset |
97,070 | Examples of non-sparse languages | <p>All I could find is an example of <a href="https://cs.stackexchange.com/questions/83795/nontrivial-examples-of-sparse-languages">sparse language</a>.</p>

<p>I understand that I need to design a language whose all strings generation should not be bounded by a polynomial function, but I feel all the languages... | complexity theory formal languages | 1 | Examples of non-sparse languages -- (complexity theory formal languages)
<p>All I could find is an example of <a href="https://cs.stackexchange.com/questions/83795/nontrivial-examples-of-sparse-languages">sparse language</a>.</p>

<p>I understand that I need to design a language whose all strings generation sho... | habedi/stack-exchange-dataset |
97,075 | Optimizing coin splitting - Is this algorithm as fast as I think? | <p>In a recent exam, I've been asked to solve the following problem:</p>

<blockquote>
 <h3>The problem</h3>
 
 <p>Two players play the following game: Given a sequence of coin values
 $v_1,\ldots,v_n (n \vert 2)$ the players take turns in taking one coin
 from either "end" of the seq... | algorithms graphs runtime analysis check my answer | 1 | Optimizing coin splitting - Is this algorithm as fast as I think? -- (algorithms graphs runtime analysis check my answer)
<p>In a recent exam, I've been asked to solve the following problem:</p>

<blockquote>
 <h3>The problem</h3>
 
 <p>Two players play the following game: Given a sequence of c... | habedi/stack-exchange-dataset |
97,099 | Flawed argument in the proof of function extensionality in cubical type theory? | <p>I am reading the lectures about cubical type theory in this <a href="https://github.com/mortberg/cubicaltt" rel="noreferrer">github repo</a>. In <a href="https://github.com/mortberg/cubicaltt/blob/master/lectures/lecture1.ctt#L451" rel="noreferrer">lecture 1</a> the author defines function extensionality the followi... | type theory homotopy type theory | 1 | Flawed argument in the proof of function extensionality in cubical type theory? -- (type theory homotopy type theory)
<p>I am reading the lectures about cubical type theory in this <a href="https://github.com/mortberg/cubicaltt" rel="noreferrer">github repo</a>. In <a href="https://github.com/mortberg/cubicaltt/blob/ma... | habedi/stack-exchange-dataset |
97,105 | Average-case complexity of linear search where half of the elements in the array are duplicates | <p>I know that for an array of size <em>n</em> <strong>distinct</strong> elements, the Average Case complexity for linear search is as follows:</p>

<p>A(n) = $\frac{n + 1}{2}$</p>

<p>However, I am having trouble coming up with the Average Case complexity in the case where half of the elements in the s... | algorithm analysis arrays searching average case | 1 | Average-case complexity of linear search where half of the elements in the array are duplicates -- (algorithm analysis arrays searching average case)
<p>I know that for an array of size <em>n</em> <strong>distinct</strong> elements, the Average Case complexity for linear search is as follows:</p>

<p>A(n) = $\f... | habedi/stack-exchange-dataset |
97,106 | Regular expression for capturing a "C-style" string | <p>I have started to learn automata theory and languages. I am new to regular expressions.</p>

<p>As a use case in real world, I would like to construct a regular expression to accept a c-style string: enclosed by double quotes, allows escape sequences (\" to escape a double quote, \ escape a backslash, \n esc... | regular languages finite automata regular expressions strings string matching | 1 | Regular expression for capturing a "C-style" string -- (regular languages finite automata regular expressions strings string matching)
<p>I have started to learn automata theory and languages. I am new to regular expressions.</p>

<p>As a use case in real world, I would like to construct a regular expression to... | habedi/stack-exchange-dataset |
97,107 | Processor pipelining question: number of instructions per second | <p>A question from the textbook Modern Operating Systems by A. S. Tanenbaum reads as follows:</p>

<blockquote>
 <p>A computer has a pipeline with four stages. Each stage takes the same time to do its work, namely, 1 nsec. How many instructions per second can this machine execute?</p>
</blockquote>
... | computer architecture operating systems | 1 | Processor pipelining question: number of instructions per second -- (computer architecture operating systems)
<p>A question from the textbook Modern Operating Systems by A. S. Tanenbaum reads as follows:</p>

<blockquote>
 <p>A computer has a pipeline with four stages. Each stage takes the same time to do ... | habedi/stack-exchange-dataset |
97,108 | Are multiple interrupts generated when I hold down a key on my keyboard? | <p>When I hold down a key on my keyboard, a continuous stream of characters is generated and is displayed on the text editor, Now since pressing a key is a hardware interrupt, so does holding down a key continuously generates interrupt signals and does CPU constantly keeps acknowledging those ? or something else happen... | computer architecture operating systems cpu | 1 | Are multiple interrupts generated when I hold down a key on my keyboard? -- (computer architecture operating systems cpu)
<p>When I hold down a key on my keyboard, a continuous stream of characters is generated and is displayed on the text editor, Now since pressing a key is a hardware interrupt, so does holding down a... | habedi/stack-exchange-dataset |
97,111 | Maximizing quantities/length in buckets to match each other | <p>I have a use case, real one, and I'm trying to come up with an efficient maximization algorithm to solve it. </p>

<p>I'll try to simplify, with a simple analogue, and after will explain the real world use case:</p>

<h3>Analogue:</h3>

<p>Imagine you have 2 buckets of springs.
Each spri... | algorithms optimization matching | 1 | Maximizing quantities/length in buckets to match each other -- (algorithms optimization matching)
<p>I have a use case, real one, and I'm trying to come up with an efficient maximization algorithm to solve it. </p>

<p>I'll try to simplify, with a simple analogue, and after will explain the real world use case... | habedi/stack-exchange-dataset |
97,120 | Defining a graph decision problem not in NP | <p>I have been doing some research online looking for graph problems that are decidable but not in NP. I have found the concept of succinct graphs, which if I understand properly, consist of making the input small (i.e. supplying the graph structure using a small amount of memory) so that when properties of the graph a... | np hard decision problem | 1 | Defining a graph decision problem not in NP -- (np hard decision problem)
<p>I have been doing some research online looking for graph problems that are decidable but not in NP. I have found the concept of succinct graphs, which if I understand properly, consist of making the input small (i.e. supplying the graph struct... | habedi/stack-exchange-dataset |
97,122 | How to implement recursive solution with large number or parameterized possible next steps | <p>I was looking at a recursive solution for the robot on a grid problem which basically states that there is a robot on the top left corner on a grid and you are supposed to find a path to the bottom right of the grid and the robot can only move down or to the right. Below is the recursive function for the solution</... | dynamic programming recursion | 1 | How to implement recursive solution with large number or parameterized possible next steps -- (dynamic programming recursion)
<p>I was looking at a recursive solution for the robot on a grid problem which basically states that there is a robot on the top left corner on a grid and you are supposed to find a path to the ... | habedi/stack-exchange-dataset |
97,124 | Karp hardness of searching for a matching erosion | <p>First, read the previous question: <a href="https://cs.stackexchange.com/questions/97055/karp-hardness-of-searching-for-a-matching-cut/">Karp hardness of searching for a matching cut</a></p>

<p>As mentioned in the supposed-to-be-comment answer in that question, without the requirement of cardinality $k$, th... | complexity theory graphs np complete np hard np | 1 | Karp hardness of searching for a matching erosion -- (complexity theory graphs np complete np hard np)
<p>First, read the previous question: <a href="https://cs.stackexchange.com/questions/97055/karp-hardness-of-searching-for-a-matching-cut/">Karp hardness of searching for a matching cut</a></p>

<p>As mentione... | habedi/stack-exchange-dataset |
97,135 | Maximum induced tree | <p>Given an undirected graph $G(V, E)$, how hard is it to decide whether it has an induced tree consisting of $k$ vertices, where $k$ is also given in the input?</p>
 | np complete np hard | 1 | Maximum induced tree -- (np complete np hard)
<p>Given an undirected graph $G(V, E)$, how hard is it to decide whether it has an induced tree consisting of $k$ vertices, where $k$ is also given in the input?</p>
 | habedi/stack-exchange-dataset |
97,148 | Can data be compressed to size smaller than Shannon data compression limit? | <p>I was reading about data compression algorithms and the theoretical limit for data compression. Recently I encountered a compression method called "Combinatorial Entropy Encoding", the main idea of this method is to encode the file as the characters presented in the file, their frequencies and the index of these cha... | information theory data compression | 1 | Can data be compressed to size smaller than Shannon data compression limit? -- (information theory data compression)
<p>I was reading about data compression algorithms and the theoretical limit for data compression. Recently I encountered a compression method called "Combinatorial Entropy Encoding", the main idea of th... | habedi/stack-exchange-dataset |
97,162 | Is there an efficient algorithm for solving tiling puzzles? | <p>As an example of the type of problem, consider Stewart Coffin's Cruiser puzzle:</p>

<blockquote>
 <p>Let R be a 48 × 31 rectangle. Let T be a 30°-60°-90° triangle with
 hypotenuse 34.565 (so legs are 17.2825 and 29.934). Let P be a right
 trapezoid with base 20.0982 and altitudes 27.1178 and ... | algorithms tiling | 1 | Is there an efficient algorithm for solving tiling puzzles? -- (algorithms tiling)
<p>As an example of the type of problem, consider Stewart Coffin's Cruiser puzzle:</p>

<blockquote>
 <p>Let R be a 48 × 31 rectangle. Let T be a 30°-60°-90° triangle with
 hypotenuse 34.565 (so legs are 17.2825 and 29.... | habedi/stack-exchange-dataset |
97,167 | Karp hardness of searching for a matching split | <h2>UPDATE: In 2 days, if no more convincing answer is posted, then bounty of 50 rep. will go to xskxzr. Due to lack of connectedness and a clean & clear cut, the bounty is still open for 2 days. (<em>UTC is now 17 Sep 01:57</em>)</h2>

<p>This is yet another follow-up question in the series:</p>

<... | complexity theory graphs np complete np hard np | 1 | Karp hardness of searching for a matching split -- (complexity theory graphs np complete np hard np)
<h2>UPDATE: In 2 days, if no more convincing answer is posted, then bounty of 50 rep. will go to xskxzr. Due to lack of connectedness and a clean & clear cut, the bounty is still open for 2 days. (<em>UTC is now 17 ... | habedi/stack-exchange-dataset |
97,194 | Is binary-search really required in Chan's convex hull algorithm? | <p>I have a little doubt about <a href="https://en.wikipedia.org/wiki/Chan%27s_algorithm" rel="nofollow noreferrer">Chan's algorithm</a>. </p>

<p>From Wikipedia's description we see that the second phase of an algorithm works with $K = \mathrm{ceil}(\frac{n}{m})$ subsets $Q_i$. The goal of the second phase is ... | algorithms algorithm analysis computational geometry convex hull | 1 | Is binary-search really required in Chan's convex hull algorithm? -- (algorithms algorithm analysis computational geometry convex hull)
<p>I have a little doubt about <a href="https://en.wikipedia.org/wiki/Chan%27s_algorithm" rel="nofollow noreferrer">Chan's algorithm</a>. </p>

<p>From Wikipedia's description ... | habedi/stack-exchange-dataset |
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