Questions tagged [correctness-proof]

Questions that ask for or about correctness proofs of algorithms.

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Confusion about assignment axiom in Hoare logic

I wanted to know if we are given the f.f.g. Hoare triple: {x = 43}x := x + 1{x = 44} How do we show that this is a valid Hoare triple? My attempt was: Using the assignment axiom: {x + 1 = 43} x := ...
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I cannot find an invariant for the following program

I have the following: (|$y=0; x=c$|) while(x > 0){y=y+a; x=x-1;} (|$y= a*c$|) This seems like a fairly simple program and I can intuitively tell that the post ...
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1answer
24 views

Proving inequalities related to Dijkstra's algorithm

Define $spdist(s,t)$ as the distance of the shortest path from vertex $s$ to $t$. Define $IN(v)$ as the set of in-neighbors of $v$. Define $w(u,v)$ as the weight of the edge $(u,v)$. I am asked to ...
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14answers
13k views

How to fool the “try some test cases” heuristic: Algorithms that appear correct, but are actually incorrect

To try to test whether an algorithm for some problem is correct, the usual starting point is to try running the algorithm by hand on a number of simple test cases -- try it on a few example problem ...
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Minimum increment/decrement to change an array into non-decreasing sequence

I was trying to solve a codeforces problem https://codeforces.com/contest/713/problem/C. Basically we have to convert a sequence in minimum increment decrement operations to a non-increasing sequence....
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2answers
180 views

Is asymptotic ordering preserved when taking log of both functions?

In one of my exercise sheets I have the following question; Let $f,g\colon \mathbb{N}\longrightarrow\mathbb{R}$ be positive functions with $f(n) \in O(g(n))$. Prove or disprove; $\ln(f(n)) \in O(\ln(...
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1answer
192 views

Prove that this solution to the Closest 3-Sum problem always works

I was working on a Leetcode problem, 3Sum Closest. I came up with a solution but struck it down because I didn't think it could be correct. But, turns out it was. I want to know why. Here's the ...
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113 views

how to prove correctness of this BFS algorithm?

Given an undirected connected graph, I wrote the following algorithm based on BFS. The algorithm detects wether this graph contains a cycle. If it contains a cycle then prints it. I'm pretty sure that ...
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44 views

Making change optimally

Consider that a currency system has $k$ denominations $d_0, d_1, ... d_{k-1}$. $d_0, d_1, ... d_{k-1}$ are such that $d_0 < d_1 < ... < d_{k-1}$ and $d_i$ divides $d_j$ for all $0<=i<j&...
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How to prove program solves the problem?

I am preparing for the exam on Theory of Programming class. Now I am trying to solve the task from the sample paper: Task description starts here Given the following problem: A problem is given by ...
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44 views

Interval partitioning problem different approach - arrange lectures in minimum number of classrooms

The problem of scheduling lectures in minimum number of classrooms is as follows: Find minimum number of classrooms to schedule all lecture so that no two occur at the same time in the same room. The ...
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22 views

Using induction vs invariants to prove correctness of algorithms

In my algorithms class I have generally been proving algorithms by induction. So for example, given some algorithm $A(n)$ that computes $x$, I show that the algorithm works for some base case, say $A(...
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1answer
41 views

Finding invariant when detecting a cycle

Let consider a connected graph $G = (V, E)$ which is not oriented. One way to detect a cycle in such a graph is : Create an array : seen of size $\mid V \mid$ with seen[i] = false for all $i$ ...
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1answer
608 views

Prove Recursive formula (Dynamic programming) N(C,i)

I've been asked to prove the correctness of the following recursive formula. The formula is trying to define, how many ways you can spend your money C on the i amount of beers. I did the following ...
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1answer
17 views

Loop invariant initialisation confusion

Consider the algorithm LastMatch below, which returns the offset (shift) of the last occurrence of the pattern P in text T, or -1 if P does not occur in T: ...
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2answers
177 views

Why is this pushdown automaton for some palindromes right?

$B = \{w \in \{0,1\}^* | w^R = w, w \text{ length is odd} \}$ Solution: For example: $111$ should be accepted steps are $q_1 \to q_2$ stack: [$\$$] $q_2 \to q_2$ stack: $[\$, 1, 1]$ (using up $11$...
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1answer
44 views

Check if possible to perform n tasks, each between moment b(i) and e(i) and taking 1 time unit

I have such a task at university: we have $n$ tasks, the $i$-th of them can be done between moment $b(i)$ and $e(i)$. If we decide to perform a task in moment $x$, we finish performing it in moment $x+...
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1answer
115 views

Proving correctness of the Newton's Method for finding the square root of a number

I'm trying to prove the correctness of this simple square root calculation algorithm using SPARK: ...
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0answers
41 views

Proof of the average case of the Heap Sort algorithm

Consider the following python implementation of the Heap Sort algorithm: ...
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1answer
53 views

Minimize cost of recursive pairwise sums: how to prove the greedy solution works?

The problem is in this other question. Why does this always work? It's not clear to me how one would use induction. For $n = 3$, a quick calculation shows it works, however, I don't think it ...
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1answer
794 views

How to prove a greedy algorithm that uses the longest increasing subsequence?

Here is the thing, I am solving an problem, and I think, say, I am pretty sure that I have the correct algorithm but I haven't been able to prove it because of my lack of practice prooving greedy ...
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32 views

Using Hoare logic to show an invariant holds or using induction?

I want to know if given a while loop: x = 0 while(x < 5){ x = x + 1 } I want to show that x (at the a ith iteration of the loop), the value of i is ...
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33 views

Where to get proofs of open problems checked? [closed]

I have found what I believe is a proof of related to an open problem in computer science. I've written it up in LaTeX, and I am looking for a way to get it evaluated. I know that cs.se/cstheory.se ...
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21 views

Linearithmic solution to finding closest pairs in an array of N elements

I am reading Algorithms 4ed by Sedgewick and Wayne. I came across this algorithm design question that asks the following: Write a program that given an array of N integers, finds a closest pair: two ...
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1answer
109 views

Prove that the greedy algorithm to remove k digits from a n-digit positive integer is optimal

Given a positive n-digit integer, such as 1214532 (n=7), remove k digits (for example k=4) such that the resulting integer is the smallest one. A greedy algorithm for this would keep removing digits ...
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1answer
81 views

Finding minimum spanning tree of a special form graph

I'm trying to find an efficient algorithm that will find me the minimum spanning tree of an undirected, weighted graph of this particular form: My idea was a recursive solution: Suppose the algorithm ...
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1answer
61 views

In-place matrix transposition using rotations

Some time ago I invented an algorithm for in-place matrix transposition using rotations. A matrix of size N×M is represented as a one-dimensional array of size N*M, which is also the future storage ...
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1answer
28 views

Dubins TSP NP-hardness proof detail

In Le Ny et al.'s paper On the Dubins Traveling Salesman Problem (https://tinyurl.com/y59f7d8x) the authors prove, among other works, that the Dubins Traveling Salesman Problem (DTSP) is NP-hard. I ...
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3answers
57 views

Why is subarray $A[p..k-1]$ empty when $k=p$?

I'm working through a proof of correctness for merge sort. I'm given a loop invariant for a for loop, which makes reference to a subarray $A[p..k-1]$. During the initialization step of the ...
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1answer
124 views

Mistake in a proof of termination phase of Simplex algorithm in CLRS?

There is a pseudo-code for Simplex algorithm in CLRS: The proof consists from three-part loop invariant: Proof We use the following three-part loop invariant: At the start of each iteration ...
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20 views

How to solve Hoare's problem when precondition contains meta symbols?

This is the proram for which I have to prove correctness using Hoare's Axioms: {X = |x|} if(x<=0) x:=-x; else skip; {X=x} This is my solution so ...
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1answer
118 views

Understanding Correctness of Bidirectional Dijkstra

I'm trying to understand the correctness of the bidirectional version of Dijkstras algorithm as mentioned here on slide 10: https://www.cs.princeton.edu/courses/archive/spr06/cos423/Handouts/EPP%...
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1answer
62 views

Do the minimum spanning trees of a graph have the same number of edges with a given weight?

I'm asking about the answer here: Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight? I didn't understand the best answer here Choose edge $e \in ...
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55 views

Confused with the proof that Edmonds-Karp always monotically increases the shortest-paths

The proof for the lemma from "Introduction to Algorithms by Cormen et. al." is not clear for me. I can't comprehend a few things. Here is a lemma and its proof. My questions are below. The notation ...
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70 views

Alternative proof of the fact that heapify can be linear-time

As an exercise, I'm trying to prove by myself that constructing a binary heap from an array in-place can be $O(N)$. I've come up with an idea, but I'm not sure about its correctness. Firstly, I ...
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0answers
73 views

Hoare triple: Loop invariant and partial correctness

Below there is Hoare triple in which variable $a$ is an array of integers, $len$, $x, i$ are integer-valued variables, and $r$ is a Boolean-valued variable. I have to provide a loop invariant (using ...
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2answers
3k views

Why is this a proof by contradiction for this algorithm? Isn't this a direct proof instead?

First Slide: Find Max(A) // INPUT: A[1..n] - an array of integers // OUTPUT: an element m of A such that m >= A[j], for all 1 <= j <= A.length max = A[j==1] for j = 2 to A.length if max < A[...
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Can we ignore the postcondition in the Hoare conditional rule when there is a return statement?

I'm proving the correctness of naive string matching using Hoare logic. I have the following pseudocode: ...
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2answers
619 views

Prove that the total distance is minimised (when travelling across the longest path)

Here is the problem: Given a tree $T$, I need to visit every node in the tree once. I can start and end anywhere I want. I would like to travel the least distance possible when doing so. I don't have ...
2
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0answers
83 views

Comparing locally maximal and localy minimal Hamiltonian paths [closed]

Let $K_n$ be a weighted complete graph on $n$ vertices. Two Hamiltonian paths are formed as follows. The first one, $H$, is formed by starting at an arbitrary vertex, and at each stage proceeding from ...
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0answers
71 views

Correct invariant of BFS

I am trying to find a correct invariant of BFS. If we represent a queue as $ Q = [a_0;...; a_n]$ such that : $Q.pop() = a_n$ then I found the following invariant which I think is correct (we denote ...
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1answer
631 views

Proof of correctness of algorithm

Can someone help me prove the correctness of this algorithm: ...
2
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1answer
36 views

Correctness of an algorithm difference between recursive and iteraive

I know that the general strategy to do the correction of an algorithm is as follow : if the algorithm is recursive then prove the correctness using induction if the algorithm is iterative (...
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1answer
56 views

Find an invariant in the minimum algorithm

I have the following simple algorithm to find the smallest element of an array $A$ of numbers: ...
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2answers
3k views

Why do these recurrences determine the number of ways of tiling a 3xN rectangle with 2x1 dominoes?

http://www.algorithmist.com/index.php/UVa_10918 The above link is a solution to UVa 10918 Problem. The problem is based on Dynamic Programming. I am not able to understand this approach to the ...
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2answers
307 views

Proof of QuickSort algorithm correctness

Recently I’ve studied QuickSort and understood its general idea. Basically, we do the following: Pick an element from the array (no matter which one and how in this context) Rearrange elements in ...
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2answers
1k views

Proof of correctness of algorithms (induction)

I am reading Algorithm's Design Manual by S.Skiena and I have a hard time understanding and proving the correctness of algorithms. I should use proof by induction and when we talk about summations and ...
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2answers
134 views

Must the champion of an entire tournament beat the champion of a possible tournament among other players?

I have a list of "players" of a "tournament". Any two adjacent players may "compete", which results in the loser being thrown out of the tournament. Winning is not transitive. The winner of a given ...
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1answer
515 views

Is my proof of my greedy algorithm to find subsequence correct?

Credit to KleinBerg and Taros Book Some of your friends have gotten into the burgeoning field of time-series data mining, in which one looks for patterns in sequences of events that occur over time. ...