Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [correctness-proof]

Questions that ask for or about correctness proofs of algorithms.

2
votes
0answers
43 views

Why is it true that given a monotonic heuristic function, A* can be seen as Dijkstra's algorithm where no node needs to be processed more than once?

Maybe I am missing something very easy and obvious. But, I don't understand why estimate cost of source vertex is subtracted from the overall estimate cost, if heuristic function $h$ is monotonic: $$...
0
votes
1answer
58 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 ...
0
votes
0answers
18 views

A linear ordering of G that puts v before w iff there is no path from w to v [closed]

Is the following a good enough proof? Let $G'$ be the graph $G$ with an edge $e$ added from $v$ to $w$. We note that a linear ordering of the form we are looking for must respect all edges of $G$ ...
2
votes
1answer
63 views

Confused about the correctness proof of Dijkstra's algorithm

In the proof of the correctness of Dijkstra algorithm, there is a lemma stating as follow: Let u be v's predecessor on a shortest path P:s->...->u->v from s to v. Then, If d(u) = δ(s,u) and edge (...
0
votes
1answer
22 views

How this proof of fractional knapsack works?

I don't understand a step in my book proving the fractional knapsack problem: Let value of items $v_1\ge v_2\ge \dots\ge v_n$, and assume $X=\langle x_1, \dots,x_n\rangle$ are the solution by ...
1
vote
1answer
43 views

Proving correctness of inefficient algorithm - Path between two vertices

Consider the following inefficient algorithm that decides if there is a path between two vertices s and t of a directed graph G. Show that the algorithm is correct. In addition, analyze its complexity ...
1
vote
2answers
40 views

Two Problems in understanding the algorithm for computing shortest paths in undirected graphs with possibly negative edge weights

Section 2 of this Lecture Note: Shortest Path Algorithms Luis Goddyn, Math 408 describes an algorithm using Edmonds' Minimum Weight Perfect Matching Algorithm to solve the shortest path problem for ...
0
votes
0answers
10 views
9
votes
0answers
79 views

Can we say McCarthy and Hoare had the same objective in the 60s regarding a mathematical theory of computation?

I don't think there's any way to ask a very precise question here, so this might be considered opinion based. Nevertheless, it seems the question is clear enough because I'm asking whether these two ...
0
votes
1answer
29 views

Is this a valid induction proof example ?

Learning induction proof now, found a "simple" example, which is a bit confusing to me (not sure if it is a valid example). If so, why the IH( suppose a root of rank k has at least $2^k$ vertices in ...
-1
votes
0answers
13 views

Fairly partitioning workload using network flows

I have the following question to answer: A fraternity has n student members. In Fall’18, m courses are being offered and for each course i, some subset $S_i$ of the fraternity members are taking the ...
0
votes
1answer
28 views

Showing that algorithm has STOP property and finding its computational complexity function

The task is to show that given algorithm has STOP property and to find its computational complexity function. $\alpha:$ $n \ge 0$ ...
1
vote
1answer
53 views

proving correctness of algorithm about graphs with DFS

I need to prove/disprove the correctness of the following algorithm: Let G be a simple, undirected and connected graph. The task is to find if the graph contains an odd cycle. The algorithm goes that ...
2
votes
1answer
27 views
1
vote
1answer
34 views

Why is this pushdown automata 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 $11$...
2
votes
0answers
24 views

Challenging exercises for proof correctness [closed]

I would like to know where I can find challenging exercises that ask to prove the correctness of an algorithm. The invariant of most of the exercises I’ve found on the internet are quite easy (...
1
vote
1answer
96 views

Confirmation of alternative correctness proof for Floyd-Warshall's all-pair shortest-path algorithm

The most common proof for Floyd-Warshall's algorithm is an induction proof on the outer-most loop, which says $\delta^k(i,j)=\begin{cases} \min\{\delta^{k-1}(i,j),\delta^{k-1}(i,k)+\delta^{k-1}(k,j)\}...
-1
votes
0answers
25 views

Perfectly Imperfect Masterpieces

I was looking through this problem and I think I understand how the implementation works, but do not understand how I am supposed to get the algorithm and analyze its time complexity. I am new to ...
1
vote
1answer
27 views

Define a length function over $A^{*} \leftarrow{N}$ such that $length(l)$ outputs the length of $l$

Consider the following definitions LIST: $\overline{nil} \ \ \ \ \ \frac{l}{a \ l}$ $a \in A$ $A^* = \mu \widehat{LIST}, \ A^{\infty} = v \widehat{LIST}$ NAT: $\overline{0} \ \ \ \ \ \frac{x}{s(...
1
vote
1answer
72 views

Proving correctness of search algorithms

I've seen correctness proofs for other searching algorithms; however, for this particular algorithm: search in a row-wise and column wise sorted matrix, I'm not able to generate a proper proof. ...
1
vote
1answer
62 views

Correctness proof for greedy algorithm based on ratio

I've an issue stated as follows: We have 10000 jobs to do, each with some length $l_i$ and weight (importance) $w_i$. Our goal is to arrange the schedule of doing these jobs (in other words, ...
3
votes
1answer
35 views

Any finite Graph G with all V have at least degree of 2, is it true that every vertex is necessarily contained IN a cycle?

As title, (note: this questions is asking weather or not all vertices are contained IN a cycle not asking if the G contains a cycle. My attempt is that: So this graph would be an counter example ...
3
votes
1answer
100 views

Relating a proof to a Haskell program

I am trying to relate the following integer square root theorem $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ and its proof to its role as a specification of ...
1
vote
1answer
248 views

Proof of a greedy algorithm concerning “Buy and Resell Problem”

"Buy and Resell Problem" can be described in the following way: There are $n$ cities. For each city, the price of products in this city is given (a positive number). Now a person will travel from ...
1
vote
2answers
40 views

Array contains elements that differ by K correctness proof

I have been puzzling over an algorithm that decides whether a sorted array of numbers contains two numbers that differ by k. I do not intuitively understand why ...
0
votes
1answer
57 views

Is Loop Invariant Proof a form of Induction?

As far as I see, what computer scientists refer to as loop invariant proofs are exact replicas of induction proof. Is it true? Can I state that loop invariant proof implies an induction? Is there a ...
5
votes
2answers
415 views

Find all critical edges for minimum spanning tree

This is a problem from the textbook "Algorithms, 4th edition" by Robert Sedgewick and Kevin Wayne. 4.3.26 Critical edges. An MST edge whose deletion from the graph would cause the MST weight to ...
0
votes
1answer
143 views

proof of correctness for greedy knapsack algorithm

I don't really understand why is statement 1 ≥ statement 2 in the attached picture. From what I understand the negative term in statement 2 must be greater than or equal the negative term in statement ...
0
votes
1answer
178 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 ...
2
votes
0answers
26 views

Maximum boundary edges amount of union of rectangles

I've read that the maximum boundary edges amount of union of $n$ rectangles, named $p$, is bounded by $p \leq n^2 + 4n$ I tried to prove this by induction, but it's seems too difficult to me, can ...
0
votes
1answer
64 views

Any proof of correctness of the Toom-Cook algorithm?

I found the toom-cook algorithm here: http://www.cs.cmu.edu/~ab/Desktop/15-211%20Archive/res00037/Multiplication_1_print.pdf and have been trying to chase down proof of it being correct, but can't ...
0
votes
2answers
189 views

Question regarding Hoare's partitioning scheme and a slight modification to it

This is the pseudocode on wikipedia for Hoare's partitioning scheme: ...
4
votes
1answer
59 views

Prove correctness of the iterative algorithm

Description: Given an array nums and a value val, remove all instances of that value in-place and return the new length. Do not allocate extra space for another array, you must do this by modifying ...
1
vote
1answer
35 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, 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+1. We can ...
2
votes
1answer
136 views

Correctness of lower bound proof

I am working on this exercise with the purpose of learning how to provide proper proofs and I would like to know if my proof for the following problem is correct. Given a sorted array $A$ (of $n$ ...
0
votes
0answers
210 views

Proof by induction of recurrence relation of dynamic programming

I am currently solving the problem using dynamic programming: Description: saying that there are inputs in 2D-array (width = 8, height = 6): ...
2
votes
1answer
126 views

How to prove the correctness of the MinDistance algorithm?

My question is about how to prove the correctness of this algorithm, I know it is not a good algorithm, it is not efficient and can be improved. How could I prove it still returns the desired value? ...
2
votes
1answer
81 views

Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?

I am trying to prove the algorithm for Question 5 in this practice exam. I am trying to prove this algorithm with the following three claims: Suppose we have a graph G, a minimum spanning tree T, ...
1
vote
0answers
60 views

Prove an algorithm. Give directed graph edge weights such that weight of every cycle is 0

I need to construct a graph with the following properties: $w(u, v)$ = $-w(v, u)$, for every edge $(u, v) \in E$ Weight of all $u \leadsto v$ paths is equal, for every $u, v \in V$ (this is zero ...
0
votes
0answers
82 views

Proof of correctness recursive reverse digit function

This is an attempt to understand better recursion. The following recursive function returns the integer obtained by reversing the digits of an input integer. I'm trying to prove its correctness: <...
4
votes
2answers
115 views

Proving an algorithm wrong

So I have this algorithm that outputs the largest value of an array: Input: $A[1,\dots,n]$, $n\geq 1$ Output: Largest value of an array ...
2
votes
2answers
50 views

Limit repetitions in randomized list with each unique element occurring n times

I have a set of 3 elements and need to generate a randomized sequence containing each element n times with the condition that one element can only occur m times in a row. So with elements [0,1,2] n = ...
3
votes
0answers
76 views

Are there are satisfying explanations for why genetic algorithms work?

The following commentator writes: Having studied this extensively back when they were called Genetic Algorithms, I would like to offer a few insights. One of the biggest reasons they fell out ...
0
votes
0answers
23 views

Compute the longest continuous horizontal segment of integer numbers

The following programme computes the longest sequence of integers before it stops at 0. I'm trying to identifying a loop invariant condition for the programme. Clearly, we loop as long as num_i!=0 and ...
0
votes
0answers
40 views

Loop invariant for

The programme returns the number of digits of an integer $n>0$. I still have some difficulties to understand the difference between the loop invariant condition and what the loop should actually ...
3
votes
1answer
192 views

Alternative algorithm for minimum spanning tree construction

Let $\textit{G(V,E)}$ be an undirected connected graph with distinct costs on its edges. Initialize $\textit{T}$ to be any spanning tree of $\textit{G}$. Consider an algorithm which replaces an ...
0
votes
1answer
59 views

Loop invariant condition IsPrime program

I'm new to the concept of loop invariant and I'm trying to figure out the loop invariant for a program that returns if an integer is prime and, if not, one possible factorization. My intuition is that ...
0
votes
0answers
54 views

Proving correctness for computing spans with a stack

Was researching about computing stock spans with a Stack and how the running time of it is $O(n)$. However, how does one prove correctness on it? ...
2
votes
1answer
216 views

Proving correctness of an iterative Fibonacci algorithm

One of the questions in the problem sets that I'm struggling in is this specific number that asks me to prove an iterative Fibonacci algorithm. The algorithm is written below: ...