Questions tagged [correctness-proof]

Questions that ask for or about correctness proofs of algorithms.

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7
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2answers
5k views

Updating an MST $T$ when the weight of an edge not in $T$ is decreased

Given an undirected, connected, weighted graph $G = (V,E,w)$ where $w$ is the weight function $w: E \to \mathbb{R}$ and a minimum spanning tree (MST) $T$ of $G$. Now we decrease the weight by $k$ of ...
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1answer
263 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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3answers
196 views

Can algorithm discovery be brute forced?

Ultimately, when you compile them to machine code, algorithms are just 1's and 0's. So then - could you brute force the genesis, and thus the discovery of new, useful algorithms? Say, in pursuit of ...
17
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3answers
3k views

Can I use the following method to prove an algorithm is correct?

I'm trying to show that a solution I have obtained via an algorithm is correct. The way I plan on doing this is first by showing that an optimal solution does indeed exist. Then, I plan on showing ...
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0answers
438 views
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3answers
59 views

Optimality of a Greedy Algorithm

If you designed a greedy algorithm to obtain an optimal solution and the algorithm can produce different combinations of values but still, any of theses combination is an optimal solution. How you ...
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0answers
22 views

Is it the right way to find preorder successor in a binary search tree?

I am confused about the case when the given node is a leaf. The code below does seem to work, but if someone asked me to prove the correctness I'd probably fail. Basically, can someone tell me why <...
0
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1answer
51 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
659 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
41 views

Semantic, total and partial correctness

I encountered the following question: Provide a definition of the semantic correctness of algorithm $A$ with respect to pre-condition $\alpha$ and post-condition $\beta$. A well-presented ...
4
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1answer
2k views

Proof of 0/1 knapsack optimal substructure

I'm trying to understand why exactly the 0/1 knapsack problem actually has the optimal substructure property. Let $E$ be the set of items to consider and $v$ and $w$ the value and weight functions ...
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0answers
17 views

How to write the invariants for one version of binary search insertion point (or leftmost entry) algorithm?

If we compare the binary search algorithm (leftmost or insertion point) on Wikipedia: Algorithm 1: ...
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0answers
59 views

Loop invariant of a search algorithm

I have to come up with a proof of correctness of the following algorithm: GuardedSearch(A; v) Input: an array A of n numbers and a number v Output: an index i such that A[i] = v, or NotFound if no ...
0
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1answer
66 views

Show that a problem is NP-Complete

The problem is, K_longestPath: We are given a graph in which some of the vertices are "cities". No two cities have an edge between them, thus every city must be at distance at least 2 from each ...
1
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1answer
144 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: ...
0
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1answer
106 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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0answers
16 views

Proving a Hoare Triple does not hold

I have this program specification annotated with a Hoare Triple. Trying to prove that is the case. ...
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0answers
38 views

Prove grade-school multiplication algorithm applied to binary numbers

I want to prove that the basic multiplication algorithm is correct when applied to binary numbers. I try to follow the steps described here and here but didn't succeed. The basic implementation ...
0
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1answer
133 views

Reduction from Vertex Cover to Dominating Set

I am trying to reduce the vertex cover (decision) problem to the dominating set (decision) problem in order to prove that the latter is NP-hard. After some research online, I found that many articles ...
2
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1answer
205 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 ...
0
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1answer
77 views

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 := ...
0
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1answer
29 views

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 ...
2
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1answer
25 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 ...
114
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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 ...
4
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2answers
210 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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0answers
53 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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0answers
97 views

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 ...
2
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0answers
59 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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0answers
24 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
19 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
375 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 $11$...
2
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1answer
48 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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47 views
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1answer
58 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 ...
3
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1answer
855 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 ...
0
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0answers
33 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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0answers
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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0answers
27 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 ...
3
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1answer
117 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 ...
3
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1answer
82 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 ...
1
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1answer
76 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 ...
1
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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 ...
3
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3answers
58 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 ...
4
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1answer
125 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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0answers
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 ...
2
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1answer
184 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%...
1
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1answer
73 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 ...
0
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0answers
90 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 ...
2
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0answers
94 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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