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# Questions tagged [correctness-proof]

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### Minimizing inversions in an array with a single swap

This was asked in the (very) recently concluded Hackerrank Worldcup. Paraphrased: Given a permutation $a$ of integers from $1$ to $N$, how can I minimize the number of inversions by a single swap ...
215 views

### How to prove if an algorithm is reentrant?

I think, maybe some formalism could exist for the task which makes it significantly easier. My problem to solve is that I invented a reentrant algorithm for a task. It is relative simple (its pure ...
1k views

### Algorithm for constructing BST from post-order traversal

Given a post-order traversal of Binary Search tree with $k$ nodes, find an algorithm that constructs the BST. My Algortihm Let $n$ represent the next element to be inserted. Let $P(y)$ ...
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### Is it possible to prove thread safety?

Given a program consisting of variables and instructions which modify these variables, and a synchronization primitive (a monitor, mutex, java's synchronized or C#'s lock), is it possible to prove ...
105 views

### Proving equality between foldl recursive and iterative fold

Hi I have two definitions of fold. I will call them foldl which is recursive and fold$_{itr}$ which is iterative. I am looking for an algebraic proof that the two definitions are equal ideally ...
8k views

### Proof of an Optimal substructure in Dynammic Programming?

Could someone please explain how exactly the proof of optimal substructure property in dynamic programming problems works? They usually say: Let's say the global optimal solution is A, and B is ...
656 views

### Understanding Log(n) Loop Invariant

When attempting to find the following loop invariant for: ...
46 views

### prove sufficient number of comparisons for the merge problem

It is given two subsequences. Their length are following: $2$ and $5$. I can show that lower bound of comparisons is $5$. My problem is that I can't show that $5$ is sufficient number of comparisons ...
1k views

### Why does this sort algorithm work?

The following O(n^2) sorting algorithm works but I can't figure out why. ...
518 views

### Proving correctness of a recursive algorithm using induction

For the program mean(A,n) if n = 1 then return A[n] else return A[n]/n+mean(A,n-1)*(n-1)/n end Show that if the recursive call to ...
342 views

### Using dynamic programming to find the number ofl increasing subsequences [closed]

I got this question today and I'm nowhere near the solution, Given a sequence of real numbers (X1, X2, ..,Xn). write an algorithm as efficient there is, that finds the number of strictly increasing ...
427 views

### Greedy algorithm correctness proof for “Elegant Permuted Sum” (UVa 11158)

Given a sequence of $2 \leq n \leq 50$ numbers $s = (s_1,s_2,...,s_n)$, find a permutation $a = (a_1,a_2,...,a_n)$ of $s$ such that $$\sum_{i=1}^{n-1} |a_i - a_{i+1}|$$ is maximized. I found many ...
64 views

### Which element is at its final position after the partitioning step in Quicksort?

In Algorithms, 4th Edition, I read that after the partitioning step one element is in its final position. The entry a[j] is in its final place in the array, for some j. No entry in a[lo] ...
455 views

### Finding a good loop invariant

I want to prove that the following program is correct. The code takes an array i of length N and a number ...
2k views

### Prove that Ford-Fulkerson can decide if there is more than one min cuts

Probelm: Deciding whether a network flow graph has more than one min cut. Optimal running time: O(V^2*E). I trying to prove the correctness of the next algorithm: run Dinitz to find max-flow and ...
913 views

### Counter example to graph coloring heuristic using BFS

I am considering the following heuristic for the graph coloring problem (i.e. to color a graph $G$ using a minimal number of colors so that no two adjacent vertices have the same color): Explore ...
253 views

### How is Chinese Remainder Theorem used in the proof of correctness for RSA

Question At the very end of (most) proofs of RSA's correctness we have something like $$m^{ed}\equiv m\pmod p$$ $$m^{ed}\equiv m\pmod q$$ Therefore by the Chinese Remainder Theorem (CRT) m^{ed}\...
110 views

### Seemingly non sequitur in proof

I'm trying to understand a small proof in an article about computing lumpability on Markov chains. There is a small detail that I cannot understand, i.e. I don't think it follows from the argument. ...
2k views

### People crossing a bridge (a proof for a greedy algorithm)

The problem Some people are crossing a bridge. Each one takes a different time to pass. Assume the people are sorted by their passing time increasingly. These are the conditions of crossing the ...
444 views

### provability of while loop vs for loop [closed]

I'm abit afraid to ask this question here seeing as I asked it on programmers SE already, the thing is I think the question is more about the underlying theory than the use in practice (or call it ...
533 views

### Given the “programs as proofs” isomorphism, how do we know that the program isn't lying?

I've been studying constructive type theory (CTT) and one of the things that I'm not clear on is the proof part: Proving the correctness of a program in a form of a proof that's nothing but the ...
1k views

### Suurballe's Algorithm: Proof of Correctness

I was reading about Suurballe's algorithm on Wikipedia, for the shortest edge-disjoint paths problem, i.e. given nodes $s$ and $t$ finding a pair of paths between these nodes, whose accumulated weight ...
307 views

### Proving correctness of an AVL-Tree colouring algorithm

I came up with the following recursive algorithm to colour the nodes of an AVL tree so that the resulting tree is red-black. The logic is that the algorithm first colours the root and, recursively, ...
953 views

### Binary Indexed Trees: Why does i & -i work?

I already read this related question on the intuition behind binary indexed trees, and while the answer explains how the tree structure works, it does not really explain how this correlates back to ...
2k views

### Are there programs that never halt and have no non-termination proof?

Like black holes in computer science. We can only know they exist but when we have one of them we will never know it's one of them.
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### Is there a flaw in this Wikipedia proof of cycle property of Minimum Spanning Tree?

On wikipedia, there is a proof for the cycle property of the Minimum Spanning Tree as follows: Cycle Property: For any cycle C in the graph, if the weight of an edge e of C is larger than the ...
815 views

### How to go about proving an algorithm correct?

The algorithm (called as rmax(1,n)) finds the maximum of a list of numbers contained in an array S[1..n]. ...
346 views

### Is there a proof of the recursive algorithm for generating all permutations of a sequence?

For clarity, I attach below a concise implementation of the algorithm in Python. I understand that it checks all possible element swaps, but I don't see how that necessarily means that all possible ...
774 views

### Understanding a proof in the sweep line algorithm when finding all line segment intersections

You have a set of line segments and you want to find all intersections. First sweep line approach: Use a priority queue Q for the events as they come, where each ...
2k views

### Rigorous Proof of Insertion Sort

Currently I self study CLRS book (Outside of any course, so I got no access to an instructor) And I am stuck proving Insertion Sort, The proof in CLRS book is not so formal. Here's the algorithm: <...
4k views

### Proving optimality of a dynamic programming algorithm

We have a string $s$ containing $n \leq 100$ bits. The move we can make on it is erasing from $s$ some substring $x$, but only if $x$ is directly preceded by $x^R$, where $x^R$ means string $x$ ...
5k views

### How to prove correctness of a shuffle algorithm?

I have two ways of producing a list of items in a random order and would like to determine if they are equally fair (unbiased). The first method I use is to construct the entire list of elements and ...
185 views

### How to show that this algorithm for evaluating polynomials works?

I'm having trouble showing how to solve this problem in particular the part where it asks "To Show that the following pseudo-code fragment finds the value of the polynomial..." How do I exactly show ...
392 views

### Hoare Calculus Incorrect Assignment Axiom

I'm currently preparing for an exam and recently came across the following exercise and would like to know whether my solution would be correct. Exercise: Prove that the following axiom is not ...
128 views

### why negative cycle exists if we can relax the edges one more time after running the Bellman Ford Algorithm

We know Bellman Ford is an algorithm to find the negative cycle. And here is the algorithm for Bellman Ford Input: Given a graph G(V,E) and w(e) is weight Output: Return Yes if negative cycle exists. ...
124 views

### Concept used in the proof [closed]

In the paper "Resolution for Quantified Boolean Formulas", I am unable to understand the proof of theorem 3.4. Please help me with the basic concept used on page 4: The concept that I am referring to ...
367 views

### Could program verification techniques prevent bugs of the genre of Heartbleed from occurring?

On the matter of the Heartbleed bug, Bruce Schneier wrote in his Crypto-Gram of 15th April: '"Catastrophic" is the right word. On the scale of 1 to 10, this is an 11.' I read several years ago that a ...
Definition: A preserved invariant of a state machine is a predicate, $P$, on states, such that whenever $P(q)$ is true of a state, $q$, and $q \rightarrow r$ for some state, $r$, then $P(r)$ holds. ...