# Tagged Questions

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### How does the induction proof work in this example?

Refer to answer 1.1 of this file: http://www.dei.unipd.it/~geppo/AA/DOCS/NPC.pdf From my understanding and this thread, http://math.stackexchange.com/a/928412, we need 3 steps for that proof. ...
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### Is the inverse of MST cycle property always true? Why?

I am trying to find an algorithm which would check for each edge in a given weighted undirected graph whether it belongs to any of the graph's Minimum Spanning Trees. I have found many potential ...
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### Algorithm for length of longest common subsequence

The case of multiple strings. A slight modification of the dynamic programming algorithm for two strings is used as a subroutine. Here is the pseudo code: ...
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### Is bounded waiting ensured in given version of Dekker's solution for critical section problem?

William Stallings discuss various step by step process in developing Dekker's algorithm in his Operating Systems book. In process, he reaches to following version of algorithm (which is incomplete as ...
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### Greedy algorithm correctness proof (UVA 10716)

Given an input string, not necessarily a palindrome, compute the number of swaps necessary to transform the string into a palindrome. By swap we mean reversing the order of two adjacent symbols (UVA ...
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### How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
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### Checking for 4-cycles in a graph

I was reviewing some selected problems on algorithms and time complexity and the notes had the following exercise (ex. 4.3 from book Algorithms by Dasgupta, Vazirani, Papadimitriou): Design and ...
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### Is there a minimum spanning tree including $e$ after removing at most $k$ edges?

Let an undirected, connected graph $G=(V,E)$ with the weight funciton $w:E\to \mathbb{R}$, an edge $e$, and $0<k\in\mathbb{N}$. Describe an algorithm determines if there are at most $k$ edges could ...
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### Proving correctness of an exponentiation routine

I have the following exponentiation routine, which takes $O(\log n)$ steps ...
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### What is the point of the “respect” requirement in cut property of minimum spanning tree?

The cut property stated in terms of Theorem 23.1 in Section 23.1 of CLRS (2nd edition) is as follows. Theorem 23.1 Let $G = (V, E)$ be a connected, undirected graph with a real-valued weight ...
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### 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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### I need help understanding how to prove partial correctness

Please help me understand how I would prove the partial correctness of the below pseudocode with respect to the following predicates: Pre: {n>=0} Post: ...
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### How to use the concept of loop invariant to reduce errors in loops?

Most of time while writing loops I usually write wrong boundary conditions(eg: wrong outcome) or my assumptions about loop terminations are wrong(eg: infinitely running loop). Here is an small example ...
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### How to find loop invariant from weakest precondition?

Consider this code: Precondition: Postcondition: rv == i <==> ∃i, 0 ≤ i ≤ a.length-1, a[i] == key ...
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### Finding a good loop invariant for a powering procedure

Consider the following algorithm for computing integer powers: ...
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### Modal logic axiom S4, transitive and reflexive frame, tableaux solver

I have a difficult problem to solve which as mentioned in the title is related to modal logic axiom S4. So, here is some background knowledge that can be useful: S4 axiom is a class of transitive ...
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### Proof of correctness of A star search algorithm

I've been looking for the proof of correctness for the A star (A*) algorithm but none of the texts and websites offer it. Mostly they are talking about the proof of optimality of the A* algorithm. I'm ...
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### simple iterate algorithm proof by induction

Suppose I have a function where it calculates which bit is larger called LargerBinary. Let's say I have an input 110111;101001, the output will be 110111 and if the input is 110110:110110, the output ...
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### Validate that a threaded binary tree works as intended

I am attempting to validate that my threaded binary tree’s insertion and deletion works as intended. Would it be safe to assume that the following procedure would have tested all corner cases at ...
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### Proof by Reduction: From Empty Language to Halting Problem on Empty Input

Question: Let language $E$ = {$\langle M \rangle$ | $M$ accepts no inputs whatsoever} Let language $H$ = { $\langle M \rangle$ | $M$ halts on an empty string input}. Is it possible to show that $H$ ...
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### Quicksort $T(n)_{best}=\Omega(n\log n)$ proof

About the proof that quicksort has $T(n)_{best}=\Omega(n\log n)$. I can't find this specific aspect anywhere online which is strange. I'm going through a proof for this in a book and I don't ...
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### Structural induction on generic list

In preparation for an exam, I've come upon the following problem. Given the constructors : ...
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### 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 ...
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### Why does this sort algorithm work?

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