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Undergraduate
By
JeffE
on June 6, 2012 | Updated Dec. 6, 2017
Longest increasing digital subsequence
Let \(S[1..n]\) be a sequence of integers between \(0\) and \(9\). A digital subsequence of \(S\) is a sequence \(D[1..k]\) of integers such that each integer \(D[i]\) is the numerical value of a sub…
Computer Science
Algorithms
dynamic programming
0
Undergraduate
By
rizwanhudda
on Aug. 24, 2012 | Updated Dec. 6, 2017
Second Minimum spanning tree
Given an weighted undirected graph \( G = (V, E)\), and \(w : E \mapsto R^+\). Let T be MST i,e minimum spanning tree of graph G. Second MST is a Tree T' different from T, and its weight is less t…
Computer Science
Mathematics
Algorithms
Graph Theory
trees
0
Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Simple path containing three given nodes
Let \(G=(V,E)\) be an undirected graph. Describe a linear time algorithm that given \(G\) and three distinct nodes \(u,v,w\) decides whether there is a simple path in \(G\) that contains all of them.
Computer Science
Mathematics
Algorithms
Graph Theory
linear time algorithms
0
Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Diameter and low-degree vertex
Let \(G = (V,E)\) be an undirected connected graph. Suppose \(G\) has a pair of nodes \(s,t\) that are distance \(d\) apart. Show that there is a vertex \(v\in G\) such that the degree of \(v\) is at…
Computer Science
Mathematics
Algorithms
Graph Theory
counting
0
Undergraduate
By
diego
on June 8, 2012 | Updated Dec. 6, 2017
The cube of a connected graph is hamiltonian
Prove that the vertices of any connected graph \(G\) can be listed in a cyclic order so that the distance in \(G\) of every two consecutive vertices is at most \(3\). Moreover, show that this can be …
Computer Science
Mathematics
Algorithms
Graph Theory
hamiltonian cycle
0
Undergraduate
By
JeffE
on June 7, 2012 | Updated Dec. 6, 2017
Maintaining fitstrings
Every non-negative integer can be represented as the sum of distinct positive Fibonacci numbers. (As a warmup exercise, prove this claim!) In other words, instead of a string of bits, we can represe…
Computer Science
Algorithms
amortized analysis
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Counting intersections of chords
Consider \(n\) chords on a circle, each defined by its endpoints. Describe an \(O(n{\log}n)\)-time algorithm to determine the number of pairs of chords that intersect inside the circle. For example, i…
Computer Science
Algorithms
Data Structures
counting
sorting
0
Undergraduate
By
Shiva Kintali
on June 8, 2012 | Updated Dec. 6, 2017
Linear time algorithms on trees
Let \(T(V,E)\) be a tree. Design linear time (i.e., \(O(|V|)\) time) algorithms for the following problems : Find an optimal vertex cover in \(T\). Find a maximum matching in \(T\). Find a maximum i…
Computer Science
Mathematics
Algorithms
Graph Theory
independent set
linear time algorithms
matching
trees
vertex cover
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Randomized QuickSort
Consider Randomized-Quicksort operating on a sequence of \(n\) distinct input numbers. Prove that the expected running time of Randomized-Quicksort is \(O(n{\log}n)\) Prove that for any constant …
Computer Science
Algorithms
Randomized Algorithms
expectation
sorting
1
Undergraduate
By
Shiva Kintali
on Oct. 17, 2012 | Updated Dec. 6, 2017
Comparison-based merging
You are given two sorted lists A and B of length \(m\) and \(n\) (where \(m \geq n\)). Your goal is to merge them into a new sorted list C. Prove that any comparison-based algorithm for merging A …
Computer Science
Algorithms
lower bound
sorting
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