hwk1 - What can you say about its running time Problem 4...

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Spring 2011 CMSC 451: Homework 1 Clyde Kruskal Due at the start of class Thursday, September 22, 2011. Problem 1. Let G = ( V, E ) be a directed graph. The reversal of G is a graph G R = ( V, E R ) where the directions of the edges have been reversed (i.e. E R = { ( i, j ) | ( j, i ) E } ). (a) Assuming that G is represented by an adjacency matrix A [1 ..n, 1 ..n ], give an O ( n 2 )-time algorithm to compute the adjacency matrix A R for G R . (b) Assuming that G is represented by an adjacency list Adj[1 ..n ], give an O ( n + e )- time algorithm to compute an adjacency list representation of G R . Problem 2. Consider a rooted DAG (directed, acyclic graph with a vertex – the root – that has a path to all other vertices). Give a linear time ( O ( | V | + | E | )) algorithm to Fnd the length of the longest simple path from the root to each of the other vertices. (The length of a path is the number of edges on it.) Problem 3. Give an algorithm to Fnd all possible topological sorts of a directed graph.
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Unformatted text preview: What can you say about its running time? Problem 4. Consider the incorrect algorithm that tries to Fnd strongly connected compo-nents by processing the nodes in the second pass in order of Fnish time. (a) Brie±y give our intuition of why this is a good idea. (b) Show that the algorithm is incorrect (by giving a counterexample). (c) Brie±y explain why our intuition was incorrect. Problem 5. Do Exercise 6 on page 108 of Kleinberg and Tardos. Problem 6. A bridge in a connected, undirected graph G = ( V, E ) is an edge whose removal diconnects G . Give an e²cient algorithm to Fnd all of the bridges in G . Write the pseudo-code. Try to make your algorithm clean and elegant. Justify the correctness of your algorithm. 1...
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