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Unformatted text preview: CS 473: Algorithms, Fall 2009 HBS 1 1. a b c d e f g h Draw the DFS tree rooted at A for the above graph. Use alphabetic ordering to break ties. Label the vertices of the tree with their pre(v) : post(v) time. Add in the remaining edges of the graph and label them as forward (F), backward (B), and cross (C) edges. Sort the vertices by their post visit order. 2. Let G be a directed graph and G SCC its strong connected component meta-graph (which is a DAG). Prove or disprove the following. For any DFS of G the vertex with smallest post-visit number is in a sink component of G SCC . 3. Let G = ( V,E ) be an undirected graph with n vertices ( | V | = n ) and m edges ( | E | = m ). Give an O ( n ) time algorithm to check if G has at least two distinct cycles and output them if it does. Assume that the graph is represented using adjacency lists. Note that m can be much larger than n so the algorithm should not check all edges. Hint: What is the structure of a minimal connected graph that has two cycles? Use DFS.of a minimal connected graph that has two cycles?...
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This note was uploaded on 01/22/2011 for the course CS 473 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08