sln-HW02

sln-HW02 - CSE450/598 Design And Analysis of Algorithms...

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CSE450/598 Design And Analysis of Algorithms HW02 Grading Keys Please note that you have to typeset your assignment using either L A T E X or Microsoft Word. Hand-written assignment will not be graded. Also, it is recommended that you save a copy of your assignment on general.asu.edu . 1. (10 pts) Exercise 3.2 (p.107) Solution: We can run either BFS or DFS on G to compute all connected components of G in O ( m + n ) time. G does not contain a cycle if and only if none of its connected components contains a cycle. Therefore we may assume that the graph is connected as we can work with each of the connected components. Starting from any node, we run BFS to compute a BFS tree T . If T contains all edges in G , G does not contain a cycle. Otherwise, there must be an edge ( u, v ) in G that is not used by T . We can ±nd the smallest common ancestor of u and v in T in linear time (If u and v are on the same layer, we back o² one layer from both nodes at each step. If u has a higher layer, we ±rst ±nd the ancestor of u which is on the same layer as v , then back o².). Let it be w . Then the w u path, the edge ( u, v ), and the v w path form a cycle of G . It takes O ( m + n ) time to compute all connected components of G . For each connected component of G , our algorithm requires time proportional to the summation of the number of nodes and number of edges in that component. Therefore the algorithm has O ( m + n ) time complexity. Grading Keys:
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sln-HW02 - CSE450/598 Design And Analysis of Algorithms...

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