hw_sol_02 - CS 473: Fundamental Algorithms, Spring 2011 HW...

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CS 473: Fundamental Algorithms, Spring 2011 HW 2 Homework is due by Monday, 23:59:59, February 7 Problem 1 is due by Sunday, 23:59:59, February 6 This homework contains four problems. Read the instructions for submitting homework on the course webpage . Collaboration Policy: For this homework, Problems 2–4 can be worked in groups of up to three students. Problem 1 should be answered in Compass as part of the assessment HW2-Online and should be done individually. 1. (30 pts.) Short questions to be answered on compass individually. 2. (30 pts.) (15 pts.) Four cycles. Design and analyze an algorithm that takes as input an undirected graph G = ( V,E ) and determines whether G contains a simple cycle (that is, a cycle which does not intersect itself) of length four. Its running time should be at most O ( | V | 3 ). You may assume that the input graph is represented either as an adjacency matrix or with adjacency lists, whichever makes your algorithm simpler. Solution : We report that G has a simple cycle of length four if and only if there exist a pair of distinct vertices x and y that have at least two common neighbors. Clearly there are O ( | V | 2 ) pair of distinct vertices. We check if x 6 = y have at least two common neighbors or not in O ( | V | ) time as follows: Consider each vertex z Adj ( x ) and check if z Adj ( y ), which can be done in O (1) time using hashing. The total running time is clearly O ( | V | 3 ). Rubrik : 10pts: If the writeup gives a correct algorithm. If the algorithm is incomplete but gives a key idea that could lead to a correct algorithm, it gets 5pts. 5pts: If the algorithm has a running time O ( | V | 3 ).
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hw_sol_02 - CS 473: Fundamental Algorithms, Spring 2011 HW...

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