AlgMidTermFall2006

AlgMidTermFall2006 - Algorithms CSE5811 MidTerm Fall2006...

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Algorithms CSE5811 MidTerm Fall2006 Points 40 Time 70 min 1. Describe what does the following algorithm produce, and analyze its asymptotic time- complexity. [For analyzing complexity, presume n = 2 k , for some positive integer k .] Algorithm Unknown (rational a , positive int n ) If n = = 1 return a Else If n %2 = = 0 then Return Unknown ( a , n /2)* Unknown ( a , n /2) Else Return a * Unknown ( a , ( n -1)/2)* Unknown ( a , ( n -1)/2) End algorithm. 2. For a weighted undirected graph G a culprit set C of arcs has the following property: if all the elements of C are removed from G it becomes acyclic (tree), and a min-culprit set M of arcs is such that the aggregate weight of arcs in M is minimum for all such sets C . Write a greedy algorithm to find M given a graph G . [Hint: First try to find appropriate G M .] 3a. The following is a directed graph G : [ a: b , d ], [ b: c ], [ c: a ], [ d: e ], [ e: f ], [ f: d ], where [ a: b , d ] means there is a directed arc in G from a to b and from a to d . Draw G . You are to find the strong connected components (SCCs) of the graph by running the relevant algorithm. Show (i) the first DFS-spanning tree of G starting with the node a , along with the post-order traversal numbers; (ii) the post-order traversal numbers shown on the nodes of the reverse graph G’ ; and (iii) the second set of DFS spanning trees on the reverse graph G’ , which identifies the SCCs of

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AlgMidTermFall2006 - Algorithms CSE5811 MidTerm Fall2006...

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