HW2_April2018_solutions.pdf

# 2 describe an algorithm to determine if g has an

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2. Describe an algorithm to determine if G has an infinite path. Prove that it is correct. Solution: Claim 1. Inf ( p ) is a subset of a strongly-connected component We need to prove both ”existence” and ”uniqueness” of such a strongly- connected component: Existence: Let u, v be any arbitrary pair of vertices in the path p . There must be a path from u to v and v to u . Hence, the there must exist a strongly connected component C with size | C | ≥ Inf ( p ) Uniqueness: Suppose there exist one more strongly component contain- ing the vertices in Inf ( p ) Let C 0 be the a different strongly component which containing vertices Inf ( p ), u C v C 0 similarly, there must be paths be- tween u and v . Hence C 0 and C are also strongly connected and C = C 0 . contradiction. Hence, proven. Part(2): 1. Reverse G to G R O ( E ) = O ( V 2 ) 2. Do DFS on G R and select the node v with highest postvisit number O ( V + E ) O ( V 2 ) 2

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3. Explore from v and add the reachable nodes set S into SCCs 4. Remove S from G R : G R = G R - S and repeat 2 until G R = 5. If SCC 6 = return True (Any strongly connected component will contain Infinite path (from part1) ) 4. Suppose we have a directed graph with non-negative edge weights. It is possible for this graph to have multiple shortest paths between two nodes. In class, we saw how to find the length of the shortest path from a node s to all other nodes. Suppose that in addition to finding the length of these shortest paths, we also want to know how many shortest paths there are. Describe
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