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Unformatted text preview: Fall 2010 ECE608, Homework #12 Solution (1) CLR 34.1-1 (a) If Longest-Path ∈ P, then we can come up with an algorithm Longest-Path- Length ( G, u, v ). Longest-Path-Length ( G, u, v ) 1. for k = | V | − 1 to 1 2. do Ans = Longest-Path ( G, u, v, k ) 3. if ( Ans = “yes”) 4. return k /* path exists */ 5. return 0 /* path does not exist */ If Longest-Path runs in polynomial time, so does Longest-Path-Length . (b) If Longest-Path-Length ∈ P, then we also can come up with an algorithm Longest-Path ( G, u, v, k ). Longest-Path ( G, u, v, k ) 1. Opt ← Longest-Path-Length ( G, u, v ) 2. if k > OPT 3. return “No” 4. else return “Yes” Similarly, if Longest-Path-Length runs in polynomial time, so does Longest- Path . From (a) and (b) , we can conclude that Longest-Path ∈ P if and only if Longest- Path-Length ∈ P. (2) CLR 34.1-6 Assume that L, L 1 , L 2 ∈ P . The following statements hold: • L 1 ∪ L 2 ∈ P since we can decide if x ∈ L 1 ∪ L 2 by deciding if x ∈ L 1 and then if x ∈ L 2 . If either holds then x ∈ L 1 ∪ L 2 otherwise it is not. • L 1 ∩ L 2 ∈ P since we can decide if x ∈ L 1 and then if x ∈ L 2 . If both hold then x ∈ L 1 ∩ L 2 otherwise it is not. • ¯ L ∈ P since x ∈ ¯ L ⇐⇒ x / ∈ L • L 1 L 2 ∈ P . Given a string x of length n denote its substring from index i to j by x ij . We can then decide x by deciding x 1 k ∈ L 1 and x ( k +1) n ∈ L 2 for all the n possible values of k 1 • L ∗ ∈ P . We can prove this by showing that the result holds for L k for all k and thus for ∪ k i =0 L k . We will use induction on k . If k = 0 we only consider the empty language and the result is trivial. Assume that L k ∈ P and consider L k +1 = LL k . The above result on concatenation gives us that LL k ∈ P . (3) CLR 34.2-1 Suppose the certificate y used to verify the language GRAPH ISOMORPHISM is in the form of two lists of vertices v 1 , v 2 , v 3 , ..., v n in G 1 and u 1 , u 2 , u 3 ,..., u n in G 2 with v i corresponding to u i ( i = 1 , 2 , ..., n ). Use DFS (or other polynomial-time algorithms) to explore all the edges in G 1 . Once an edge ( v i , v j ) is found in G 1 , we check in G 2 if there is an edge between u i and u j . If there is, we just go on. If not, G 1 and G 2 are not isomorphic because there isn’t an corresponding edge between u i and u j in G 2 for ( v i , v j ) in G 1 ; we stop and return false. After we explore all the edges in G 1 and all the edges in G 1 have their corresponding edges in G 2 , we do the same thing again by exploring all the edges in G 2 and checking if every edge in G 2 has its corresponding edge in G 1 . If every edge in G 1 ( G 2 ) has its corresponding edge in G 2 ( G 1 ), we can make the conclusion that G 1 and G 2 are isomorphic and return TRUE. Otherwise, they are not isomorphic with the certificate y , and this algorithm will return FALSE....
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This note was uploaded on 01/10/2012 for the course ECE 565 taught by Professor Pai during the Fall '11 term at Purdue.
- Fall '11