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Reachability as Transitive Closure

Reachability as Transitive Closure. Algorithm : Design & Analysis [17]. In the last class …. Undirected and Symmetric Digraph UDF Search Skeleton Biconnected Components Articulation Points and Biconnectedness Biconnected Component Algorithm Analysis of the Algorithm.

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Reachability as Transitive Closure

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  1. Reachability as Transitive Closure Algorithm : Design & Analysis [17]

  2. In the last class… • Undirected and Symmetric Digraph • UDF Search Skeleton • Biconnected Components • Articulation Points and Biconnectedness • Biconnected Component Algorithm • Analysis of the Algorithm

  3. Reachability as Transitive Closure • Transitive Closure by DFS • Transitive Closure by Shortcuts • Washall’s Algorithm for Transitive Closure • All-Pair Shortest Paths

  4. Fundamental Questions • For all pair of vertices in a graph, say, u, v: • Is there a path from u to v? • What is the shortest path from u to v? • Reachability as a (reflexive) transitive closure of the adjacency relation, which can be represented as a bit matrix.

  5. Computing the Reachability by DFS • With each vertex vi, constructing a DFS tree rooted at vi. • For each vertex vj encountered, R[i][j] is set to true. • At most, m edges are processed. • It is possible to set one in more than one row during each DFS. (In fact, if vk is on the path from vi to vj, when vj is encountered, not only R[i][j], but also R[k][j] can be set)

  6. Computing Reachability by Condensation Graph • Note: the elements of the sub-matrix of R for a strong component are all true. • The steps: • Find the strong components of G, and get G, the condensation graph of G. ((n+m)) • Computing reachability for G. • Expand the reachability for G to that for G. (O(n2))

  7. 1 1 1 2 2 2 5 5 5 3 3 3 4 4 4 Transitive Closure by Shortcuts • The idea: if there are edges sisk, sksj, then an edge sisj, the “shortcut” is inserted. Pass two Pass one Note: the triple (1,5,3) is considered more than once

  8. Transitive Closure by Shortcuts: the algorithm • Input: A, an nnboolean matrix that represents a binary relation • Output: R, the boolean matrix for the transitive closure of A • Procedure • void simpleTransitiveClosure(boolean[][] A, intn, boolean[][] R) • int i,j,k; • Copy A to R; • Set all main diagonal entries, rii, to true; • while (any entry of R changed during one complete pass) • for (i=1; in; i++) • for(j=1; jn; j++) • for(k=1; kn; k++) • rij=rij(rikrkj) The order of (i,j,k) matters

  9. Change the order: Washall’s Algorithm • void simpleTransitiveClosure(boolean[][] A, intn, boolean[][] R) • int i,j,k; • Copy A to R; • Set all main diagonal entries, rii, to true; • while (any entry of R changed during one complete pass) • for (k=1; kn; k++) • for(i=1; in; i++) • for(j=1; jn; j++) • rij=rij(rikrkj) k varys in the outmost loop Note: “false to true” can not be reversed

  10. Highest-numbered intermediate vertex The highest intermediate vertex in the intervals (sisk), (sksj) are both less than sk sj sk si A specific order is assumed for all vertices Vertical position of vertices reflect their vertex numbers

  11. Correctness of Washall’s Algorithm • Notation: • The value of rij changes during the execution of the body of the “fork…” loop • After initializations: rij(0) • After the kth time of execution: rij(k)

  12. Correctness of Washall’s Algorithm • If there is a simple path from si to sj(ij) for which the highest-numbered intermediate vertex is sk, then rij(k)=true. • Proof by induction: • Base case: rij(0)=true if and only if sisjE • Hypothesis: the conclusion holds for h<k(h0) • Induction: the simple sisj-path can be looked as sisk-path+sksj-path, with the indices h1, h2 of the highest-numbered intermediate vertices of both segment strictly(simple path) less than k. So, rij(h1)=true, rij(h2)=true, then rij(k-1)=true, rij(k-1)=true(Remember, false to true can not be reversed). So, rij(k)=true

  13. Correctness of Washall’s Algorithm • If there is no path from si to sj, then rij=false. • Proof • If rij=true, then only two cases: • rij is set by initialization, then sisjE • Otherwise, rij is set during the kth execution of (fork…) when rij(k-1)=true, rij(k-1)=true, which, recursively, leads to the conclusion of the existence of a sisj-path. (Note: If a sisj-path exists, there exists a simple sisj-path)

  14. All-pairs Shortest Path • Non-negative weighted graph • Shortest path property: If a shortest path from x to z consisting of path P from x to y followed by path Q from y to z. Then P is a shortest xz-path, and Q, a shortest zy-path.

  15. Computing the Distance Matrix • Basic formula: • D(0)[i][j]=wij • D(k)[i][j]=min(D(k-1)[i][j], D(k-1)[i][k]+ D(k-1)[i][j]) • Basic property: • D(k)[i][j]dij(k) where dij(k) is the weight of a shortest path from vi to vj with highest numbered vertex vk.

  16. All-Pairs Shortest Paths • Floyd algorithm • Only slight changes on Washall’s algorithm. • Routing table

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