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Unformatted text preview: Chapter 9 The Transitive Closure, All Pairs Shortest Paths This chapter studies two related problems that answers the following questions. 1. Can I get there from here? (is there a path from u to v?) 2. What is the shortest path from u to v? Kleene, Warshall, and Floyd all studied these problems. Kleene – the kleene closure, regular languages Warshall – transitive closure Floyd – all pairs of shortest paths There are many applications  networks, air routes, computer connections, flowcharts, electrical connections, passwords, etc. 9.2.1 Definition of transitive closure and adjacency matrix. Definitions: * S a finite set of elements. * binary relation on S is a subset of S X S, call it A. s i is related to s j with the notation s i As j * Can be represented by an adjacency matrix, which is an important relation in itself. { j i As s if true otherwise false ij a = * Equivalence relation and partial orders are additional examples of interesting relations. * The relation can be viewed as a directed graph as we looked at in the previous chapter. G = (S,A). S is the vertices, A as the ordered pairs of edges. * zero matrix – all entries 0. * Notation – ۸ and, ۷ or, ¬ not, + binary or, ∑ multiway or (not exclusive or, which is not used in this chapter) * identity matrix – all entries 0 except the diagonal which is 1. * A relation is transitive iff for all x, y, z in S xAy, yAz implies xAz. * Transitive  from any value s j As i , s i As k implies s j As k (gets to another value). * Reflexive Transitive closure – Let S be a set and let A be a binary relation on S. Let G = (S,A). RTC is a binary relation R defined by: s i Rs j iff there is a path from s i to s j in G * Reflexive since there is a path from each vertex to itself of length zero. * We are to study methods of finding the transitive closure. * Assume the number of elements of S is n. * Assume the number of elements of A is m. * A relation and its transitive closure. A = 00010 00100 01000 00010 01001 R = 01111 01110 01110 01110 11111 9.2.2 Finding the Reachability Matrix by Depthfirst search * Let R ultimately be the reachability matrix.* Let R ultimately be the reachability matrix....
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This note was uploaded on 07/10/2011 for the course COT 4400 taught by Professor Eggen,r during the Fall '08 term at UNF.
 Fall '08
 EGGEN,R
 Algorithms

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