Section V.6: Warshall s Algorithm to find Transitive Closure
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1 Secton V.6: Warshall s Algorthm to fnd Transtve Closure Defnton V.6.: Let S be the fnte set {v,... n }, R a relaton on S. The adacency matrx A of R s an n x n Boolean (zero-one) matrx defned by f the dgraph D has an edge from v A, = f the dgraph D has no edge from v (Ths s a specal case of the adacency matrx M of a drected graph n Epp p Her defnton allows for more than one edge between two vertces. But the dgraph of a relaton has at most one edge between any two vertces). Warshall s algorthm s an effcent method of fndng the adacency matrx of the transtve closure of relaton R on a fnte set S from the adacency matrx of R. It uses propertes of the dgraph D, n partcular, walks of varous lengths n D. The defnton of walk, transtve closure, relaton, and dgraph are all found n Epp. Defnton V.6.2: We let A be the adacency matrx of R and T be the adacency matrx of the transtve closure of R. T s called the reachablty matrx of dgraph D due to the property that T, = f and only f v can be reached from v n D by a sequence of arcs (edges). Dgraph Implementaton Defnton V.6.: If a,... n, b s a walk n a dgraph D, a v, b v n, n > 2, then v,... and v n are the nteror vertces of ths walk (path). [] In Warshall s algorthm we construct a sequence of Boolean matrces A = W, [] [n] W, W,..., W =T, where A and T are as above. Ths can be done from dgraph D as follows. [] [ W ], = f and only f there s a walk from v wth elements of a subset of {v }as nteror vertces. [ W ] = f and only f there s a walk from {v } as nteror vertces. v wth elements of a subset of Contnung ths process, we generalze to [ k ] [ W ] = f and only f there s a walk from v wth elements of a subset of { v, k } as nteror vertces.
2 Note: In constructng W [k ] from W [ k ] we shall ether keep zeros or change some zeros to ones. No ones ever get changed to zeros. Example V.6. llustrates ths process. Example V.6.: Get the transtve closure of the relaton represented by the dgraph below. Use the method descrbed above. Indcate what arcs must be added to ths dgraph to get the dgraph of the transtve closure, and draw the dgraph of the transtve closure. v v2 v 4 v Soluton: A = Walks {v 2 }, {v 2 4 } and { v } have elements of {v }as nteror [] [] [] vertces. Therefore [ W ] 2, 2 =, [ W ] 2, 4 =, and [ W ], 2 = W [] = The new ones are underlned. We need only consder walks wth v 2 or v and v 2 as nteror vertces snce walks wth v as an nteror vertex have already been consdered. Walks {v }, {v }, and { v } have elements of {v } as nteror vertces. Therefore [ W ], =, [ W ], =, and [ W ], =. There are other walks wth elements of subsets of {v }as nteror vertces, but they do not contrbute any new ones to W. 2
3 W [] 2 = The new ones are underlned. There are walks wth elements of subsets of {v } as nteror vertces. We need only consder walks wth v (possbly along wth v or v 2 or both) as nteror vertces snce walks wth v (but not v ) as nteror vertces have already been consdered. [] However, none of these walks create any new ones n W. We contnue ths process [4] to obtan W. However, any walks we construct wth v 4 as an nteror vertex [4] [] contrbutes no new ones. Therefore, T = W = W = W. Therefore, T = One must add arcs from v 2 2, and v. The graph of the transtve closure s drawn below. v v2 v 4 v PseudoCode Implementaton No algorthm s practcal unless t can be mplemented for a large data set. The followng verson of Warshall s algorthm s found n Bogart s text (pp ). The algorthm mmedately follows from defnton V.6.4. Defnton V.6.4: If A s an m x n matrx, then the Boolean OR operaton of row and row s defned as the n-tuple x = ( x, x2,, xn ) where each xk = ak a k. We do componentwse OR on row and row. Notaton: Let a and a denote the -th and -th rows of A, respectvely. Then we say x = a a. (Italc x represents an n-tuple)
4 Algorthm Warshall Input: Adacency matrx A of relaton R on a set of n elements Output: Adacency matrx T of the transtve closure of R. Algorthm Body: T := A [ntalze T to A] for := to n for := to n f T = then a := a a [form the Boolean OR of row and row, store t n a ] next next end Algorthm Warshall [ ] Note: The matrx T at the end of each teraton of s the same as W n the dgraph mplementaton of Warshall s algorthm. Example V.6.2: Let A = T = Trace the pseudocode mplementaton of Warshall s algorthm on A, showng the detals of each Boolean OR between rows Soluton: = = T = no acton = 2 T = no acton [] = T, = no acton Therefore W = T = A = 2 = T = ( ) OR ( ) = (,, ) = ( ) row of T becomes ( ) = 2 T = row 2 OR row 2 s computed and put nto row 2 however, row 2 OR row 2 = row 2 = T, = no acton At ths stage we have T = W = 4
5 = = T = no acton = 2 T = ( ) OR ( ) = (,, ) = ( ) result s put nto row 2, whch s unchanged = T, = no acton At ths stage T = [] W = W above. We now have the transtve closure. Exercses: () For each of the adacency matrces A gven below, (a) draw the correspondng dgraph and (b) fnd the matrx T of the transtve closure usng the dgraph mplementaton of Warshall s algorthm. Show all work (see example V.6.). (c) Indcate what arcs must be added to the dgraph for A to get the dgraph of the transtve closure, and draw the dgraph of the transtve closure. () A = () A = () A = (2) For the matrx A n example V.6., compute all the Boolean OR operatons that occur n the pseudocode verson of Warshall s algorthm. Wrte separately the matrces that result from a OR operaton n the nner loop. Also convnce yourself that the matrx T at [ ] the end of each teraton of s the same as W n the dgraph mplementaton of Warshall s algorthm. 5
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