11/13/17. directed graphs. CS 220: Discrete Structures and their Applications. relations and directed graphs; transitive closure zybooks

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1 dirctd graphs CS 220: Discrt Strctrs and thir Applications rlations and dirctd graphs; transiti closr zybooks G=(V, E) rtics dgs dgs rtics/ nods Edg (, ) gos from rtx to rtx. in-dgr of a rtx: th nmbr of dgs pointing into it. ot-dgr of a rtx: th nmbr of dgs pointing ot of it. ttp://datamining.typpad.com/gallry/blog-map-gallry.html G=(V, E) rtics dgs dgs dirctd graphs in-dgr of a rtx: th nmbr of dgs pointing into it. in-dgr() = { (, ) E } ot-dgr of a rtx: th nmbr of dgs pointing ot of it. trminology A dirctd graph (or digraph) is a pair (V, E). V is a st of rtics, and E, a st of dirctd dgs, is a sbst of V V. Th rtx is th tail of th dg (, ) and rtx is th had. If th had and th tail of an dg ar th sam rtx, th dg is calld a slf-loop. ot-dgr() = { (, ) E } rtics/ nods Exampl: Th wb. What ar th rtics/dgs? 1

2 matrics An n m matrix or a st S is an array of lmnts from S with n rows and m colmns. Th ntry in row i and colmn j is dnotd by A i,j. A matrix is calld a sqar matrix if th nmbr of rows is qal to th nmbr of colmns. adjacncy matrix A dirctd graph G with n rtics can b rprsntd by an n n matrix or th st {0, 1} calld th adjacncy matrix for G. If matrix A is th adjacncy matrix for a graph G, thn A i,j = 1 if thr is an dg from rtx i to rtx j in G. Othrwis, A i,j = 0. adjacncy matrix What ar th missing als in th following adjacncy matrix? walks A walk from 0 to l in a dirctd graph G is a sqnc of altrnating rtics and dgs that starts and nds with a rtx: 0,( 0, 1 ), 1,( 1, 2 ), 2,..., l 1,( l 1, l ), l 4 3 A 2,1 =? A 4,3 =? 2 1 2

3 walks walks, circits, paths, cycls A walk from 0 to l in a dirctd graph G is a sqnc of altrnating rtics and dgs that starts and nds with a rtx: 0,( 0, 1 ), 1,( 1, 2 ), 2,..., l 1,( l 1, l ), l A walk can also b dnotd by th sqnc of rtics: 0, 1,..., l. Th sqnc of rtics is a walk only if ( i-1, i ) E for i = 1, 2,...,l. Th lngth of a walk is l, th nmbr of dgs in th walk. A circit is a walk in which th first rtx is th sam as th last rtx. A sqnc of on rtx, dnotd <a>, is a circit of lngth 0. A walk is a path if no rtx is rpatd in th walk. A circit is a cycl if thr ar no othr rpatd rtics, xcpt th first and th last. Lt R b a rlation from A to B, and lt S b a rlation from B to C. Th composit S!R of R and S is dfind as: S!R = {(a,c) : b sch that arb and bsc} Exampl: Lt R b th rlation sch that arb if a is a parnt of b. What is th rlation R!R? Lt R b a rlation from A to B, and lt S b a rlation from B to C. Th composit S!R of R and S is dfind as: S!R = {(a,c) : b sch that arb and bsc} Exampl: 3

4 Composit rlation on a st: d a a c b b R R 2 c d Th powrs R n of rlationship R can b dfind rcrsily: R 1 = R and R n+1 = R n!r Th statmnt of six-dgrs of sparation can b sccinctly xprssd as ar 6 b for all a,b whr R is th rlation on th st of popl sch that arb if a knows b paths and rlations Th dg st E of a dirctd graph G can b iwd as a rlation. E k : th rlation E composd with itslf k tims. G k : th dirctd graph whos dg st is E k. Th Graph Powr Thorm: Lt G b a dirctd graph. Lt and b any two rtics in G. Thr is an dg from to in G k if and only if thr is a walk of lngth k from to in G. Th transiti closr of a graph G: G + = G 1 G 2 G 3 G 4... In th nion, thr is only on copy of th rtx st and th nion is takn or th dg sts of th graphs. (, ) is an dg in G + if rtx can b rachd from rtx in G by a walk of any lngth. Similarly w can dfin of a rlation R: R + = R 1 R 2 R 3 R

5 Exampls: Lt R b th rlation btwn stats in th US whr arb if a and b shar a common bordr. What is R +? What is R + for th parnt rlation? A = {1, 2, b}. What is for: R = {(1, 1), (b, b)} S = {(1, 2), (2, b), (1, b)} T = {(2, b), (b, 2), (1, 1)} Th transiti closr of a graph G: G + = G 1 G 2 G 3 G 4... If th graph has n rtics: G + = G 1 G 2 G 3... G n Th sam holds for a rlation R. Lt R b a rlation on a finit domain with n lmnts. Thn R + = R 1 R 2 R 3... R n Lmma: Lt G b a graph with n rtics. If thr is a path from to in G, thn thr is sch a path with lngth not xcding n. an algorithm for Lt R b a rlation or a st A. Rpat th following stp ntil no pair is addd to R: ü If thr ar x, y, z A sch that (x, y) R, (y, z) R and (x, z) R, thn add (x, z) to R. 5