Ma/CS 6a Class 15: Flows and Bipartite Graphs
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1 //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d Sink
2 //206 Flow in a Nwork Givn a flow nwork G = (V, E,,, c), a flow in G i a funcion f: E N ha aifi Evry E aifi f c. Evry v V, aifi u,v E Toal flow nring v. f u, v = v,w E f v, w Toal flow xiing v. Exampl: Flow Th capacii ar in rd. Th flow i in blu. 2/7 2/ 0/ a 2/ c 0/ b 2/ d / / / / 2
3 //206 Rmindr: Cu A cu i a pariioning of h vric of h flow nwork ino wo S, T uch ha S and T. Th iz of a cu i h um of h capacii of h dg from S o T. 7 a b c d Rmindr: Max Flow Min Cu Max flow min cu horm. In vry flow nwork, h iz of h minimum cu i qual o h iz of h maximum flow. 7 a b c d
4 //206 Edg in Min Cu Problm. Givn a flow nwork V, E,,, c and an dg E, dcrib an algorihm for chcking whhr i in a la on min cu of h nwork. Soluion. Find a max flow. Dcra h capaciy of by on and find a max flow in h nw nwork. By h max-flow min-cu horm, i in a min cu iff h iz of h maximum flow dcrad. Edg in Min Cu Problm. Givn a flow nwork V, E,,, c and an dg E, dcrib an algorihm for chcking whhr i in vry min cu of h nwork. Soluion. Find a max flow. Incra h capaciy of by on and find a max flow in h nw nwork. By h max-flow min-cu horm, i in vry min cu iff h iz of h maximum flow incrad. 4
5 //206 Rmindr: Maching A maching of an undircd graph G i a of vrx-dijoin dg of G. Th iz of a maching i h numbr of dg in i. A maximum maching of G i a maching of maximum iz. Flow and Bipari Maching Problm. givn a bipari graph G = V V 2, E, u a max flow algorihm o find a maximum maching of G. V V 2
6 //206 Flow and Bipari Maching (2) Dirc h dg from V o V 2. Add a ourc and dg from i o vry vrx of V. Similarly, add a ink. Dirc dg o h righ and capacii o. V V 2 Flow and Bipari Maching () Thr i a bijcion bwn h maching of G and h flow of h nwork. A max maching corrpond o a max flow. V V 2 6
7 //206 Machin and Tak Problm. W ar givn n machin and m ak ha h machin nd o do. Th i h machin prform a mo n i ak. Evry machin ha a ub of h ak ha i i abl o do. Dcrib an algorihm for aigning all of h ak (or aing ha no aignmn xi). Soluion n n 2 n n 4 Tak Machin W wih o find a flow of iz m. 7
8 //206 Indgr and Oudgr Conidr a digraph G = V, E. Th indgr of a vrx v V i h numbr of dg of E ha ar dircd ino i. Th oudgr of a vrx v V i h numbr of dg of E ha ar dircd ou of i. Indgr: 2 Oudgr: Subgraph wih Boundd Dgr Problm. Givn a dircd graph G = V, E, dcrib an algorihm for finding a ub of h dg E E o ha vry vrx of V ha an indgr and an oudgr of xacly in V, E (or announc ha uch a do no xi). 8
9 //206 Soluion Build a bipari graph G = U W, E o ha V = U = W. Wri V = v,, v n, U = u,, u n, and W = w,, w n. For vry dg v i, v j E, w add u i, w j E. Exampl: Building G u w v2 v u 2 w 2 v v v 4 u u 4 w w 4 u w 9
10 //206 Soluion Build a bipari graph G = U W, E o ha V = U = W. Wri V = v,, v n, U = u,, u n, and W = w,, w n. For vry dg v i, v j u i, w j E. E, w add Chck whhr hr xi a prfc maching in G. If o, rurn h dg of h maching. Ohrwi, no valid ub xi. Corrcn An dg v i, v j E conribu o h oudgr of v i and o h indgr of v j. In h bipari graph, i corrpond o u i, w j, conribuing o h dgr of u i, w j. Conidr E E and h dgr ha h dg of E induc. A vrx v i ha indgr and oudgr iff boh u i and w i hav a dgr of. Evry dgr in h bipari ubgraph i iff vry indgr and oudgr i in h inducd ubgraph. 0
11 //206 Dircing a Graph wih Dgr Conrain Problm. Givn an undircd graph G = V, E, which migh no b impl, dcrib an algorihm ha dirc ach of h dg of G, o ha no oudgr i largr han. 2 d ou Soluion W build a flow nwork: v v 2 2 v v 4 4 v A vrx for vry lmn of V. An dg for vry vrx-dg pair ha ar adjacn in G. A vrx for vry lmn of E.
12 //206 Soluion (con.) v v 2 2 v v 4 4 v Thr i a valid orinaion of h dg if and only if h iz of h max flow i E. Corrcn (Skch) Thr i a flow of iz E in h nwork. If and only if Thr i a flow whr vry vrx on h righ id of h nwork rciv a flow of from on of i wo nighbor. If and only if Thr i an orinaion of h dg of G uch ha vry oudgr i a mo. 2
13 //206 Back o MST Problm. Conidr a conncd undircd graph G = V, E, a wigh funcion w: E N, an dg E, and an ingr k > 0. Dcrib an fficin algorihm for chcking whhr w can rmov a mo k dg from G o ha i in vry MST of h ruling graph. Rricion. G ha o rmain conncd afr h rmoval. Sinc k i larg, w ar no allowd o ry all ~ E k of a mo k dg. Warm-up S = u, v. Whn i in vry MST of G? If in vry ohr pah bwn u and v hr i an dg wih a wigh largr han w(). Non of h pah can b h pah bwn u and v in an MST. u v
14 //206 Warm-up #2 S = u, v. Whn i in vry MST of G afr rmoving a mo on dg of E? If afr rmoving a mo on dg vry rmaining pah bwn u and v conain an dg wih a wigh largr han w(). u v Soluion S = u, v. W wih o rmov up o k dg, o ha vry ohr pah bwn u and v conain an dg havir han w(). How can w do ha? Rmov vry E wih w > w() (Th ar ignord; no par of h chon ub). W g a problm ha wa olvd in h prviou lcur: Finding h minimum ub of dg uch ha i rmoval diconnc from (ignoring h wigh). 4
15 //206 A Small Iu Acually, w only olvd h problm of finding a minimum numbr of diconncing dg in dircd graph. In h ca of an undircd graph: Rplac vry dg wih a pair of ani-paralll dg. Alo from prviou cla: Thr xi a max flow ha do no u any pair of ani-paralll dg. Th End
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