CS 410/584, Algorithm Design & Analysis, Lecture Notes 5

Transcription

1 CS 4/584,, Ford-Fulkeron Mehod Flow maximizaion in a nework (graph) wih capaciie Baic idea: Find a pah from ource o arge ha ill ha flow capaciy (augmening pah) Add he maximum flow allowed along hi pah Repea unil David Maier 1 Iue 1. How do we accoun for flow by. Doe adding an augmening pah lead o a legal flow?. Will hi proce converge? 4. If o, will i lead o David Maier 1

2 CS 4/584,, Problem Formulaion Direced graph G = (N, E) Two pecial node: Aume for any node v N, here are pah Capaciy c(v,w) 0 If (v, w) no an edge, hen Flow f(v, w) can be f:c v w v w David Maier Deail Aume no uele flow beween node Poiive flow in only one direcion 1:0 :0 v w v w 5: : David Maier 4

3 CS 4/584,, Legal Flow f 1. f(v, w) c(v, w). f(v, w) = -f(w, v). For any v, Σ Σf(v, w) = 0 David Maier 5 Inpu = Oupu Conider node f(,1) + f(,1) + f(,8) + f(,) 4:8 : 1 8 1: :5 1 7:8 : 1 8 : 5:5 1 David Maier

4 CS 4/584,, Value of Flow f Toal flow ou of ource f = Σf(, w) :18 : 8 4:4 9 David Maier 7 Reidual Capaciy for a Flow Reidual Capaciy beween node v and w: r(u,v) = c(u,v) f(u,v) : 7 r(,7) = c(,7) f(,7) r(7,) = c(7,) f(7,) David Maier 8 4

5 CS 4/584,, Reidual Graph for Flow f R = (N, E ) E = {(v,w) r(v,w) } Capaciie i are reidual capaciie i for f David Maier 9 :5 0:0 1 0:15 :1 Example 0: :4 : 7 0: 8:0 4:15 :5 7: 5 1: 4: : David Maier 5

6 CS 4/584,, Capaciy of a Pah Minimum capaciy edge Add a flow of along he pah :5 0:0 1 :1 0:15 0: :4 : 0: 7 8:0 4:15 :5 7: 5 1: 4: : 8 David Maier 11 Capaciy? I he Reul a Legal Flow? Skew ymmery? Conervaion? :4 0:15 David Maier 1

7 CS 4/584,, :5 :0 1 :15 :1 Example 0: 4:4 : 7 0: :0 4:15 :5 7: 5 1: 4: : David Maier 1 :5 :0 1 :15 :1 Example 0: 4:4 : 7 0: :0 4:15 :5 7: 5 1: 4: : David Maier 14 7

8 CS 4/584,, Would eem o: I i a Maximum Flow? Have a group of edge ha divide from and :5 :0 1 :15 :1 1: 4:4 : 0: :0 7 5:15 :5 7: 5 : 5: : 8 David Maier 15 Cu of a Graph Divide node of N ino wo group S, T :5 :0 1 :1 4:4 :15 1: : 7 0: :0 5:15 :5 7: 5 : 5: : 8 Ne flow acro cu Σf(v, w) Capaciy acro cu Σc(v, w) David Maier 1 8

9 CS 4/584,, Reul Ne flow acro any cu f i bounded above by Max-flow/min-cu heorem 1. f i maximum flow. reidual graph ha no. f i capaciy of David Maier 17 Conider reidual graph R wih no augmening pah S = {v } T = N - S Mu have Claim ha for (v,w) wih v S, T, mu have Suppoe no. Then r(v,w) > 0. Then R ha v w David Maier 18 9

10 CS 4/584,, Baic Implemenaion Sar wih 0 flow Repea Add flow along an augmening pah Doe i alway converge? Ye, if capaciie are ineger. Flow grow by a lea How long doe i ake? If you pick augmening pah arbirarily O( E f* ) David Maier 19 Sar wih 0 flow Edmond-Karp Algorihm Repea Add flow along an augmening pah wih Time complexiy no longer depend on value of maximum flow O( N E ) ime Inuiion: Lengh of hore pah o a node v in reidual graph Each addiion of flow increae diance o one node Diance o a node v can be increaed a mo David Maier 0

11 CS 4/584,, Edmond-Karp Example David Maier 1 11