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1 Annoncemen CSEP Applied Algorihm Richard Anderon Lecre 9 Nework Flow Applicaion Reading for hi week Nework flow applicaion Nex week: Chaper 8. NP-Compleene Final exam, March 8, 6:0 pm. A UW. hor In cla (CSE 0 / CSE 0) Comprehenie 67% po miderm, % pre miderm Nework Flow Reiew Nework flow definiion Flow example Agmening Pah Reidal Graph Ford Flkeron Algorihm C Maxflow-MinC Theorem Nework Flow Definiion Find a maximm flow Flowgraph: Direced graph wih diingihed erice (orce) and (ink) Capaciie on he edge, c(e) >= 0 Problem, aign flow f(e) o he edge ch ha: 0 <= f(e) <= c(e) Flow i conered a erice oher han and Flow coneraion: flow going ino a erex eqal he flow going o The flow leaing he orce i a large a poible a d 0/ 0/ / 0/ / 0/0 /0 b e 0/ 0/ 0/ /0 c f 0/0 g h 0/ / i 0/0 /

2 Reidal Graph Reidal Graph Flow graph howing he remaining capaciy Flow graph G, Reidal Graph G R G: edge e from o wih capaciy c and flow f G R : edge e from o wih capaciy c f G R : edge e from o wih capaciy f /0 /0 0/0 /0 0 0 Agmening Pah Lemma Le P =,,, k be a pah from o wih minimm capaciy b in he reidal graph. b ni of flow can be added along he pah P in he flow graph. /0 /0 /0 0/0 0 0 Ford-Flkeron Algorihm (96) while no done Conrc reidal graph G R Find an - pah P in G R wih capaciy b > 0 Add b ni along in G If he m of he capaciie of edge leaing S i a mo C, hen he algorihm ake a mo C ieraion C in a graph C: Pariion of V ino dijoin e S, T wih in S and in T. Cap(S,T): m of he capaciie of edge from S o T Flow(S,T): ne flow o of S Sm of flow o of S min m of flow ino S Ford Flkeron MaxFlow MinC Theorem There exi a flow which ha he ame ale of he minimm c Show ha a c i he dal of he flow Proe ha he agmening pah algorihm find a maximm flow Gie an algorihm for finding he minimm c Flow(S,T) <= Cap(S,T)

3 Beer mehod of for conrcing a nework flow Improed mehod for finding agmening pah or blocking flow Goldberg Preflow-Ph algorihm Tex, ecion 7. Applicaion of Nework Flow Problem Redcion Redce Problem A o Problem B Coner an inance of Problem A o an inance of Problem B Ue a olion of Problem B o ge a olion o Problem A Pracical Ue a program for Problem B o ole Problem A Theoreical Show ha Problem B i a lea a hard a Problem A Problem Redcion Example Redce he problem of finding he pah in a direced graph o he problem of finding a hore pah in a direced graph Conrc an eqialen minimizaion problem Undireced Nework Flow Undireced graph wih edge capaciie Flow may go eiher direcion along he edge (bjec o he capaciy conrain) 0 0 Mli-orce nework flow Mli-orce nework flow Sorce,,..., k Sink,,..., j Sole wih Single orce nework flow 0 0 Conrc an eqialen flow problem

4 Biparie Maching A graph G=(V,E) i biparie if he erice can be pariioned ino dijoin e X,Y A maching M i a be of he edge ha doe no hare any erice Find a maching a large a poible Applicaion A collecion of eacher A collecion of core And a graph howing which eacher can each which core RA PB CC 0 6 DG AK 0 Conering Maching o Nework Flow Finding edge dijoin pah Conrc a maximm cardinaliy e of edge dijoin pah Theorem Finding erex dijoin pah The maximm nmber of edge dijoin pah eqal he minimm nmber of edge whoe remoal eparae from Conrc a maximm cardinaliy e of eriex dijoin pah

5 Nework flow wih erex capaciie Balanced allocaion Problem 9, Page 9 To make a long ory hor: N injred people K hopial Aign each peron o a hopial wih 0 mine drie Aign N/K paien o each hopial Baeball eliminaion Baeball eliminaion Can he Dinoar win he leage? Remaining game: AB, AC, AD, AD, AD, BC, BC, BC, BD, CD W L An Bee Cockroache Dinoar Can he Fri Flie win or ie he leage? Remaining game: AC, AD, AD, AD, AF, BC, BC, BC, BC, BC, BD, BE, BE, BE, BE, BF, CE, CE, CE, CF, CF, DE, DF, EF, EF W L An 7 Bee 6 7 Cockroache 6 7 Dinoar Earhworm 0 Fri Flie A eam win he leage if i ha ricly more win han any oher eam a he end of he eaon A eam ie for fir place if no eam ha more win, and here i ome oher eam wih he ame nmber of win Ame Fri Flie win remaining game Fri Flie are ied for fir place if no eam win more han 9 game Allowable win An () Bee () Cockroache () Dinoar () Earhworm () 8 game o play AC, AD, AD, AD, BC, BC, BC, BC, BC, BD, BE, BE, BE, BE, CE, CE, CE, DE W L An 7 Bee 6 8 Cockroache 6 9 Dinoar Earhworm Fri Flie 9 Remaining game AC, AD, AD, AD, BC, BC, BC, BC, BC, BD, BE, BE, BE, BE, CE, CE, CE, DE AC AD BC BD BE CE DE A B C D E T

6 Soling problem wih a minimm c Image Segmenaion Open Pi Mining / Tak Selecion Problem Image Segmenaion Separae foregrond from backgrond Redcion o min-c problem S, T i a c if S, T i a pariion of he erice wih in S and in T The capaciy of an S, T c i he m of he capaciie of all edge going from S o T S, T i a c if S, T i a pariion of he erice wih in S and in T The capaciy of an S, T c i he m of he capaciie of all edge going from S o T Image analyi a i : ale of aigning pixel i o he foregrond b i : ale of aigning pixel i o he backgrond p ij : penaly for aigning i o he foregrond, j o he backgrond or ice era A: foregrond, B: backgrond Q(A,B) = S {i in A} a i + S {j in B} b j - S {(i,j) in E, i in A, j in B} p ij Pixel graph o flow graph Minc Conrcion a p p b 6

7 Open Pi Mining Applicaion of Min-c Open Pi Mining Problem Tak Selecion Problem Redcion o Min C problem S, T i a c if S, T i a pariion of he erice wih in S and in T The capaciy of an S, T c i he m of he capaciie of all edge going from S o T Open Pi Mining Mine Graph Each ni of earh ha a profi (poibly negaie) Geing o he ore below he rface reqire remoing he dir aboe Te drilling gie reaonable eimae of co Plan an opimal mining operaion Deermine an opimal mine Generalizaion Precedence graph G=(V,E) Each in V ha a profi p() A e F if feaible if when w in F, and (,w) in E, hen in F. Find a feaible e o maximize he profi

8 Min c algorihm for profi maximizaion Conrc a flow graph where he minimm c idenifie a feaible e ha maximize profi Precedence graph conrcion Precedence graph G=(V,E) Each edge in E ha infinie capaciy Add erice, Each erex in V i aached o and wih finie capaciy edge Show a finie ale c wih a lea wo erice on each ide of he c The ink ide of a finie c i a feaible e No edge permied from S o T If a erex i in T, all of i anceor are in T Infinie Finie If p() > 0, cap(,) = p() cap(,) = 0 If p() < 0 cap(,) = -p() cap(,) = 0 If p() = 0 cap(,) = 0 cap(,) = 0 Seing he co - 0 Enmerae all finie, c and how heir capaciie - - 8

9 Minimm c gie opimal olion Why? Comping he Profi Co(W) = S {w in W; p(w) < 0} -p(w) Benefi(W) = S {w in W; p(w) > 0} p(w) Profi(W) = Benefi(W) Co(W) - - Maximm co and benefi C = Co(V) B = Benefi(V) Expre Cap(S,T) in erm of B, C, Co(T), Benefi(T), and Profi(T) Smmary Conrc flow graph Infinie capaciy for precedence edge Capaciie o orce/ink baed on co/benefi Finie c gie a feaible e of ak Minimizing he c correpond o maximizing he profi Find minimm c wih a nework flow algorihm 9