Network Flow. Data Structures and Algorithms Andrei Bulatov
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1 Nework Flow Daa Srucure and Algorihm Andrei Bulao
2 Algorihm Nework Flow 24-2 Flow Nework Think of a graph a yem of pipe We ue hi yem o pump waer from he ource o ink Eery pipe/edge ha limied capaciy Flow occur when we pump waer hrough he yem. A flow i amoun of waer flowing hrough each pipe How much waer can we pump hrough he yem wihou blowing up any pipe? 20 u
3 Algorihm Nework Flow 24-3 The Formalim Flow Nework: - a digraph G = (V;G) - eery edge e ha capaciy, a nonnegaie number - here i a ingle ource node V - here i a ingle ink node V Node oher han and are called inernal c e
4 Algorihm Nework Flow 24-4 The Formalim (cnd) Flow : A flow i a funcion f: E + R uch ha (1) (Capaciy condiion) For each e E, we hae (2) (Coneraion condiion, Kirchhoff principle) for each node excep and The alue of he flow i e ou of f ( e) f ( e) = e ino 0 f ( e) e ou of f ( e) c e Noe ha Why? f ( e) = e ino e ou of f ( e)
5 Algorihm Nework Flow 24-5 The Problem The Maximum Flow Problem Inance: A flow nework G,, Objecie: Find a flow of maximal alue.
6 Algorihm Nework Flow 24-6 Algorihm: Simple Flow and Reidual Graph Conider a flow nework u Naural idea: puh a flow along a 30 pah Howeer, he flow canno be u improed hi way, bu can be improed in a differen way
7 Algorihm Nework Flow 24-7 Reidual Graph Gien a flow nework G, and a flow f, conruc he reidual graph wih repec o f - he node e of i he ame a G - for each edge e of G wih f ( e ) < include e in wih capaciy c e f (e) (forward edge) - for each edge e = (u,) in G wih f(e) > 0 include e = (,u) wih capaciy f(e) (backward edge) Capaciy of an edge in he reidual graph i called reidual capaciy c e u u
8 Algorihm Nework Flow 24-8 Reidual Graph Saring wih he zero flow - puh a flow along (,u), (u,), (,) uch ha f(,u) = f(u,) = f(,) = 20 - conruc he reidual graph w.r.. f - puh a flow along (,), (,u), (u,).. g(,) = g(,u) = g(u,) = 10 - conruc he reidual graph w.r.. g - we canno puh any flow anymore. - i f + g maximal? u u u
9 Algorihm Nework Flow 24-9 Augmening a Flow Le P be an - pah in boleneck(p,f) denoe he minimal reidual capaciy of he edge of P Augmen(f,P) e b:=boleneck(p,f) for each edge (u,) P do if e=(u,) i a forward edge hen increae f(e) by b ele decreae f(e) by b endfor reurn f Any - pah in i called an augmening pah
10 Algorihm Nework Flow Augmening a Flow (cnd) Le f be he funcion obained afer augmening Lemma f i a flow Proof Capaciy condiion: I uffice o conider arc of P Le e = (u,) P By conrucion boleneck(p,f) i a mo he reidual capaciy of e If e i a forward edge, hen 0 f ( e) f '( e) = f ( e) + boleneck( P, f ) f ( e) + ( c f ( e)) = e c e
11 Algorihm Nework Flow Augmening a Flow (cnd) Proof (cnd) If e i a forward edge, hen 0 f ( e) f '( e) = f ( e) + boleneck( P, f ) f ( e) + ( c f ( e)) = e c e Coneraion condiion: I uffice o obere ha for eery node he addiional amoun of flow, 0 or boleneck(p,f) enering he node equal he addiional amoun of flow, 0 or boleneck(p,f), leaing he node. QED
12 Algorihm Nework Flow Algorihm Ford-Falkeron Max-Flow(G) e f(e):=0 for all e in G while here i an - pah in he reidual graph do le P be a imple - pah in e f :=Augmen(f,P) e G = G e f:=f endwhile reurn f f : f '
13 Algorihm Nework Flow Terminaion We find a parameer ha increae eery ime Augmen i applied. Clearly, i i he alue, (f), of he flow Lemma A eery age of he algorihm, he flow alue are ineger Lemma Le f be a flow in G, and le P be a imple - pah in. Then (f ) = (f) + boleneck(p,f). Since boleneck(p,f) > 0, we hae (f ) > (f). Proof The fir arc of P leae, and P doe no reii again. Moreoer, i i a forward arc. Hence (f ) = (f) + boleneck(p,f) > (f)
14 Algorihm Nework Flow Terminaion (cnd) Corollary Le C be he oal capaciy of arc leaing, i.e. C = Then if all capaciie in he flow nework are ineger, Ford-Falkeron erminae in a mo C ieraion of he while loop. c e e ou of Proof Since all capaciie are ineger, eery ieraion increae he alue by a lea 1. QED
15 Algorihm Nework Flow Running Time Theorem If all he capaciie are ineger hen he Ford-Falkeron algorihm can be implemened o run in O(mC) ime Proof The algorihm execue he while loop a mo C ime. The reidual graph conain a mo 2m edge. Uing BFS we find an - pah in i in O(m + n) = O(m) ime Augmening ake O(n) = O(m) ime QED
16 Algorihm Nework Flow Ford-Falkeron: Analyi Theorem If all he capaciie are ineger hen he Ford-Falkeron algorihm find a maximal flow. To be proed laer.
17 Algorihm Nework Flow Cu A cu i a pariion of G ino wo e, A and B, o ha A and B The capaciy of he cu i c( A, B) = Alo f ou ( A) = in c e e ou of A f ( e), f ( A) = e ou of A e in A f ( e) Lemma For any flow f we hae ou in ( f ) = f ( A) f ( A)
18 Algorihm Nework Flow Cu and Flow Value Proof ou By definiion ( f ) = f ( ) Since f in ou in ( ) = 0 we alo hae ( f ) = f ( ) f ( ) ou in Furhermore, f ( ) f ( ) = 0 for, ou in Thu ( f ) = f ( ) f ( ) If boh end of e belong o A, i conribue 0 o he um aboe If he beginning of e i in A, i conribue poiily If he end of e i in A, i conribue negaiely Hence A ( f ) f ( ) f ( ) = f ( e) f ( e) = f ( A) f ( A) A ou in = e ou of A e in A ou in
19 Algorihm Nework Flow Cu and Flow Value Corollary Le f be a flow and (A,B) a cu. Then in ou ( f ) = f ( ) f ( ) Corollary Le f be a flow, and (A,B) a cu. Then (f) c(a,b)
20 Algorihm Nework Flow Max Flow. Min Cu Le f be he flow reurned by he Ford-Falkeron algorihm. We find a cu (A,B) uch ha (f) = c(a,b) By he Corollary aboe hi mean ha (f) i maximal poible, and ha c(a,b) i he alue of he maximal flow Lemma Le f be a flow uch ha here i no - pah in he reidual graph Then here i a cu (A,B) in G uch ha (f) = c(a,b) Proof Le A be he e of all erice uch ha i reachable from in Le B be he remaining erice
21 Algorihm Nework Flow Max Flow. Min Cu (cnd) u y Fir, how ha (A,B) i a cu Obiouly, A Since here i no - pah in we hae A Second, uppoe ha e =(u,) i an edge in G, for which u A, B Then f ( e) =. c e Indeed, oherwie e would be a forward edge in A conradicion wih he choice of A
22 Algorihm Nework Flow Max Flow. Min Cu (cnd) u y x Third, uppoe ha e =(x,y) i an edge in G, for which x B, y A Then f(e) = 0 Indeed, oherwie he edge e = (y,x) would be a backward edge in A conradicion wih he choice of A Thu ( f ) = f ou = c ( A) f in e = e ou of A ( A) = 0 c( A, B) f ( e) e ou of A e in A f ( e)
23 Algorihm Nework Flow Max Flow. Min Cu (cnd) Corollary The flow reurned by he Ford-Falkeron algorihm i a maximal flow Corollary (Max Flow Min Cu Theorem) In eery flow nework he maximum alue of a flow equal he minimum capaciy of a cu Corollary Gien a flow of maximal alue, we can compue a cu of minimum capaciy in O(m) ime Corollary If all capaciie in a flow nework are ineger, hen here i a maximum flow f for which eery f(e) i an ineger
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