7.6 Disjoint Paths. 7. Network Flow Applications. Edge Disjoint Paths. Edge Disjoint Paths

Transcription

1 . Nework Flow Applcaon. Djon Pah Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne Edge Djon Pah Edge Djon Pah Djon pah problem. Gven a dgraph G = (V, E) and wo node and, fnd he ma number of edgedjon pah. Def. Two pah are edgedjon f hey have no edge n common. E: communcaon nework. Djon pah problem. Gven a dgraph G = (V, E) and wo node and, fnd he ma number of edgedjon pah. Def. Two pah are edgedjon f hey have no edge n common. E: communcaon nework.

2 Edge Djon Pah Edge Djon Pah Ma flow formulaon: agn un capacy o every edge. Ma flow formulaon: agn un capacy o every edge. Theorem. Ma number edgedjon pah equal ma flow value. Pf. Suppoe here are k edgedjon pah P,..., P k. Se f(e) = f e parcpae n ome pah P ; ele e f(e) =. Snce pah are edgedjon, f a flow of value k. Theorem. Ma number edgedjon pah equal ma flow value. Pf. $ Suppoe ma flow value k. Inegraly heorem % here e flow f of value k. Conder edge (, u) wh f(, u) =. by conervaon, here e an edge (u, v) wh f(u, v) = connue unl reach, alway choong a new edge Produce k (no necearly mple) edgedjon pah. Nework Connecvy Edge Djon Pah and Nework Connecvy Nework connecvy. Gven a dgraph G = (V, E) and wo node and, fnd mn number of edge whoe removal dconnec from. Menger Theorem (9). The ma number of edgedjon pah equal o he mn number of edge whoe removal dconnec from. Def. A e of edge F # E dconnec from f all pah ue a lea one edge n F. Pf. " Suppoe he removal of F # E dconnec from, and F = k. All pah ue a lea one edge of F. Hence, he number of edgedjon pah a mo k. 8

3 Djon Pah and Nework Connecvy Menger Theorem (9). The ma number of edgedjon pah equal o he mn number of edge whoe removal dconnec from.. Bpare Machng Pf. $ " Suppoe ma number of edgedjon pah k. " Then ma flow value k. " Maflow mncu % cu (A, B) of capacy k. " Le F be e of edge gong from A o B. " F = k and dconnec from. A 9 Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne Machng Bpare Machng Machng. Inpu: undreced graph G = (V, E). M # E a machng f each node appear n a mo edge n M. Ma machng: fnd a ma cardnaly machng. Bpare machng. Inpu: undreced, bpare graph G = (L & R, E). M # E a machng f each node appear n a mo edge n M. Ma machng: fnd a ma cardnaly machng. machng,, L R

4 Bpare Machng Bpare Machng Bpare machng. Inpu: undreced, bpare graph G = (L & R, E). M # E a machng f each node appear n a mo edge n M. Ma machng: fnd a ma cardnaly machng. Ma flow formulaon. Creae dgraph G = (L & R & {, }, E ). Drec all edge from L o R, and agn nfne (or un) capacy. Add ource, and un capacy edge from o each node n L. Add nk, and un capacy edge from each node n R o. G ma machng,, L R L R Bpare Machng: Proof of Correcne Bpare Machng: Proof of Correcne Theorem. Ma cardnaly machng n G = value of ma flow n G. Pf. Gven ma machng M of cardnaly k. Conder flow f ha end un along each of k pah. f a flow, and ha cardnaly k. Theorem. Ma cardnaly machng n G = value of ma flow n G. Pf. $ Le f be a ma flow n G of value k. Inegraly heorem % k negral and can aume f. Conder M = e of edge from L o R wh f(e) =. each node n L and R parcpae n a mo one edge n M M = k: conder cu (L &, R & ) G G G G

5 Perfec Machng Perfec Machng Def. A machng M # E perfec f each node appear n eacly one edge n M. Noaon. Le S be a ube of node, and le N(S) be he e of node adjacen o node n S. Q. When doe a bpare graph have a perfec machng? Srucure of bpare graph wh perfec machng. Clearly we mu have L = R. Wha oher condon are neceary? Wha condon are uffcen? Obervaon. If a bpare graph G = (L & R, E), ha a perfec machng, hen N(S) $ S for all ube S # L. Pf. Each node n S ha o be mached o a dfferen node n N(S). No perfec machng: S = {,, } N(S) = {, }. L R 8 Marrage Theorem Proof of Marrage Theorem Marrage Theorem. [Frobenu 9, Hall 9] Le G = (L & R, E) be a bpare graph wh L = R. Then, G ha a perfec machng ff N(S) $ S for all ube S # L. Pf. % Th wa he prevou obervaon. L R No perfec machng: S = {,, } N(S) = {, }. Pf. ( Suppoe G doe no have a perfec machng. Formulae a a ma flow problem and le (A, B) be mn cu n G. " By maflow mncu, cap(a, B) < L. " Defne L A = L ) A, L B = L ) B, R A = R ) A. cap(a, B) = L B + R A. Snce mn cu can ue edge: N(L A ) # R A. N(L A ) R A = cap(a, B) L B < L L B = L A. Chooe S = L A. G A L A = {,, } L B = {, } R A = {, } N(L A ) = {, } 9

6 Image Segmenaon. Image Segmenaon Image egmenaon. Cenral problem n mage proceng. Dvde mage no coheren regon. E: Three people andng n fron of comple background cene. Idenfy each peron a a coheren objec. Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne Image Segmenaon Image Segmenaon Foreground / background egmenaon. Label each pel n pcure a belongng o foreground or background. V = e of pel, E = par of neghborng pel. a $ lkelhood pel n foreground. b $ lkelhood pel n background. p j $ eparaon penaly for labelng one of and j a foreground, and he oher a background. Goal. Accuracy: f a > b n olaon, prefer o label n foreground. Smoohne: f many neghbor of are labeled foreground, we hould be nclned o label a foreground. Fnd paron (A, B) ha mamze: # a + b j " A j " B foreground background # $ # p j (, j) " E AI{, j} = Formulae a mn cu problem. Mamzaon. No ource or nk. Undreced graph. Turn no mnmzaon problem. Mamzng # a + b j " A j " B equvalen o mnmzng # $ # p j (, j) " E AI{, j} = ( # " V a + b # j " V ) j or alernavely # a j + # b + # p j j " B a conan " A (, j) " E AI{, j} = $ # a $ b j " A j " B # + # p j (, j) " E AI{, j} =

7 Image Segmenaon Image Segmenaon Formulae a mn cu problem. G = (V, E). Add ource o correpond o foreground; add nk o correpond o background Ue wo anparallel edge nead of undreced edge. p j p j p j Conder mn cu (A, B) n G. A = foreground. cap(a, B) = # a j + # b + # p j j " B " A (, j) " E " A, j " B Precely he quany we wan o mnmze. f and j on dfferen de, p j couned eacly once a j a j p j j p j j b A b G G Projec Selecon. Projec Selecon Projec wh prereque. can be pove or negave Se P of poble projec. Projec v ha aocaed revenue p v. ome projec generae money: creae neracve ecommerce nerface, redegn web page oher co money: upgrade compuer, ge e lcene Se of prereque E. If (v, w) * E, can do projec v unle alo do projec w. A ube of projec A # P feable f he prereque of every projec n A alo belong o A. Projec elecon. Chooe a feable ube of projec o mamze revenue. Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne 8

8 Projec Selecon: Prereque Graph Projec Selecon: Mn Cu Formulaon Prereque graph. Include an edge from v o w f can do v whou alo dong w. {v, w, } feable ube of projec. {v, } nfeable ube of projec. Mn cu formulaon. Agn capacy o all prereque edge. Add edge (, v) wh capacy pv f pv >. Add edge (v, ) wh capacy pv f pv <. For noaonal convenence, defne p = p =. w w u v v feable pu py y pw z pv pz p v nfeable w 9 Projec Selecon: Mn Cu Formulaon Open P Mnng Clam. (A, B) mn cu ff A + { } opmal e of projec. Infne capacy edge enure A + { } feable. Ma revenue becaue: cap(a, B) = # p v + # ($ p v ) Openp mnng. (uded nce early 9) Block of earh are eraced from urface o rereve ore. Each block v ha ne value pv = value of ore proceng co. Can remove block v before w or. v" B: pv > = v" A: pv < #pv $ #pv v: pv > v " A conan w A u pu w pw py pv y z v v p

9 Baeball Elmnaon. Baeball Elmnaon "See ha hng n he paper la week abou Enen?... Some reporer aked hm o fgure ou he mahemac of he pennan race. You know, one eam wn o many of her remanng game, he oher eam wn h number or ha number. Wha are he myrad poble? Who go he edge?" "The hell doe he know?" "Apparenly no much. He pcked he Dodger o elmnae he Gan la Frday." Don DeLllo, Underworld Algorhm Degn by Éva Tardo and Jon Klenberg Copyrgh Addon Weley Slde by Kevn Wayne Baeball Elmnaon Baeball Elmnaon Agan = rj Agan = rj Team Wn w Loe l To play r Al Ph NY Mon Team Wn w Loe l To play r Al Ph NY Alana 8 8 Alana 8 8 Phlly 8 9 Phlly 8 9 New York 8 8 New York 8 8 Monreal 8 Monreal 8 Whch eam have a chance of fnhng he eaon wh mo wn? Monreal elmnaed nce can fnh wh a mo 8 wn, bu Alana already ha 8. w + r < wj % eam elmnaed. Only reaon por wrer appear o be aware of. Suffcen, bu no neceary Mon Whch eam have a chance of fnhng he eaon wh mo wn? Phlly can wn 8, bu ll elmnaed... If Alana loe a game, hen ome oher eam wn one. Remark. Anwer depend no ju on how many game already won and lef o play, bu alo on whom heyre agan. " "

10 Baeball Elmnaon Baeball Elmnaon: Ma Flow Formulaon Baeball elmnaon problem. Se of eam S. Dnguhed eam * S. Team ha won w game already. Team and y play each oher r y addonal me. I here any oucome of he remanng game n whch eam fnhe wh he mo (or ed for he mo) wn? Can eam fnh wh mo wn? Aume eam wn all remanng game % w + r wn. Dvvy remanng game o ha all eam have w + r wn. game lef eam can ll wn h many more game r = w + r w game node eam node 8 Baeball Elmnaon: Ma Flow Formulaon Baeball Elmnaon: Eplanaon for Spor Wrer Theorem. Team no elmnaed ff ma flow aurae all arc leavng ource. Inegraly heorem % each remanng game beween and y added o number of wn for eam or eam y. " Capacy on (, ) arc enure no eam wn oo many game. Team Wn w Loe To play Agan = r j l NY Bal Bo Tor NY Balmore 8 Boon 9 8 Torono Dero 9 8 r De game lef r = w + r w eam can ll wn h many more game AL Ea: Augu, 99 Whch eam have a chance of fnhng he eaon wh mo wn? Dero could fnh eaon wh 9 + = wn. game node eam node 9

11 Baeball Elmnaon: Eplanaon for Spor Wrer Baeball Elmnaon: Eplanaon for Spor Wrer Team Wn w Whch eam have a chance of fnhng he eaon wh mo wn? Dero could fnh eaon wh 9 + = wn. Cerfcae of elmnaon. R = {NY, Bal, Bo, Tor} Have already won w(r) = 8 game. Mu wn a lea r(r) = more. Loe To play Agan = r j l NY Bal Bo Tor NY Balmore 8 Boon 9 8 Torono Dero 9 8 Average eam n R wn a lea / > game. r AL Ea: Augu, 99 De Cerfcae of elmnaon. If # remanng game # wn 8 8 T " S, w(t) := $ w, g(t) := $ g y, #T, y #T LB on avg # game won 8 w(t)+ g(t) T > w z + g z hen z elmnaed (by ube T). Theorem. [HoffmanRvln, 9] Team z elmnaed f and only f here e a ube T ha elmnae z. Proof dea. Le T = eam node on ource de of mn cu. Baeball Elmnaon: Eplanaon for Spor Wrer Baeball Elmnaon: Eplanaon for Spor Wrer Pf of heorem. Ue ma flow formulaon, and conder mn cu (A, B). Defne T* = eam node on ource de of mn cu. Oberve y * A ff boh * T* and y * T*. nfne capacy edge enure f y * A hen * A and y * A f * A and y * A bu y * T, hen addng y o A decreae capacy of cu Pf of heorem. Ue ma flow formulaon, and conder mn cu (A, B). Defne T* = eam node on ource de of mn cu. Oberve y * A ff boh * T* and y * T*. " g(s " {z}) > cap(a, B) capacy of game edge leavng capacy of eam edge leavng 8 8 = g(s " {z})" g(t*) + $ (w z + g z " w ) #T* = g(s " {z})" g(t*) " w(t*) + T* (w z + g z ) game lef y eam can ll wn h many more game " Rearrangng erm: w z + g z < w(t*)+ g(t*) T* r = y w z + r z w