Struktur Data & Algoritme (Data Structures & Algorithms) Ide Algoritma Ford-Fulkerson. Motivation. 1-Source, 1-Target Problem.

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1 rukur D & Algorime (D ruure & Algorihm) Mximum Flow uryn eiwn eiwn@.ui..i Fkul Ilmu Kompuer Univeri Inonei emeer Genp 2/2 Verion. Inernl Ue Only oure, rge Prolem Flow nework Grph Verek p ymmeril iree grph ihuungkn engn iiii yng erup lurn engn [u,v] kpi ei erenu nr verek u n v [u,v] Jik [u,v] erri ik ii yng menghuungkn u n v f[u,v] lirn yng erji nr verek u n v f[v,u] = f[u,v] f[u,v] [u,v] r[u,v] kpi ei i/reiul yng ik igunkn nr verek u n v r[u,v] = [u,v] f[u,v] Du verek ieu egi oure n rge/ink DA/Mx Flow/ur/V./ Moivion Bnyk mlh grph lm uni ny erkin engn lirn meril Cirn: lm uu jringn pip yng menglirkn minyk, erp kpi ei (volume yng menglir per un wku) minyk mellui jringn ereu? Llu lin: lm uu jringn jln, erp y ukung jringn jln ereu? Proe: i gin mn erjiny olenek proe ereu? Ie Algorim ForFulkeron Menpkn ph reiul nr n, jik mk liri mellui ph ereu engn ei eui engn reiu erkeil i lm ph ereu Upe eip f[u,v] unuk eip (u,v) lm ph eui engn ei ereu Hiung reiul r[u,v] = [u,f] f[u,v] unuk eip (u,v) lm ph egi reiul erru Ulngi hingg ik ph reiul nr n Grph MxFlow lh grph f[u,v] erkhir engn hny mengmil ii (u,v) jik f[u,v] >. DA/Mx Flow/ur/V./2 DA/Mx Flow/ur/V./ Funmenl D ype: ring, Mh, Cing

2 Algorim ForFulkeron Conoh Memu grph imeri, jik [u,v] emenr [v,u] ik engn memu [v,u] = Iniilii f[u,v] = f[v,u] = unuk eip (u,v) lm grph Iniilii r[u,v] = [u,v] unuk eip (u,v) lm grph While (ph=finphdlmr(,)) { mhei = minimum {r[u,v] unuk eip (u,v) lm ph} For eip ii (u,v) lm ph { f[u,v] += r[u,v]; f[v,u] = f[u,v]; } Upe r[u,v] = [u,v] f[u,v] unuk eip (u,v) lm ph DA/Mx Flow/ur/V./ Kpi v 6 V v 2 v 6 v v DA/Mx Flow/ur/V./ Berny Alirn Mximum Conoh: Iniilii Grph MxFlow ik erif unik (i eerp olui) epi erny ei lirn unik (mkimum) Alirn mximum lm grph iperoleh engn menjumlhkn ol flow yng kelur ri u ol flow yng mukke Alirn f V v v Kpi eri r v 6 V 2 v 2 V V v DA/Mx Flow/ur/V./6 DA/Mx Flow/ur/V./ Funmenl D ype: ring, Mh, Cing 2

3 Conoh: penrin ph 2 v 6 v 2 Conoh: Penrin ph 2 v 2 2 v Milny iperoleh ph vv Dei: min{6,2,,,} = Upe eip f[u,v] += n f[v,u] = f[u,v], unuk eip (u,v) lm ph Milny iperoleh ph vv Dei: min{2,,,,2}= Upe eip f[u,v] += n f[v,u] = f[u,v], unuk eip (u,v) lm ph DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./ Conoh: ieri Conoh: ieri 2 Alirn w v v Kpi eri r v v Alirn f v v Kpi eri r v v 2 V V V V 2 V V V v V v DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./2 Funmenl D ype: ring, Mh, Cing

4 Conoh: Penrin ph v v Conoh: Penrin ph v 2 v Milny iperoleh ph v eri ph Dei: min{,,,}= Upe eip f[u,v] += n f[v,u] = f[u,v], unuk eip (u,v) lm ph Dei: min{,,}= Upe eip f[u,v] += n f[v,u] = f[u,v], unuk eip (u,v) lm ph DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./ Conoh: ieri Conoh: ieri Alirn f v v Kpi eri r v v Alirn w v v Kpi eri r v v 2 V 2 V V 2 V V V 2 2 V v V v DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./6 Funmenl D ype: ring, Mh, Cing

5 Conoh: Penrin ph Mlh Algorim v 2 v ik lgi ph ri ke mk elei engn grph lirn mkimum inykn lm f[u,v] engn mengmil hrg yng poiif 2 Penrin ph merupkn hl yng pling kriikl lm lgorim ini. Ku eruruk jik erji iui. Bil ph yng iemukn ellu mellui u mk ieri kn erji lm jumlh yng er eip ieri ei lm ph yng iemukn hny v DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./ COnoh: mximum flow hil Alirn mkimum ri ke lh 2 2 v Penrin ph Vrin Algorim EmonKrp Dengn BF: hore ph engn ri jumlh ii lm ph yng pling eiki Dengn DF 2 v DA/Mx Flow/ur/V./ DA/Mx Flow/ur/V./2 Funmenl D ype: ring, Mh, Cing

6 Anlii Algorim Penrin flow ph ilkukn p mkimum E uh ii, ji O(E). Bnyk ieri lh m engn jumlh flow ph yng iemukn. Jik jumlh lh f* mk kompleki wku lh O(E f* ) Penmhn ummy verex u/ verek oure engn ii mengrh ke eip verek oure emul Penmhn ummy verex u/ verex rge engn ii mengrh ri eip verek rge emul eip ii yng imhkn erkpi ng er (k erhingg!) v DA/Mx Flow/ur/V./2 DA/Mx Flow/ur/V./2 Mulioure, Mulirge Prolem Dlm mlh yng leih ny oure i erl ri ejumlh verek menuju ke ejumlh verek linny. Apkh lgorim ForFulkeron i igunkn? Conoh: Mximum Biprie Mhing (MBM) Biprie Grph: lm uu uniree grph G={V,E}, V erprii u ue V n V2 n E E n eip (u,v) E erpenuhi u V n v V2. Biprie Mhing: lh E engn E E, imn eip (u,v) E er unuk v v n u u, ik (u,v ) u (u,v) E. Crinliy ri Biprie Mhing lh E Mximum Biprie Mhing lh mhing engn rinliy erer v DA/Mx Flow/ur/V./22 DA/Mx Flow/ur/V./2 Funmenl D ype: ring, Mh, Cing 6

7 Conoh v v v v Biprie Grph, verek erprii v n Biprie Mhing, ik mkiml (rinliy 2) Mximl Biprie Grph (rinliy ) DA/Mx Flow/ur/V./2 DA/Mx Flow/ur/V./2 Penrin MBM egi Prolem MxFlow Conoh Ku mhkn verek oure n iiii errh yng mengrh ke eip verek lm V engn kpi mhkn verek rge n iiii errh yng mengrh ri eip verek lm V2 engn kpi eip ii lm G lh ii engn kpi n errh engn rh ri verek i V ke verek i V2 DA/Mx Flow/ur/V./26 Kpi lm repreeni mrik egi Cpiy Nework Flow Grph (oo ik igmrkn kren erhrg u ) DA/Mx Flow/ur/V./2 Funmenl D ype: ring, Mh, Cing

8 Funmenl D ype: ring, Mh, Cing DA/Mx Flow/ur/V./2 f Jik ph perm r DA/Mx Flow/ur/V./ f Jik ph keu r DA/Mx Flow/ur/V./ ik lgi ph lm r ri ke, mk mxflow grph Mk mximum mhing lh n