CMPS 6610/4610 Fall Flow Networks. Carola Wenk Slides adapted from slides by Charles Leiserson

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1 CMP 6610/4610 Fall 2016 Flow Nework Carola Wenk lide adaped rom lide by Charle Leieron

2 Max low and min c Fndamenal problem in combinaorial opimizaion Daliy beween max low and min c Many applicaion: Biparie maching Image egmenaion Airline chedling Nework reliabiliy rey deign Baeball eliminaion Gene ncion predicion 2

3 Flow nework Deiniion. A low nework i a direced graph G = V E wih wo diingihed erice: a orce and a ink. Each edge E ha a nonnegaie capaciy c. I E hen c = 0. We reqire ha i E hen E. 2 Example:

4 Flow nework Deiniion. A poiie low on G i a ncion : V V R aiying he ollowing: Capaciy conrain: For all V 0 c. Flow coneraion: For all V \{ } V V The ale o a low i he ne low o o he orce: V V 0 4

5 A low on a nework low capaciy 1:3 2:2 2:3 1:3 1:1 2:2 2:3 1:2 Flow coneraion like Kircho crren law: Flow ino i = 3. Flow o o i = 3. The ale o hi low i = 3. 5

6 The maximm-low problem Maximm-low problem: Gien a low nework G ind a low o maximm ale on G. 2:3 2:2 2:3 0:3 1:1 2:2 3:3 2:2 The ale o he maximm low i 4. 6

7 7 C Deiniion. A c T o a low nework G = V E i a pariion o V ch ha and T. I i a low on G hen he ne low acro he c i The capaciy o he c i T T T T c T c

8 C 2:3 0:3 2:2 1:1 2:3 T 2:2 2:2 3:3 T = = 4 ct = = 9 8

9 9 Anoher characerizaion o low ale Lemma. For any low and any c T we hae = T. } \{ T T T T T V V V V V V V V Proo: 0

10 10 Upper bond on he maximm low ale Theorem. The ale o any low i bonded rom aboe by he capaciy o any c: ct.. Proo. T c c T T T T T

11 Flow ino he ink 2:3 2:2 2:3 0:3 1:1 2:2 3:3 2:2 = {} V\{} = V\{} = 4 11

12 Reidal nework Deiniion. Le be a low on G = V E. The reidal nework G V E i he graph wih reidal capaciie c ie c = i E 0 oherwie Edge in E admi more low. Lemma. E 2 E. 12

13 Agmening pah Deiniion. Le p be a pah rom o in G. The reidal capaciy o p i c p min { c }. p I c p > 0 hen p i called an agmening pah in G wih repec o. The low ale can be increaed along an agmening pah p by c p. Ex.: G: 3:5 2:6 4:5 2:3 2:5 c p = 2 G : p

14 Agmening pah con. G: c p = 2 G : p 3:5 2: :5 2: :5 3 G: 3 5:5 2 4:6 1. 2:5 0:3 1 4:5 2 G :

15 Max-low min-c heorem Theorem. The ollowing are eqialen: 1. = c T or ome c T. min-c 2. i a maximm low. 3. admi no agmening pah. Proo. 1 2: ince c T or any c T he ampion ha c T implie ha i a maximm low. 2 3: I here wa an agmening pah he low ale cold be increaed conradicing he maximaliy o. 15

16 Proo conined 3 1: Deine = { V : here exi an agmening pah in G rom o } and le T = V \. ince admi no agmening pah here i no pah rom o in G. Hence and T and h T i a c. Conider any erice and T. pah in G T We m hae c = 0 ince i c > 0 hen no T a amed. Th = c i E ince c = c. And oherwie =0. mming oer all and T yield T = c T and ince = T he heorem ollow. 16

17 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G:

18 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 0:10 9 0:10 9 0:1 0:10 9 0:

19 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 0:10 9 0:10 9 0:1 0:10 9 0:

20 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 0:10 9 1:1 0:10 9 1:

21 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 0:10 9 1:1 0:10 9 1:

22 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 1:10 9 0:1 1:10 9 1:

23 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 1:10 9 0:1 1:10 9 1:

24 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 2:10 9 1:10 9 1:1 1:10 9 2: billion ieraion on a graph wih 4 erice! 24

25 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Rnime: Le * be he ale o a maximm low and ame i i an inegral ale. The iniializaion ake O E ime There are a mo * ieraion o he loop Find an agmening pah wih DF in O V + E ime Each agmenaion ake O V ime O E * ime in oal 25

26 Edmond-Karp algorihm Edmond and Karp noiced ha many people implemenaion o Ford-Flkeron agmen along a breadh-ir agmening pah: a hore pah in G rom o where each edge wih poiie capaciy ha weigh 1. Thee implemenaion wold alway rn relaiely a. ince a breadh-ir agmening pah can be ond in O V + E ime heir analyi which proided he ir polynomial-ime bond on maximm low oce on bonding he nmber o low agmenaion. In independen work Dinic alo gae polynomial-ime bond. 26

27 Rnning ime o Edmond- Karp One can how ha he nmber o low agmenaion i.e. he nmber o ieraion o he while loop i O V E. Breadh-ir earch rn in O V + E ime All oher bookkeeping i O V per agmenaion. The Edmond-Karp maximm-low algorihm rn in O V E 2 ime. 27

28 Monooniciy lemma Lemma. Le = be he breadh-ir diance rom o in G. Dring he Edmond- Karp algorihm increae monoonically. Proo. ppoe ha i a low on G and agmenaion prodce a new low. Le =. We ll how ha by indcion on. For he bae cae = 0. For he indcie cae conider a breadh-ir pah in G. We m hae + 1 ince bpah o hore pah are hore pah. Cerainly E and now conider wo cae depending on wheher E. 28

29 Cae 1 Cae: E. We hae + 1 riangle ineqaliy + 1 indcion = breadh-ir pah and h monooniciy o i eablihed. 29

30 Cae 2 Cae: E. ince E he agmening pah p ha prodced rom m hae inclded. Moreoer p i a breadh-ir pah in G : p =. Th we hae 1 breadh-ir pah 1 indcion 2 breadh-ir pah < hereby eablihing monooniciy or hi cae oo. 30

31 Coning low agmenaion Theorem. The nmber o low agmenaion in he Edmond-Karp algorihm Ford-Flkeron wih breadh-ir agmening pah i O V E. Proo. Le p be an agmening pah and ppoe ha we hae c = c p or edge p. Then we ay ha i criical and i diappear rom he reidal graph aer low agmenaion. Example: G : c p = 2 31

32 Coning low agmenaion Theorem. The nmber o low agmenaion in he Edmond-Karp algorihm Ford-Flkeron wih breadh-ir agmening pah i O V E. Proo. Le p be an agmening pah and ppoe ha we hae c = c p or edge p. Then we ay ha i criical and i diappear rom he reidal graph aer low agmenaion. Example: G :

33 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: 33

34 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: = 5 = 6 34

35 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: = 5 = 6 35

36 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:

37 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:

38 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:

39 Rnning ime o Edmond- Karp Diance ar o nonnegaie neer decreae and are a mo V 1 nil he erex become nreachable. Th occr a a criical edge O V ime becae increae by a lea 2 beween occrrence. ince he reidal graph conain O E edge he nmber o low agmenaion i O V E. Corollary. The Edmond-Karp maximm-low algorihm rn in O V E 2 ime. Proo. Breadh-ir earch rn in O E ime and all oher bookkeeping i O V per agmenaion. 39

40 Be o dae The aympoically ae algorihm o dae or maximm low de o King Rao and Tarjan rn in O V E log E / V log V V ime. I we allow rnning ime a a ncion o edge weigh he ae algorihm or maximm low de o Goldberg and Rao rn in ime Omin{ V 2/3 E 1/2 } E log V 2 / E + 2 log C where C i he maximm capaciy o any edge in he graph. 40

Flow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001

Flow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001 CS 545 Flow Nework lon Efra Slide courey of Charle Leieron wih mall change by Carola Wenk Flow nework Definiion. flow nework i a direced graph G = (V, E) wih wo diinguihed verice: a ource and a ink. Each