7. NETWORK FLOW I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 11/22/17 6:11 AM

Size: px
Start display at page:

Download "7. NETWORK FLOW I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 11/22/17 6:11 AM"

Transcription

1 7. NETWORK FLOW I max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework Lecure lide by Kevin Wayne Copyrigh 2005 Pearon-Addion Weley hp:// La updaed on 11/22/17 6:11 AM

2 7. NETWORK FLOW I SECTION 7.1 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

3 Flow nework A flow nework i a uple G = (V, E,,, c). Digraph (V, E) wih ource V and ink V. Non-negaive capaciy c(e) for each e E. Inuiion. Maerial flowing hrough a ranporaion nework; maerial originae a ource and i en o ink. capaciy

4 Minimum-cu problem Def. An -cu (cu) i a pariion (A, B) of he verice wih A and B. Def. I capaciy i he um of he capaciie of he edge from A o B. cap(a, B) = e A c(e) capaciy = = 30 4

5 Minimum-cu problem Def. An -cu (cu) i a pariion (A, B) of he verice wih A and B. Def. I capaciy i he um of he capaciie of he edge from A o B. cap(a, B) = e A c(e) 10 8 don include edge from B o A capaciy = =

6 Minimum-cu problem Def. An -cu (cu) i a pariion (A, B) of he verice wih A and B. Def. I capaciy i he um of he capaciie of he edge from A o B. cap(a, B) = c(e) e A Min-cu problem. Find a cu of minimum capaciy capaciy = = 28 6

7 v Maximum-flow problem Def. An -flow (flow) f i a funcion ha aifie: For each e E : 0 f(e) c(e) [capaciy] For each v V {, } : f(e) = f(e) [flow conervaion] e v e v flow capaciy inflow a v = = 10 5 / 9 ouflow a v = = / 10 0 / 4 5 / 15 0 / 15 5 / 10 5 / 5 5 / 8 10 / / 15 0 / 4 0 / 6 0 / / / 16 7

8 Maximum-flow problem Def. An -flow (flow) f i a funcion ha aifie: For each e E : 0 f(e) c(e) [capaciy] For each v V {, } : f(e) = f(e) [flow conervaion] e v e v Def. The value of a flow f i: val(f) = f(e) f(e) e e 5 / 9 10 / 10 0 / 4 5 / 15 0 / 15 5 / 10 5 / 5 5 / 8 10 / / 15 0 / 4 0 / 6 0 / / 10 value = = / 16 8

9 Maximum-flow problem Def. An -flow (flow) f i a funcion ha aifie: For each e E : 0 f(e) c(e) [capaciy] For each v V {, } : f(e) = f(e) [flow conervaion] e v e v Def. The value of a flow f i: val(f) = f(e) f(e) e e Max-flow problem. Find a flow of maximum value. 8 / 9 10 / 10 0 / 4 2 / 15 0 / 15 8 / 10 5 / 5 8 / 8 10 / / 15 0 / 4 3 / 6 0 / / 10 value = = / 16 9

10 7. NETWORK FLOW I SECTION 7.1 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

11 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f flow capaciy 0 / 4 0 / 10 0 / 2 0 / 8 0 / 6 0 / 10 value of flow 0 / 10 0 / 9 0 /

12 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f 0 / / 10 0 / / 8 0 / 6 0 / / 10 0 / 9 0 / = 8 12

13 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f 0 / / / 2 8 / 8 0 / 6 0 / / 10 0 / 9 8 / = 10 13

14 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f 0 / 4 10 / 10 2 / 2 8 / / / / 10 2 / 9 10 / = 16 14

15 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f ending flow value = 16 0 / 4 10 / 10 2 / 2 8 / 8 6 / 6 6 / 10 6 / 10 8 / 9 10 /

16 Toward a max-flow algorihm Greedy algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P where each edge ha f (e) < c(e). Augmen flow along pah P. Repea unil you ge uck. flow nework G and flow f bu max-flow value = 19 3 / 4 10 / 10 0 / 2 7 / 8 6 / 6 9 / 10 9 / 10 9 / 9 10 /

17 Why he greedy algorihm fail Q. Why doe he greedy algorihm fail? A. Once greedy algorihm increae flow on an edge, i never decreae i. Ex. The max flow i unique; flow on edge (v, w) i zero. Greedy algorihm could chooe v w for fir augmening pah. flow nework G v 2 Boom line. Need ome mechanim o undo bad deciion w 17

18 Reidual nework Original edge. e = (u, v) E. Flow f (e). Capaciy c(e). Revere edge. e revere = (v, u). original flow nework G u 6 / 17 flow capaciy v Undo flow en. Reidual capaciy. reidual nework Gf reidual capaciy c f (e) = c(e) f(e) e E f(e) e E u 11 6 v revere edge edge wih poiive reidual capaciy Reidual nework. G f = (V, E f,,, c f ). E f = {e : f (e) < c(e)} {e revere : f (e) > 0}. Key propery: f ʹ i a flow in G f iff f + f ʹ i a flow in G. where flow on a revere edge negae flow on correponding forward edge 18

19 Augmening pah Def. An augmening pah i a imple pah in he reidual nework G f. Def. The boleneck capaciy of an augmening pah P i he minimum reidual capaciy of any edge in P. Key propery. Le f be a flow and le P be an augmening pah in G f. Then, afer calling AUGMENT, he reuling f ʹ i a flow and val( f ʹ ) = val( f ) + boleneck(gf, P). AUGMENT ( f, c, P) b boleneck capaciy of pah P. FOREACH edge e P IF (e E ) f [e] f [e] + b. ELSE f [e revere ] f [e revere ] b. RETURN f. 19

20 Ford Fulkeron algorihm Ford Fulkeron augmening pah algorihm. Sar wih f (e) = 0 for each edge e E. Find an pah P in he reidual nework G f. Augmen flow along pah P. Repea unil you ge uck. FORD FULKERSON (G) FOREACH edge e E : f [e] 0. Gf reidual nework of G wih repec o f. WHILE (here exi an pah P in Gf ) f AUGMENT ( f, c, P). Updae Gf. RETURN f. augmening pah 20

21 7. NETWORK FLOW I Secion 7.2 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

22 Relaionhip beween flow and cu Flow value lemma. Le f be any flow and le (A, B) be any cu. Then, he value of he flow f equal he ne flow acro he cu (A, B). val(f) = e A f(e) e A f(e) ne flow acro cu = = 25 5 / 9 10 / 10 0 / 4 5 / 15 0 / 15 5 / 10 5 / 5 5 / 8 10 / 10 value of flow = / 15 0 / 4 0 / 6 0 / / / 16 22

23 Relaionhip beween flow and cu Flow value lemma. Le f be any flow and le (A, B) be any cu. Then, he value of he flow f equal he ne flow acro he cu (A, B). val(f) = e A f(e) e A f(e) ne flow acro cu = = 25 5 / 9 10 / 10 0 / 4 5 / 15 0 / 15 5 / 10 5 / 5 5 / 8 10 / 10 value of flow = / 15 0 / 4 0 / 6 0 / / / 16 23

24 Relaionhip beween flow and cu Flow value lemma. Le f be any flow and le (A, B) be any cu. Then, he value of he flow f equal he ne flow acro he cu (A, B). val(f) = e A f(e) e A f(e) ne flow acro cu = ( ) ( ) = 25 5 / 9 edge from B o A 10 / 10 0 / 4 5 / 15 0 / 15 5 / 10 5 / 5 5 / 8 10 / 10 value of flow = / 15 0 / 4 0 / 6 0 / / / 16 24

25 Relaionhip beween flow and cu Flow value lemma. Le f be any flow and le (A, B) be any cu. Then, he value of he flow f equal he ne flow acro he cu (A, B). val(f) = f(e) f(e) e A e A Pf. val(f) = f(e) f(e) e e by flow conervaion, all erm excep for v = are 0 = f(e) f(e) v A e v e v = e A f(e) e A f(e) 25

26 Relaionhip beween flow and cu Weak dualiy. Le f be any flow and (A, B) be any cu. Then, val( f ) cap(a, B). Pf. val(f) = f(e) f(e) e A e A flow-value lemma e A f(e) e A c(e) = cap(a, B) 8 / 9 10 / 10 0 / 4 2 / 15 0 / 15 8 / / 5 7 / 8 9 / / 15 0 / 4 2 / 6 0 / / / 16 value of flow = 27 capaciy of cu = 30 26

27 Cerificae of opimaliy Corollary. Le f be a flow and le (A, B) be any cu. If val( f ) = cap(a, B), hen f i a max flow and (A, B) i a min cu. weak dualiy Pf. For any flow f ʹ: val( f ʹ) cap(a, B) = val( f ). For any cu (Aʹ, Bʹ): cap(aʹ, Bʹ) val( f ) = cap(a, B). 8 / 9 10 / 10 0 / 4 2 / 15 0 / 15 8 / / 5 8 / 8 10 / / 15 0 / 4 3 / 6 0 / / / 16 value of flow = 28 = capaciy of cu = 28 27

28 Max-flow min-cu heorem Augmening pah heorem. A flow f i a max flow iff no augmening pah. Max-flow min-cu heorem. Value of a max flow = capaciy of a min cu. Pf. The following hree condiion are equivalen for any flow f : i. There exi a cu (A, B) uch ha cap(a, B) = val( f ). ii. f i a max flow. iii. There i no augmening pah wih repec o f. [ i ii ] Suppoe ha (A, B) i a cu uch ha cap(a, B) = val( f ). Then, for any flow f ʹ: val( f ʹ) cap(a, B) = val( f ). Thu, f i a max flow. weak dualiy by aumpion rong dualiy if Ford Fulkeron erminae, hen f i max flow 28

29 Max-flow min-cu heorem Augmening pah heorem. A flow f i a max flow iff no augmening pah. Max-flow min-cu heorem. Value of a max flow = capaciy of a min cu. Pf. The following hree condiion are equivalen for any flow f : i. There exi a cu (A, B) uch ha cap(a, B) = val( f ). ii. f i a max flow. iii. There i no augmening pah wih repec o f. [ ii iii ] We prove conrapoiive: ~iii ~ii. Suppoe ha here i an augmening pah wih repec o f. Can improve flow f by ending flow along hi pah. Thu, f i no a max flow. 29

30 Max-flow min-cu heorem [ iii i ] Le f be a flow wih no augmening pah. Le A be e of node reachable from in reidual nework Gf. By definiion of cu A: A. By definiion of flow f: A. val(f) = e A f(e) e A f(e) original flow nework G A edge e = (v, w) wih v B, w A mu have f(e) = 0 B flow-value lemma = c(e) e A = cap(a, B) edge e = (v, w) wih v A, w B mu have f(e) = c(e) 30

31 7. NETWORK FLOW I SECTION 7.3 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

32 Analyi of Ford Fulkeron algorihm (when capaciie are inegral) Aumpion. Capaciie are ineger beween 1 and C. Inegraliy invarian. Throughou he algorihm, he flow f (e) and he reidual capaciie c f (e) are ineger. Theorem. The algorihm erminae in a mo val( f *) n C ieraion, where f * i a max flow. Pf. Each augmenaion increae he value of he flow by a lea 1. Corollary. The running ime of Ford Fulkeron i O(m n C). Corollary. If C = 1, he running ime of Ford Fulkeron i O(m n). Inegraliy heorem. Then exi a max flow f * for which every flow f * (e) i an ineger. Pf. Since algorihm erminae, heorem follow from invarian. 32

33 Bad cae for Ford Fulkeron Q. I generic Ford Fulkeron algorihm poly-ime in inpu ize? m, n, and log C A. No. If max capaciy i C, hen algorihm can ake C ieraion. v w w v v w w v v w w v each augmening pah end only 1 uni of flow (# augmening pah = 2C) v C C 1 C C w 33

34 Chooing good augmening pah Ue care when elecing augmening pah. Some choice lead o exponenial algorihm. Clever choice lead o polynomial algorihm. Pahology. If capaciie are irraional, algorihm no guaraneed o erminae (or converge o correc anwer)! Goal. Chooe augmening pah o ha: Can find augmening pah efficienly. Few ieraion. 34

35 Chooing good augmening pah Chooe augmening pah wih: Max boleneck capaciy ( fae ). Sufficienly large boleneck capaciy. Fewe edge. Theoreical Improvemen in Algorihmic Efficiency for Nework Flow Problem JACK EDMONDS Univeriy of Waerloo, Waerloo, Onario, Canada AND RICHARD M. KARP Univeriy of California, Berkeley, California ABSTRACT. Thi paper preen new algorihm for he maximum flow problem, he Hichcock ranporaion problem, and he general minimum-co flow problem. Upper bound on he number of ep in hee algorihm are derived, and are hown o compale favorably wih upper bound on he number of ep required by earlier algorihm. Edmond-Karp 1972 (USA) Diniz 1970 (Sovie Union) invened in repone o a cla exercie by Adel on-vel kiĭ 35

36 Capaciy-caling algorihm Inuiion. Chooe augmening pah wih highe boleneck capaciy: i increae flow by max poible amoun in given ieraion. Don worry abou finding exac highe boleneck pah. Mainain caling parameer Δ. Le G f (Δ) be he par of he reidual nework coniing of only hoe arc wih capaciy Δ Gf Gf (Δ), Δ =

37 Capaciy-caling algorihm CAPACITY-SCALING (G) FOREACH edge e E : f [e] 0. Δ large power of 2 C. WHILE (Δ 1) Gf (Δ) Δ-reidual nework of G wih repec o flow f. WHILE (here exi an pah P in Gf (Δ)) f AUGMENT ( f, c, P). Updae Gf (Δ). Δ Δ / 2. RETURN f. 37

38 Capaciy-caling algorihm: proof of correcne Aumpion. All edge capaciie are ineger beween 1 and C. Inegraliy invarian. All flow and reidual capaciie are inegral. Theorem. If capaciy-caling algorihm erminae, hen f i a max flow. Pf. By inegraliy invarian, when Δ = 1 G f (Δ) = G f. Upon erminaion of Δ = 1 phae, here are no augmening pah. 38

39 Capaciy-caling algorihm: analyi of running ime Lemma 1. The ouer while loop repea 1 + log 2 C ime. Pf. Iniially C / 2 < Δ C; Δ decreae by a facor of 2 in each ieraion. Lemma 2. Le f be he flow a he end of a Δ-caling phae. Then, he max-flow value val( f ) + m Δ. Lemma 3. There are a mo 2m augmenaion per caling phae. Pf. Le f be he flow a he end of he previou caling phae. Lemma 2 max-flow value val( f ) + 2 m Δ. Each augmenaion in a Δ-phae increae val( f ) by a lea Δ. Theorem. The caling max-flow algorihm find a max flow in O(m log C) augmenaion. I can be implemened o run in O(m 2 log C) ime. Pf. Follow from Lemma 1 and Lemma 3. proof on nex lide 39

40 Capaciy-caling algorihm: analyi of running ime Lemma 2. Le f be he flow a he end of a Δ-caling phae. Then, he max-flow value val( f ) + m Δ. Pf. We how here exi a cu (A, B) uch ha cap(a, B) val( f ) + m Δ. Chooe A o be he e of node reachable from in G f (Δ). By definiion of cu A: A. By definiion of flow f: A. original flow nework edge e = (v, w) wih v B, w A mu have f(e) < Δ val(f) = e A f(e) e A f(e) A B e A (c(e) ) e A e A c(e) e A e A cap(a, B) m edge e = (v, w) wih v A, w B mu have f(e) > c(e) Δ 40

41 7. NETWORK FLOW I SECTION 17.2 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

42 Shore augmening pah Q. Which augmening pah? A. The one wih he fewe edge. can find via BFS SHORTEST-AUGMENTING-PATH (G) FOREACH e E : f (e) 0. Gf reidual nework of G wih repec o flow f. WHILE (here exi an pah in Gf ) P BREADTH-FIRST-SEARCH (Gf ). f AUGMENT ( f, c, P). Updae Gf. RETURN f. 42

43 Shore augmening pah: overview of analyi Lemma 1. Throughou he algorihm, he lengh of a hore augmening pah never decreae. Lemma 2. Afer a mo m hore-pah augmenaion, he lengh of a hore augmening pah ricly increae. Theorem. The hore-augmening-pah algorihm run in O(m 2 n) ime. Pf. O(m + n) ime o find hore augmening pah via BFS. O(m) augmenaion for pah of lengh k. If here i an augmening pah, here i a imple one. 1 k < n O(m n) augmenaion. 43

44 Shore augmening pah: analyi Def. Given a digraph G = (V, E) wih ource, i level graph i defined by: (v) = number of edge in hore pah from o v. L G = (V, E G ) i he ubgraph of G ha conain only hoe edge (v, w) E wih (w) = (v) + 1. graph G level graph LG = 0 = 1 = 2 = 3 44

45 Shore augmening pah: analyi Def. Given a digraph G = (V, E) wih ource, i level graph i defined by: (v) = number of edge in hore pah from o v. L G = (V, E G ) i he ubgraph of G ha conain only hoe edge (v, w) E wih (w) = (v) + 1. Propery. Can compue level graph in O(m + n) ime. Pf. Run BFS; delee back and ide edge. Key propery. P i a hore v pah in G iff P i an v pah L G. level graph LG = 0 = 1 = 2 = 3 45

46 Shore augmening pah: analyi Lemma 1. The lengh of a hore augmening pah never decreae. Le f and f ʹ be flow before and afer a hore-pah augmenaion. Le L and Lʹ be level graph of G f and G f ʹ. Only back edge added o G f (any pah wih a back edge i longer han previou lengh) level graph L = 0 = 1 = 2 = 3 level graph L 46

47 Shore augmening pah: analyi Lemma 2. Afer a mo m hore-pah augmenaion, he lengh of a hore augmening pah ricly increae. The boleneck edge() i deleed from L afer each augmenaion. No new edge added o L unil lengh of hore pah ricly increae. level graph L = 0 = 1 = 2 = 3 level graph L 47

48 Shore augmening pah: review of analyi Lemma 1. Throughou he algorihm, he lengh of a hore augmening pah never decreae. Lemma 2. Afer a mo m hore-pah augmenaion, he lengh of a hore augmening pah ricly increae. Theorem. The hore-augmening-pah algorihm run in O(m 2 n) ime. Pf. O(m + n) ime o find hore augmening pah via BFS. O(m) augmenaion for pah of lengh k. If here i an augmening pah, here i a imple one. 1 k < n O(m n) augmenaion. 48

49 Shore augmening pah: improving he running ime Noe. Θ(m n) augmenaion neceary on ome flow nework. Try o decreae ime per augmenaion inead. Simple idea O(m n 2 ) [Diniz 1970] Dynamic ree O(m n log n) [Sleaor Tarjan 1983] A Daa Srucure for Dynamic Tree DANIEL D. SLEATOR AND ROBERT ENDRE TARJAN Bell Laboraorie, Murray Hill, New Jerey Received May 8, 1982; revied Ocober 18, 1982 A daa rucure i propoed o mainain a collecion of verex-dijoin ree under a equence of wo kind of operaion: a link operaion ha combine wo ree ino one by adding an edge, and a cu operaion ha divide one ree ino wo by deleing an edge. Each operaion require O(log n) ime. Uing hi daa rucure, new fa algorihm are obained for he following problem: (1) Compuing neare common anceor. (2) Solving variou nework flow problem including finding maximum flow, blocking flow, and acyclic flow. (3) Compuing cerain kind of conrained minimum panning ree. (4) Implemening he nework implex algorihm for minimum-co flow. The mo ignifican applicaion i (2); an O(mn log n)-ime algorihm i obained o find a maximum flow in a nework of n verice and m edge, beaing by a facor of log n he fae algorihm previouly known for pare graph. 49

50 7. NETWORK FLOW I SECTION 18.1 max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

51 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. level graph LG 51

52 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. advance level graph LG 52

53 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. augmen level graph LG 53

54 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. advance level graph LG 54

55 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. rerea level graph LG 55

56 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. advance level graph LG 56

57 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. augmen level graph LG 57

58 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. advance level graph LG 58

59 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. rerea level graph LG 59

60 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. rerea level graph LG 60

61 Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. end of phae level graph LG 61

62 Blocking-flow algorihm INITIALIZE (G, f ) ADVANCE (v) LG level-graph of Gf. P. GOTO ADVANCE (). RETREAT (v) IF (v = ) AUGMENT(P). Remove auraed edge from LG. P. GOTO ADVANCE (). IF (v = ) STOP. ELSE Delee v (and all inciden edge) from LG. Remove la edge (u, v) from P. GOTO ADVANCE (u). IF (here exi edge (v, w) LG) Add edge (v, w) o P. GOTO ADVANCE (w). ELSE GOTO RETREAT (v). 62

63 Blocking-flow algorihm: analyi Lemma. A phae can be implemened o run in O(m n) ime. Pf. Iniializaion happen once per phae. A mo m augmenaion per phae. (becaue an augmenaion delee a lea one edge from L G ) A mo n rerea per phae. (becaue a rerea delee one node from L G ) A mo m n advance per phae. O(m) uing BFS O(mn) per phae O(m + n) per phae O(mn) per phae (becaue a mo n advance before rerea or augmenaion) Theorem. [Diniz 1970] The blocking-flow algorihm run in O(mn 2 ) ime. Pf. By lemma, O(mn) ime per phae. A mo n phae (a in hore-augmening-pah analyi). 63

64 Chooing good augmening pah: ummary year mehod # augmenaion running ime 1955 augmening pah n C O(m n C) 1970 fae augmening pah m log (mc) O(m 2 log n log (mc)) 1972 capaciy caling m log C O(m 2 log C) 1985 improved capaciy caling m log C O(m n log C) 1970 hore augmening pah m n O(m 2 n) 1970 blocking flow m n O(m n 2 ) 1983 dynamic ree m n O(m n log n ) augmening pah algorihm wih m edge, n node and ineger capaciie beween 1 and C 64

65 Maximum-flow algorihm: heory year mehod wor cae dicovered by 1951 implex O(m 3 C) Danzig 1955 augmening pah O(m 2 C) Ford Fulkeron 1970 hore augmening pah O(m 3 ) Diniz, Edmond Karp 1970 fae augmening pah O(m 2 log m log( m C )) Diniz, Edmond Karp 1977 blocking flow O(m 5/2 ) Cherkaky 1978 blocking flow O(m 7/3 ) Galil 1983 dynamic ree O(m 2 log m) Sleaor Tarjan 1985 improved capaciy caling O(m 2 log C) Gabow 1997 lengh funcion O(m 3/2 log m log C) Goldberg Rao 2012 compac nework O(m 2 / log m) Orlin?? O(m)? max-flow algorihm for pare digraph wih m edge, ineger capaciie beween 1 and C 65

66 Maximum-flow algorihm: pracice Puh-relabel algorihm (SECTION 7.4). [Goldberg Tarjan 1988] Increae flow one edge a a ime inead of one augmening pah a a ime. A New Approach o he Maximum-Flow Problem ANDREW V. GOLDBERG Maachue Iniue of Technology, Cambridge, Maachue AND ROBERT E. TARJAN Princeon Univeriy, Princeon, New Jerey, and AT&T Bell Laboraorie, Murray Hill, New Jerey Abrac. All previouly known effcien maximum-flow algorihm work by finding augmening pah, eiher one pah a a ime (a in he original Ford and Fulkeron algorihm) or all hore-lengh augmening pah a once (uing he layered nework approach of Dinic). An alernaive mehod baed on he preflow concep of Karzanov i inroduced. A preflow i like a flow, excep ha he oal amoun flowing ino a verex i allowed o exceed he oal amoun flowing ou. The mehod mainain a preflow in he original nework and puhe local flow exce oward he ink along wha are eimaed o be hore pah. The algorihm and i analyi are imple and inuiive, ye he algorihm run a fa a any oher known mehod on dene. graph, achieving an O(n)) ime bound on an n-verex graph. By incorporaing he dynamic ree daa rucure of Sleaor and Tarjan, we obain a verion of he algorihm running in O(nm log(n /m)) ime on an n-verex, m-edge graph. Thi i a fa a any known mehod for any graph deniy and faer on graph of moderae deniy. The algorihm alo admi eficien diribued and parallel implemenaion. A parallel implemenaion running in O(n log n) ime uing n proceor and O(m) pace i obained. Thi ime bound mache ha of he Shiloach-Vihkin algorihm, which alo ue n proceor bu require O(n ) pace. Caegorie and Subjec Decripor: F.2.2 [Analyi of Algorihm and Problem Complexiy]: Non- 66

67 Maximum-flow algorihm: pracice Warning. Wor-cae running ime i generally no ueful for predicing or comparing max-flow algorihm performance in pracice. Be in pracice. Puh relabel mehod wih gap relabeling: O(m 3/2 ). On Implemening Puh-Relabel Mehod for he Maximum Flow Problem Bori V. Cherkaky 1 and Andrew V. Goldberg 2 1 Cenral Iniue for Economic and Mahemaic, Kraikova S. 32, , Mocow, Ruia cher@eemi.mk.u 2 Compuer Science Deparmen, Sanford Univeriy Sanford, CA 94305, USA goldberg ~c. anford, edu Abrac. We udy efficien implemenaion of he puh-relabel mehod for he maximum flow problem. The reuling code are faer han he previou code, and much faer on ome problem familie. The peedup i due o he combinaion of heuriic ued in our implemenaion. We alo exhibi a family of problem for which he running ime of all known mehod eem o have a roughly quadraic growh rae. ELSEVIER European Journal of Operaional Reearch 97 (1997) Theory and Mehodology EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Compuaional inveigaion of maximum flow algorihm Ravindra K. Ahuja a, Murali Kodialam a b, Ajay K. Mihra c, Jame B. Orlin d,. Deparmen ~'lndurial and Managemen Engineering. Indian Iniue of Technology. Kanpur, , India b AT& T Bell Laboraorie, Holmdel, NJ 07733, USA c KA'F-Z Graduae School of Buine, Univeriy of Piburgh, Piburgh, PA 15260, USA d Sloun School of Managemen, Maachue Iniue of Technology. Cambridge. MA USA Received 30 Augu 1995; acceped 27 June

68 Maximum-flow algorihm: pracice Compuer viion. Differen algorihm work beer for ome dene problem ha arie in applicaion o compuer viion. An Experimenal Comparion of Min-Cu/Max-Flow Algorihm for Energy Minimizaion in Viion Yuri Boykov and Vladimir Kolmogorov Abrac VERMA, BATRA: MAXFLOW REVISITED 1 MaxFlow Reviied: An Empirical Comparion of Maxflow Algorihm for Dene Viion Problem Afer [15, 31, 19, 8, 25, 5] minimum cu/maximum flow algorihm on graph emerged a an increaingly ueful ool for exac or approximae energy minimizaion in low-level viion. The combinaorial opimizaion lieraure provide many min-cu/max-flow algorihm wih differen polynomial ime complexiy. Their pracical efficiency, however, ha o dae been udied mainly ouide he cope of compuer viion. The goal ofhipaperioprovidean experimenal comparion of he efficiency of min-cu/max flow algorihm for applicaion in viion. We compare he running ime of everal andard algorihm, a well a a Tanmay Verma anmay08054@iiid.ac.in Dhruv Bara dbara@ic.edu Abrac IIIT-Delhi Delhi, India TTI-Chicago Chicago, USA new algorihm ha we have recenly developed. The algorihm we udy include boh Goldberg-Tarjan yle puh-relabel mehod and algorihm baed on Ford-Fulkeron yle augmening pah. We benchmark hee algorihm on a number of ypical graph in he conex of image reoraion, ereo, and egmenaion. In many cae our new algorihm work everal ime faer han any of he oher mehod making near real-ime performance poible. An implemenaion of our max-flow/min-cu algorihm i available upon reque for reearch purpoe. Algorihm for finding he maximum amoun of flow poible in a nework (or maxflow) play a cenral role in compuer viion problem. We preen an empirical comparion of differen max-flow algorihm on modern problem. Our problem inance arie from energy minimizaion problem in Objec Caegory Segmenaion, Image Deconvoluion, Super Reoluion, Texure Reoraion, Characer Compleion and 3D Segmenaion. We compare 14 differen implemenaion and find ha he mo popularly ued implemenaion of Kolmogorov [5] i no longer he fae algorihm available, epecially for dene graph. 68

69 7. NETWORK FLOW I max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework

70 Biparie maching Q. Which max-flow algorihm o ue for biparie maching? Generic augmening pah: O(m val( f * )) = O(m n). Capaciy caling: O(m 2 log C) = O(m 2 ). Blocking flow: O(m n 2 ). Q. Sugge more ophiicaed algorihm are no o fa when C = 1. A. No, ju need more clever analyi! Nex. We prove ha hore-augmening-pah algorihm can be implemened o run in O(m n 1/2 ) ime. NETWORK FLOW AND TESTING GRAPH CONNECTIVITY* SHIMON EVEN" AND R. ENDRE TARJAN:I: Abrac. An algorihm of Dinic for finding he maximum flow in a nework i decribed. I i hen hown ha if he verex capaciie are all equal o one, he algorihm require a mo O(IV[ 1/2 IEI) ime, and if he edge capaciie are all equal o one, he algorihm require a mo O(I VI 2/3. IEI) ime. Alo, hee bound are igh for Dinic algorihm. Thee reul are ued o e he verex conneciviy of a graph in O(IVI 1/z. IEI 2) ime and he edge conneciviy in O(I V[ 5/3. IEI) ime. 70

71 Simple uni-capaciy nework Def. A flow nework i a imple uni-capaciy nework if: Every edge ha capaciy 1. Every node (oher han or ) ha eiher (i) exacly one enering edge or (ii) exacly one leaving edge (or boh). Propery. Le G be a imple uni-capaciy nework and le f be a 0 1 flow, hen Gf i a imple uni-capaciy nework. Ex. Biparie maching

72 Simple uni-capaciy nework Shore-augmening-pah algorihm. Normal augmenaion: lengh of hore pah doe no change. Special augmenaion: lengh of hore pah ricly increae. Theorem. [Even Tarjan 1975] In imple uni-capaciy nework, he horeaugmening-pah algorihm compue a maximum flow in O(m n 1/2 ) ime. Pf. Lemma 1. Each phae of normal augmenaion ake O(m) ime. Lemma 2. Afer a mo n 1/2 phae, val( f ) val( f * ) n 1/2. Lemma 3. Afer a mo n 1/2 addiional augmenaion, flow i opimal. Lemma 3. Afer a mo n 1/2 addiional augmenaion, flow i opimal. Pf. Each augmenaion increae flow value by a lea 1. 72

73 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG advance level graph LG 73

74 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG augmen level graph LG 74

75 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG advance level graph LG 75

76 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG rerea level graph LG 76

77 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG advance level graph LG 77

78 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG augmen level graph LG 78

79 Simple uni-capaciy nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. delee all edge in augmening pah from LG end of phae level graph LG 79

80 Simple uni-capaciy nework: analyi Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, advance along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. If ge uck, delee node from L G and go o previou node. Lemma 1. A phae of normal augmenaion ake O(m) ime. Pf. O(m) o creae level graph L G. O(1) per edge ince each edge ravered and deleed a mo once. O(1) per node ince each node deleed a mo once. 80

81 Simple uni-capaciy nework: analyi Lemma 2. Afer a mo n 1/2 phae, val( f ) val( f * ) n 1/2. Afer n 1/2 phae, lengh of hore augmening pah i > n 1/2. Level graph ha more han n 1/2 level. Le 1 h n 1/2 be layer wih min number of node: Vh n 1/2. level graph LG for flow f V 0 V n 1/2 V 1 V h 81

82 Simple uni-capaciy nework: analyi Lemma 2. Afer a mo n 1/2 phae, val( f ) val( f * ) n 1/2. Afer n 1/2 phae, lengh of hore augmening pah i > n 1/2. Level graph ha more han n 1/2 level. Le 1 h n 1/2 be layer wih min number of node: Vh n 1/2. Le A = {v : (v) < h} {v : (v) = h and v ha 1 ougoing reidual edge}. capf (A, B) Vh n 1/2 val( f ) val( f * ) n 1/2. reidual nework Gf reidual edge A V 0 V 1 V h V 1/2 n 82

7. NETWORK FLOW I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 11/22/17 6:11 AM

7. NETWORK FLOW I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 11/22/17 6:11 AM 7. NETWORK FLOW I max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework Lecure lide by Kevin

More information

Max Flow, Min Cut COS 521. Kevin Wayne Fall Soviet Rail Network, Cuts. Minimum Cut Problem. Flow network.

Max Flow, Min Cut COS 521. Kevin Wayne Fall Soviet Rail Network, Cuts. Minimum Cut Problem. Flow network. Sovie Rail Nework, Max Flow, Min u OS Kevin Wayne Fall Reference: On he hiory of he ranporaion and maximum flow problem. lexander Schrijver in Mah Programming, :,. Minimum u Problem u Flow nework.! Digraph

More information

Soviet Rail Network, 1955

Soviet Rail Network, 1955 Sovie Rail Nework, 1 Reference: On he hiory of he ranporaion and maximum flow problem. Alexander Schrijver in Mah Programming, 1: 3,. Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic

More information

Algorithms. Algorithms 6.4 MAXIMUM FLOW

Algorithms. Algorithms 6.4 MAXIMUM FLOW Algorihm ROBERT SEDGEWICK KEVIN WAYNE 6.4 MAXIMUM FLOW Algorihm F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE hp://alg4.c.princeon.edu inroducion Ford Fulkeron algorihm maxflow mincu heorem analyi

More information

! Abstraction for material flowing through the edges. ! G = (V, E) = directed graph, no parallel edges.

! Abstraction for material flowing through the edges. ! G = (V, E) = directed graph, no parallel edges. Sovie Rail Nework, haper Nework Flow Slide by Kevin Wayne. opyrigh Pearon-ddion Weley. ll righ reerved. Reference: On he hiory of he ranporaion and maximum flow problem. lexander Schrijver in Mah Programming,

More information

The Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm

The Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,

More information

Maximum Flow and Minimum Cut

Maximum Flow and Minimum Cut // Sovie Rail Nework, Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy. Nework Flow Flow nework.

More information

The Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm

The Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,

More information

Ford Fulkerson algorithm max-flow min-cut theorem. max-flow min-cut theorem capacity-scaling algorithm

Ford Fulkerson algorithm max-flow min-cut theorem. max-flow min-cut theorem capacity-scaling algorithm 7. NETWORK FLOW I 7. NETWORK FLOW I max-flow and min-cu problem max-flow and min-cu problem Ford Fulkeron algorihm Ford Fulkeron algorihm max-flow min-cu heorem max-flow min-cu heorem capaciy-caling algorihm

More information

Maximum Flow. Contents. Max Flow Network. Maximum Flow and Minimum Cut

Maximum Flow. Contents. Max Flow Network. Maximum Flow and Minimum Cut Conen Maximum Flow Conen. Maximum low problem. Minimum cu problem. Max-low min-cu heorem. Augmening pah algorihm. Capaciy-caling. Shore augmening pah. Princeon Univeriy COS Theory o Algorihm Spring Kevin

More information

Soviet Rail Network, 1955

Soviet Rail Network, 1955 7.1 Nework Flow Sovie Rail Nework, 19 Reerence: On he hiory o he ranporaion and maximum low problem. lexander Schrijver in Mah Programming, 91: 3, 00. (See Exernal Link ) Maximum Flow and Minimum Cu Max

More information

6/3/2009. CS 244 Algorithm Design Instructor: t Artur Czumaj. Lecture 8 Network flows. Maximum Flow and Minimum Cut. Minimum Cut Problem.

6/3/2009. CS 244 Algorithm Design Instructor: t Artur Czumaj. Lecture 8 Network flows. Maximum Flow and Minimum Cut. Minimum Cut Problem. Maximum Flow and Minimum Cu CS lgorihm Deign Inrucor: rur Czumaj Lecure Nework Max and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy.

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorihm Deign and Analyi LECTURES 17 Nework Flow Dualiy of Max Flow and Min Cu Algorihm: Ford-Fulkeron Capaciy Scaling Sofya Rakhodnikova S. Rakhodnikova; baed on lide by E. Demaine, C. Leieron, A. Smih,

More information

Today: Max Flow Proofs

Today: Max Flow Proofs Today: Max Flow Proof COSC 58, Algorihm March 4, 04 Many of hee lide are adaped from everal online ource Reading Aignmen Today cla: Chaper 6 Reading aignmen for nex cla: Chaper 7 (Amorized analyi) In-Cla

More information

Graphs III - Network Flow

Graphs III - Network Flow Graph III - Nework Flow Flow nework eup graph G=(V,E) edge capaciy w(u,v) 0 - if edge doe no exi, hen w(u,v)=0 pecial verice: ource verex ; ink verex - no edge ino and no edge ou of Aume every verex v

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorihm Deign and Analyi LECTURE 0 Nework Flow Applicaion Biparie maching Edge-dijoin pah Adam Smih 0//0 A. Smih; baed on lide by E. Demaine, C. Leieron, S. Rakhodnikova, K. Wayne La ime: Ford-Fulkeron

More information

1 Motivation and Basic Definitions

1 Motivation and Basic Definitions CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider

More information

MAXIMUM FLOW. introduction Ford-Fulkerson algorithm maxflow-mincut theorem

MAXIMUM FLOW. introduction Ford-Fulkerson algorithm maxflow-mincut theorem MAXIMUM FLOW inroducion Ford-Fulkeron algorihm maxflow-mincu heorem Mincu problem Inpu. An edge-weighed digraph, ource verex, and arge verex. each edge ha a poiive capaciy capaciy 9 10 4 15 15 10 5 8 10

More information

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

More information

Main Reference: Sections in CLRS.

Main Reference: Sections in CLRS. Maximum Flow Reied 09/09/200 Main Reference: Secion 26.-26. in CLRS. Inroducion Definiion Muli-Source Muli-Sink The Ford-Fulkeron Mehod Reidual Nework Augmening Pah The Max-Flow Min-Cu Theorem The Edmond-Karp

More information

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow. CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex

More information

Network Flows: Introduction & Maximum Flow

Network Flows: Introduction & Maximum Flow CSC 373 - lgorihm Deign, nalyi, and Complexiy Summer 2016 Lalla Mouaadid Nework Flow: Inroducion & Maximum Flow We now urn our aenion o anoher powerful algorihmic echnique: Local Search. In a local earch

More information

Matching. Slides designed by Kevin Wayne.

Matching. Slides designed by Kevin Wayne. Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node appear in a mo edge in M. Max maching: find a max cardinaliy maching. Slide deigned by Kevin Wayne. Biparie Maching Biparie

More information

Greedy. I Divide and Conquer. I Dynamic Programming. I Network Flows. Network Flow. I Previous topics: design techniques

Greedy. I Divide and Conquer. I Dynamic Programming. I Network Flows. Network Flow. I Previous topics: design techniques Algorihm Deign Technique CS : Nework Flow Dan Sheldon April, reedy Divide and Conquer Dynamic Programming Nework Flow Comparion Nework Flow Previou opic: deign echnique reedy Divide and Conquer Dynamic

More information

CS 473G Lecture 15: Max-Flow Algorithms and Applications Fall 2005

CS 473G Lecture 15: Max-Flow Algorithms and Applications Fall 2005 CS 473G Lecure 1: Max-Flow Algorihm and Applicaion Fall 200 1 Max-Flow Algorihm and Applicaion (November 1) 1.1 Recap Fix a direced graph G = (V, E) ha doe no conain boh an edge u v and i reveral v u,

More information

4/12/12. Applications of the Maxflow Problem 7.5 Bipartite Matching. Bipartite Matching. Bipartite Matching. Bipartite matching: the flow network

4/12/12. Applications of the Maxflow Problem 7.5 Bipartite Matching. Bipartite Matching. Bipartite Matching. Bipartite matching: the flow network // Applicaion of he Maxflow Problem. Biparie Maching Biparie Maching Biparie maching. Inpu: undireced, biparie graph = (, E). M E i a maching if each node appear in a mo one edge in M. Max maching: find

More information

CSE 521: Design & Analysis of Algorithms I

CSE 521: Design & Analysis of Algorithms I CSE 52: Deign & Analyi of Algorihm I Nework Flow Paul Beame Biparie Maching Given: A biparie graph G=(V,E) M E i a maching in G iff no wo edge in M hare a verex Goal: Find a maching M in G of maximum poible

More information

introduction Ford-Fulkerson algorithm

introduction Ford-Fulkerson algorithm Algorihm ROBERT SEDGEWICK KEVIN WAYNE. MAXIMUM FLOW. MAXIMUM FLOW inroducion inroducion Ford-Fulkeron algorihm Ford-Fulkeron algorihm Algorihm F O U R T H E D I T I O N maxflow-mincu heorem analyi of running

More information

introduction Ford-Fulkerson algorithm

introduction Ford-Fulkerson algorithm Algorihm ROBERT SEDGEWICK KEVIN WAYNE. MAXIMUM FLOW. MAXIMUM FLOW inroducion inroducion Ford-Fulkeron algorihm Ford-Fulkeron algorihm Algorihm F O U R T H E D I T I O N maxflow-mincu heorem analyi of running

More information

Maximum Flow. How do we transport the maximum amount data from source to sink? Some of these slides are adapted from Lecture Notes of Kevin Wayne.

Maximum Flow. How do we transport the maximum amount data from source to sink? Some of these slides are adapted from Lecture Notes of Kevin Wayne. Conen Conen. Maximum flow problem. Minimum cu problem. Max-flow min-cu heorem. Augmening pah algorihm. Capaciy-caling. Shore augmening pah. Chaper Maximum How do we ranpor he maximum amoun daa from ource

More information

Network Flows UPCOPENCOURSEWARE number 34414

Network Flows UPCOPENCOURSEWARE number 34414 Nework Flow UPCOPENCOURSEWARE number Topic : F.-Javier Heredia Thi work i licened under he Creaive Common Aribuion- NonCommercial-NoDeriv. Unpored Licene. To view a copy of hi licene, vii hp://creaivecommon.org/licene/by-nc-nd/./

More information

Network Flow. Data Structures and Algorithms Andrei Bulatov

Network Flow. Data Structures and Algorithms Andrei Bulatov Nework Flow Daa Srucure and Algorihm Andrei Bulao 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

More information

CSE 421 Introduction to Algorithms Winter The Network Flow Problem

CSE 421 Introduction to Algorithms Winter The Network Flow Problem CSE 42 Inroducion o Algorihm Winer 202 The Nework Flow Problem 2 The Nework Flow Problem 5 a 4 3 x 3 7 6 b 4 y 4 7 6 c 5 z How much uff can flow from o? 3 Sovie Rail Nework, 955 Reference: On he hiory

More information

16 Max-Flow Algorithms and Applications

16 Max-Flow Algorithms and Applications Algorihm A proce canno be underood by opping i. Underanding mu move wih he flow of he proce, mu join i and flow wih i. The Fir Law of Mena, in Frank Herber Dune (196) There a difference beween knowing

More information

Algorithmic Discrete Mathematics 6. Exercise Sheet

Algorithmic Discrete Mathematics 6. Exercise Sheet Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap

More information

CSC 364S Notes University of Toronto, Spring, The networks we will consider are directed graphs, where each edge has associated with it

CSC 364S Notes University of Toronto, Spring, The networks we will consider are directed graphs, where each edge has associated with it CSC 36S Noe Univeriy of Torono, Spring, 2003 Flow Algorihm The nework we will conider are direced graph, where each edge ha aociaed wih i a nonnegaive capaciy. The inuiion i ha if edge (u; v) ha capaciy

More information

Randomized Perfect Bipartite Matching

Randomized Perfect Bipartite Matching Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for

More information

CS4445/9544 Analysis of Algorithms II Solution for Assignment 1

CS4445/9544 Analysis of Algorithms II Solution for Assignment 1 Conider he following flow nework CS444/944 Analyi of Algorihm II Soluion for Aignmen (0 mark) In he following nework a minimum cu ha capaciy 0 Eiher prove ha hi aemen i rue, or how ha i i fale Uing he

More information

Flow networks, flow, maximum flow. Some definitions. Edmonton. Saskatoon Winnipeg. Vancouver Regina. Calgary. 12/12 a.

Flow networks, flow, maximum flow. Some definitions. Edmonton. Saskatoon Winnipeg. Vancouver Regina. Calgary. 12/12 a. Flow nework, flow, maximum flow Can inerpre direced graph a flow nework. Maerial coure hrough ome yem from ome ource o ome ink. Source produce maerial a ome eady rae, ink conume a ame rae. Example: waer

More information

Flow Networks. Ma/CS 6a. Class 14: Flow Exercises

Flow Networks. Ma/CS 6a. Class 14: Flow Exercises 0/0/206 Ma/CS 6a Cla 4: Flow Exercie Flow Nework A flow nework i a digraph G = V, E, ogeher wih a ource verex V, a ink verex V, and a capaciy funcion c: E N. Capaciy Source 7 a b c d e Sink 0/0/206 Flow

More information

Algorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov)

Algorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov) Algorihm and Daa Srucure 2011/ Week Soluion (Tue 15h - Fri 18h No) 1. Queion: e are gien 11/16 / 15/20 8/13 0/ 1/ / 11/1 / / To queion: (a) Find a pair of ube X, Y V uch ha f(x, Y) = f(v X, Y). (b) Find

More information

Reminder: Flow Networks

Reminder: Flow Networks 0/0/204 Ma/CS 6a Cla 4: Variou (Flow) Execie Reminder: Flow Nework A flow nework i a digraph G = V, E, ogeher wih a ource verex V, a ink verex V, and a capaciy funcion c: E N. Capaciy Source 7 a b c d

More information

18 Extensions of Maximum Flow

18 Extensions of Maximum Flow Who are you?" aid Lunkwill, riing angrily from hi ea. Wha do you wan?" I am Majikhie!" announced he older one. And I demand ha I am Vroomfondel!" houed he younger one. Majikhie urned on Vroomfondel. I

More information

Admin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t)

Admin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t) /0/ dmin lunch oday rading MX LOW PPLIION 0, pring avid Kauchak low graph/nework low nework direced, weighed graph (V, ) poiive edge weigh indicaing he capaciy (generally, aume ineger) conain a ingle ource

More information

3/3/2015. Chapter 7. Network Flow. Maximum Flow and Minimum Cut. Minimum Cut Problem

3/3/2015. Chapter 7. Network Flow. Maximum Flow and Minimum Cut. Minimum Cut Problem // Chaper Nework Flow Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy. Nonrivial applicaion /

More information

Introduction to Congestion Games

Introduction to Congestion Games Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game

More information

23 Maximum Flows and Minimum Cuts

23 Maximum Flows and Minimum Cuts A proce canno be underood by opping i. Underanding mu move wih he flow of he proce, mu join i and flow wih i. The Fir Law of Mena, in Frank Herber Dune (196) Conrary o expecaion, flow uually happen no

More information

They were originally developed for network problem [Dantzig, Ford, Fulkerson 1956]

They were originally developed for network problem [Dantzig, Ford, Fulkerson 1956] 6. Inroducion... 6. The primal-dual algorihmn... 6 6. Remark on he primal-dual algorihmn... 7 6. A primal-dual algorihmn for he hore pah problem... 8... 9 6.6 A primal-dual algorihmn for he weighed maching

More information

Basic Tools CMSC 641. Running Time. Problem. Problem. Algorithmic Design Paradigms. lg (n!) (lg n)! (lg n) lgn n.2

Basic Tools CMSC 641. Running Time. Problem. Problem. Algorithmic Design Paradigms. lg (n!) (lg n)! (lg n) lgn n.2 Baic Tool CMSC April, Review Aympoic Noaion Order of Growh Recurrence relaion Daa Srucure Li, Heap, Graph, Tree, Balanced Tree, Hah Table Advanced daa rucure: Binomial Heap, Fibonacci Heap Soring Mehod

More information

26.1 Flow networks. f (u,v) = 0.

26.1 Flow networks. f (u,v) = 0. 26 Maimum Flow Ju a we can model a road map a a direced graph in order o find he hore pah from one poin o anoher, we can alo inerpre a direced graph a a flow nework and ue i o anwer queion abou maerial

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual

More information

Maximum Flow in Planar Graphs

Maximum Flow in Planar Graphs Maximum Flow in Planar Graph Planar Graph and i Dual Dualiy i defined for direced planar graph a well Minimum - cu in undireced planar graph An - cu (undireced graph) An - cu The dual o he cu Cu/Cycle

More information

Please Complete Course Survey. CMPSCI 311: Introduction to Algorithms. Approximation Algorithms. Coping With NP-Completeness. Greedy Vertex Cover

Please Complete Course Survey. CMPSCI 311: Introduction to Algorithms. Approximation Algorithms. Coping With NP-Completeness. Greedy Vertex Cover Pleae Complee Coure Survey CMPSCI : Inroducion o Algorihm Dealing wih NP-Compleene Dan Sheldon hp: //owl.oi.uma.edu/parner/coureevalsurvey/uma/ Univeriy of Maachue Slide Adaped from Kevin Wayne La Compiled:

More information

Average Case Lower Bounds for Monotone Switching Networks

Average Case Lower Bounds for Monotone Switching Networks Average Cae Lower Bound for Monoone Swiching Nework Yuval Filmu, Toniann Piai, Rober Robere, Sephen Cook Deparmen of Compuer Science Univeriy of Torono Monoone Compuaion (Refreher) Monoone circui were

More information

Chapter 7. Network Flow. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.

Chapter 7. Network Flow. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. Chapter 7 Network Flow Slide by Kevin Wayne. Copyright 5 Pearon-Addion Weley. All right reerved. Soviet Rail Network, 55 Reference: On the hitory of the tranportation and maximum flow problem. Alexander

More information

Today s topics. CSE 421 Algorithms. Problem Reduction Examples. Problem Reduction. Undirected Network Flow. Bipartite Matching. Problem Reductions

Today s topics. CSE 421 Algorithms. Problem Reduction Examples. Problem Reduction. Undirected Network Flow. Bipartite Matching. Problem Reductions Today opic CSE Algorihm Richard Anderon Lecure Nework Flow Applicaion Prolem Reducion Undireced Flow o Flow Biparie Maching Dijoin Pah Prolem Circulaion Loweround conrain on flow Survey deign Prolem Reducion

More information

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

CMPS 6610/4610 Fall Flow Networks. Carola Wenk Slides adapted from slides by Charles Leiserson CMP 6610/4610 Fall 2016 Flow Nework Carola Wenk lide adaped rom lide by Charle Leieron Max low and min c Fndamenal problem in combinaorial opimizaion Daliy beween max low and min c Many applicaion: Biparie

More information

16 Max-Flow Algorithms

16 Max-Flow Algorithms A process canno be undersood by sopping i. Undersanding mus move wih he flow of he process, mus join i and flow wih i. The Firs Law of Mena, in Frank Herber s Dune (196) There s a difference beween knowing

More information

April 3, The maximum flow problem. See class notes on website.

April 3, The maximum flow problem. See class notes on website. 5.05 April, 007 The maximum flow problem See cla noe on webie. Quoe of he day You ge he maxx for he minimum a TJ Maxx. -- ad for a clohing ore Thi wa he mo unkinde cu of all -- Shakepeare in Juliu Caear

More information

u(t) Figure 1. Open loop control system

u(t) Figure 1. Open loop control system Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference

More information

Network flows. The problem. c : V V! R + 0 [ f+1g. flow network G = (V, E, c), a source s and a sink t uv not in E implies c(u, v) = 0

Network flows. The problem. c : V V! R + 0 [ f+1g. flow network G = (V, E, c), a source s and a sink t uv not in E implies c(u, v) = 0 Nework flow The problem Seing flow nework G = (V, E, c), a orce and a ink no in E implie c(, ) = 0 Flow from o capaciy conrain kew-ymmery flow-coneraion ale of he flow jfj = P 2V Find a maximm flow from

More information

7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching

7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching Chaper. Biparie Maching Nework Flow Slide by Kein Wayne. Copyrigh 00 Pearon-Addion Weley. All righ reered. Maching Biparie Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node

More information

Random Walk with Anti-Correlated Steps

Random Walk with Anti-Correlated Steps Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and

More information

7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching

7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching Chaper. Biparie Maching Nework Flow Slide by Kevin Wayne. Copyrigh PearonAddion Weley. All righ reerved. Maching Biparie Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node

More information

CHAPTER 7: SECOND-ORDER CIRCUITS

CHAPTER 7: SECOND-ORDER CIRCUITS EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha

More information

20/20 20/20 0/5 0/5 20/20 20/20 5/5 0/5 0/5 5/5 0/20 25/30 20/20 30/30 20/20 0/5 5/5 20/20 0/5 0/5 15/20 15/25 20/20 10/10

20/20 20/20 0/5 0/5 20/20 20/20 5/5 0/5 0/5 5/5 0/20 25/30 20/20 30/30 20/20 0/5 5/5 20/20 0/5 0/5 15/20 15/25 20/20 10/10 Annoncemen CSEP Applied Algorihm Richard Anderon Lecre 9 Nework Flow Applicaion Reading for hi week 7.-7.. Nework flow applicaion Nex week: Chaper 8. NP-Compleene Final exam, March 8, 6:0 pm. A UW. hor

More information

Algorithms and Theory of Computation. Lecture 11: Network Flow

Algorithms and Theory of Computation. Lecture 11: Network Flow Algorithms and Theory of Computation Lecture 11: Network Flow Xiaohui Bei MAS 714 September 18, 2018 Nanyang Technological University MAS 714 September 18, 2018 1 / 26 Flow Network A flow network is a

More information

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3

More information

Selfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos

Selfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos Selfih Rouing Tim Roughgarden Cornell Univeriy Include join work wih Éva Tardo 1 Which roue would you chooe? Example: one uni of raffic (e.g., car) wan o go from o delay = 1 hour (no congeion effec) long

More information

Longest Common Prefixes

Longest Common Prefixes Longes Common Prefixes The sandard ordering for srings is he lexicographical order. I is induced by an order over he alphabe. We will use he same symbols (,

More information

18.03SC Unit 3 Practice Exam and Solutions

18.03SC Unit 3 Practice Exam and Solutions Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care

More information

An introduction to the theory of SDDP algorithm

An introduction to the theory of SDDP algorithm An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking

More information

Geometric Path Problems with Violations

Geometric Path Problems with Violations Click here o view linked Reference 1 1 1 0 1 0 1 0 1 0 1 Geomeric Pah Problem wih Violaion Anil Mahehwari 1, Subha C. Nandy, Drimi Paanayak, Saanka Roy and Michiel Smid 1 1 School of Compuer Science, Carleon

More information

Mechtild Stoer * Frank Wagner** Abstract. fastest algorithm known. The runtime analysis is straightforward. In contrast to

Mechtild Stoer * Frank Wagner** Abstract. fastest algorithm known. The runtime analysis is straightforward. In contrast to SERIE B INFORMATIK A Simple Min Cu Algorihm Mechild Soer * Frank Wagner** B 9{1 May 199 Abrac We preen an algorihm for nding he minimum cu of an edge-weighed graph. I i imple in every repec. I ha a hor

More information

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen

More information

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given

More information

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,

More information

Chapter 7: Inverse-Response Systems

Chapter 7: Inverse-Response Systems Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem

More information

DETC2004/CIE ALGORITHMIC FOUNDATIONS FOR CONSISTENCY-CHECKING OF INTERACTION-STATES OF MECHATRONIC SYSTEMS

DETC2004/CIE ALGORITHMIC FOUNDATIONS FOR CONSISTENCY-CHECKING OF INTERACTION-STATES OF MECHATRONIC SYSTEMS Proceeding of DETC 04 ASME 2004 Deign Engineering Technical Conference and Compuer and Informaion in Engineering Conference Sal Lake Ciy, Uah, USA, Sepember 28-Ocober 2, 2004 DETC2004/CIE-79 ALGORITHMIC

More information

Stationary Distribution. Design and Analysis of Algorithms Andrei Bulatov

Stationary Distribution. Design and Analysis of Algorithms Andrei Bulatov Saionary Disribuion Design and Analysis of Algorihms Andrei Bulaov Algorihms Markov Chains 34-2 Classificaion of Saes k By P we denoe he (i,j)-enry of i, j Sae is accessible from sae if 0 for some k 0

More information

Wrap up: Weighted, directed graph shortest path Minimum Spanning Tree. Feb 25, 2019 CSCI211 - Sprenkle

Wrap up: Weighted, directed graph shortest path Minimum Spanning Tree. Feb 25, 2019 CSCI211 - Sprenkle Objecive Wrap up: Weighed, direced graph hore pah Minimum Spanning Tree eb, 1 SI - Sprenkle 1 Review Wha are greedy algorihm? Wha i our emplae for olving hem? Review he la problem we were working on: Single-ource,

More information

Network Flow Applications

Network Flow Applications Hopial problem Neork Flo Applicaion Injured people: n Hopial: k Each peron need o be brough o a hopial no more han 30 minue aay Each hopial rea no more han n/k" people Gien n, k, and informaion abou people

More information

Stability in Distribution for Backward Uncertain Differential Equation

Stability in Distribution for Backward Uncertain Differential Equation Sabiliy in Diribuion for Backward Uncerain Differenial Equaion Yuhong Sheng 1, Dan A. Ralecu 2 1. College of Mahemaical and Syem Science, Xinjiang Univeriy, Urumqi 8346, China heng-yh12@mail.inghua.edu.cn

More information

Dynamic Programming 11/8/2009. Weighted Interval Scheduling. Weighted Interval Scheduling. Unweighted Interval Scheduling: Review

Dynamic Programming 11/8/2009. Weighted Interval Scheduling. Weighted Interval Scheduling. Unweighted Interval Scheduling: Review //9 Algorihms Dynamic Programming - Weighed Ineral Scheduling Dynamic Programming Weighed ineral scheduling problem. Insance A se of n jobs. Job j sars a s j, finishes a f j, and has weigh or alue j. Two

More information

Topics in Combinatorial Optimization May 11, Lecture 22

Topics in Combinatorial Optimization May 11, Lecture 22 8.997 Topics in Combinaorial Opimizaion May, 004 Lecure Lecurer: Michel X. Goemans Scribe: Alanha Newman Muliflows an Disjoin Pahs Le G = (V,E) be a graph an le s,,s,,...s, V be erminals. Our goal is o

More information

Selfish Routing and the Price of Anarchy. Tim Roughgarden Cornell University

Selfish Routing and the Price of Anarchy. Tim Roughgarden Cornell University Selfih Rouing and he Price of Anarchy Tim Roughgarden Cornell Univeriy 1 Algorihm for Self-Inereed Agen Our focu: problem in which muliple agen (people, compuer, ec.) inerac Moivaion: he Inerne decenralized

More information

CS261: A Second Course in Algorithms Lecture #1: Course Goals and Introduction to Maximum Flow

CS261: A Second Course in Algorithms Lecture #1: Course Goals and Introduction to Maximum Flow CS61: A Second Coure in Algorihm Lecure #1: Coure Goal and Inroducion o Maximum Flo Tim Roughgarden January 5, 016 1 Coure Goal CS61 ha o major coure goal, and he coure pli roughly in half along hee line.

More information

Research Article On Double Summability of Double Conjugate Fourier Series

Research Article On Double Summability of Double Conjugate Fourier Series Inernaional Journal of Mahemaic and Mahemaical Science Volume 22, Aricle ID 4592, 5 page doi:.55/22/4592 Reearch Aricle On Double Summabiliy of Double Conjugae Fourier Serie H. K. Nigam and Kuum Sharma

More information

NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY

NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY Ian Crawford THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working paper CWP02/04 Neceary and Sufficien Condiion for Laen

More information

Introduction to SLE Lecture Notes

Introduction to SLE Lecture Notes Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will

More information

Algorithms. Algorithms 6.4 MAXIMUM FLOW

Algorithms. Algorithms 6.4 MAXIMUM FLOW Algorithms ROBERT SEDGEWICK KEVIN WAYNE 6.4 MAXIMUM FLOW Algorithms F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE http://algs4.cs.princeton.edu introduction Ford Fulkerson algorithm maxflow mincut

More information

7. NETWORK FLOW II. Soviet rail network (1950s) Max-flow and min-cut applications. "Free world" goal. Cut supplies (if cold war turns into real war).

7. NETWORK FLOW II. Soviet rail network (1950s) Max-flow and min-cut applications. Free world goal. Cut supplies (if cold war turns into real war). Sovie rail nework (9). NETWORK FLOW II "Free world" goal. Cu upplie (if cold war urn ino real war). Lecure lide by Kevin Wayne Copyrigh Pearon-Addion Weley Copyrigh Kevin Wayne hp://www.c.princeon.edu/~wayne/kleinberg-ardo

More information

Notes for Lecture 17-18

Notes for Lecture 17-18 U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up

More information

CMU-Q Lecture 3: Search algorithms: Informed. Teacher: Gianni A. Di Caro

CMU-Q Lecture 3: Search algorithms: Informed. Teacher: Gianni A. Di Caro CMU-Q 5-38 Lecure 3: Search algorihms: Informed Teacher: Gianni A. Di Caro UNINFORMED VS. INFORMED SEARCH Sraegy How desirable is o be in a cerain inermediae sae for he sake of (effecively) reaching a

More information

The multisubset sum problem for finite abelian groups

The multisubset sum problem for finite abelian groups Alo available a hp://amc-journal.eu ISSN 1855-3966 (prined edn.), ISSN 1855-3974 (elecronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 417 423 The muliube um problem for finie abelian group Amela Muraović-Ribić

More information

Notes on cointegration of real interest rates and real exchange rates. ρ (2)

Notes on cointegration of real interest rates and real exchange rates. ρ (2) Noe on coinegraion of real inere rae and real exchange rae Charle ngel, Univeriy of Wiconin Le me ar wih he obervaion ha while he lieraure (mo prominenly Meee and Rogoff (988) and dion and Paul (993))

More information

What is maximum Likelihood? History Features of ML method Tools used Advantages Disadvantages Evolutionary models

What is maximum Likelihood? History Features of ML method Tools used Advantages Disadvantages Evolutionary models Wha i maximum Likelihood? Hiory Feaure of ML mehod Tool ued Advanage Diadvanage Evoluionary model Maximum likelihood mehod creae all he poible ree conaining he e of organim conidered, and hen ue he aiic

More information

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2) Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful

More information

ARTIFICIAL INTELLIGENCE. Markov decision processes

ARTIFICIAL INTELLIGENCE. Markov decision processes INFOB2KI 2017-2018 Urech Univeriy The Neherland ARTIFICIAL INTELLIGENCE Markov deciion procee Lecurer: Silja Renooij Thee lide are par of he INFOB2KI Coure Noe available from www.c.uu.nl/doc/vakken/b2ki/chema.hml

More information

Price of Stability and Introduction to Mechanism Design

Price of Stability and Introduction to Mechanism Design Algorihmic Game Theory Summer 2017, Week 5 ETH Zürich Price of Sabiliy and Inroducion o Mechanim Deign Paolo Penna Thi i he lecure where we ar deigning yem which involve elfih player. Roughly peaking,

More information