Outline. CS38 Introduction to Algorithms 5/8/2014. Network flow. Lecture 12 May 8, 2014

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1 /8/0 Ouline CS8 Inroducion o Algorihm Lecure May 8, 0 Nework flow finihing capaciy-caling analyi Edmond-Karp, blocking-flow implemenaion uni-capaciy imple graph biparie maching edge-dijoin pah aignmen problem * lide from Kevin Wayne May 8, 0 CS8 Lecure May 8, 0 CS8 Lecure Minimum cu problem Def. A -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. Maximum flow problem Def. An -flow (flow) f i a funcion ha aifie: For each e E : [capaciy] For each v V {, } : [flow conervaion] Def. The value of a flow f i: val( f ) = Min-cu problem. Find a cu of minimum capaciy. Max-flow problem. Find a flow of maximum value. 8 / / 0 / 8 / 8 / 8 0 / / 0 / capaciy = = 8 value = = 8 / 6 Reidual graph Augmening pah Original edge: e = (u, v) E. Flow f (e). Capaciy c(e). Reidual edge. "Undo" flow en. e = (u, v) and e R = (v, u). Reidual capaciy: original graph G 6 / 7 u flow capaciy reidual graph Gf u 6 v reidual capaciy v Def. An augmening pah i a imple pah P in he reidual graph G f. Def. The boleneck capaciy of an augmening 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 f ' i a flow and val( f ' ) = val( f ) + boleneck(gf, P). AUGMENT (f, c, P) Reidual graph: G f = (V, E f ). Reidual edge wih poiive reidual capaciy. E f = {e : f (e) < c(e)} {e R : f (e) > 0}. Key propery: f ' i a flow in G f iff f + f ' i a flow in G. where flow on a revere edge negae flow on a forward edge b boleneck capaciy of pah P. FOREACH edge e P IF (e E ) f (e) f (e) + b. ELSE f (e R ) f (e R ) b. RETURN f. 6

2 /8/0 Ford-Fulkeron algorihm Ford-Fulkeron augmening pah algorihm. Sar wih f (e) = 0 for all edge e E. Find an augmening pah P in he reidual graph G f. Augmen flow along pah P. Repea unil you ge uck. FORD-FULKERSON (G,,, c) FOREACH edge e E : f (e) 0. } Gf reidual graph. WHILE (here exi an augmening pah P in Gf ) f AUGMENT (f, c, P). Updae Gf. RETURN f. 7 Capaciy-caling algorihm May 8, 0 CS8 Lecure 8 Capaciy-caling algorihm 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 ubgraph of he reidual graph coniing only of arc wih capaciy Δ. CAPACITY-SCALING(G,,, c) FOREACH edge e E : f (e) 0. Δ large power of C. WHILE (Δ ) Gf (Δ) Δ-reidual graph. WHILE (here exi an augmening pah P in Gf (Δ)) f AUGMENT (f, c, P). Updae Gf (Δ). Δ Δ /. RETURN f. Gf Gf (Δ), Δ = Capaciy-caling algorihm: proof of correcne Capaciy-caling algorihm: analyi of running ime Aumpion. All edge capaciie are ineger beween and C. Inegraliy invarian. All flow and reidual capaciy value are inegral. Theorem. If capaciy-caling algorihm erminae, hen f i a max-flow. By inegraliy invarian, when Δ = G f (Δ) = G f. Upon erminaion of Δ = phae, here are no augmening pah. Lemma. The ouer while loop repea + log C ime. Iniially C / < Δ C; Δ decreae by a facor of in each ieraion. Lemma. Le f be he flow a he end of a Δ-caling phae. Then, he value of he max-flow val( f ) + m Δ. proof on nex lide Lemma. There are a mo m augmenaion per caling phae. Le f be he flow a he end of he previou caling phae. LEMMA val( f *) val( f ) + 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 log C) ime. Follow from LEMMA and LEMMA.

3 /8/0 Capaciy-caling algorihm: analyi of running ime Lemma. Le f be he flow a he end of a Δ-caling phae. Then, he value of he max-flow val( f ) + m Δ. 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. val( f ) original nework A edge e = (v, w) wih v B, w A mu have f(e) Δ B Shore augmening pah edge e = (v, w) wih v A, w B mu have f(e) c(e) Δ May 8, 0 CS8 Lecure Shore augmening pah Q. Which augmening pah? A. The one wih he fewe number of edge. can find via BFS Shore augmening pah: overview of analyi L. Throughou he algorihm, lengh of he hore pah never decreae. L. Afer a mo m hore pah augmenaion, he lengh of he hore augmening pah ricly increae. SHORTEST-AUGMENTING-PATH(G,,, c) FOREACH e E : f (e) 0. Gf reidual graph. WHILE (here exi an augmening pah in Gf ) P BREADTH-FIRST-SEARCH (Gf,, ). f AUGMENT (f, c, P). Updae Gf. RETURN f. Theorem. The hore augmening pah algorihm run in O(m n) ime. 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. k < n O(m n) augmenaion. 6 Shore augmening pah: analyi Def. Given a digraph G = (V, E) wih ource, i level graph i defined by: l(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 l(w) = l(v) +. Shore augmening pah: analyi Def. Given a digraph G = (V, E) wih ource, i level graph i defined by: l(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 l(w) = l(v) +. graph G Propery. Can compue level graph in O(m + n) ime. 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. l= 0 l= l= l= 7 l= 0 l= l= l= 8

4 /8/0 Shore augmening pah: analyi L. Throughou he algorihm, lengh of he hore pah never decreae. Le f and f ' be flow before and afer a hore pah augmenaion. Le L and L' be level graph of G f and G f '. Only back edge added o G f ' (any pah wih a back edge i longer han previou lengh) Shore augmening pah: analyi L. Afer a mo m hore pah augmenaion, he lengh of he 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 level graph L l= 0 l= l= l= l= 0 l= l= l= level graph L' level graph L' 9 0 Shore augmening pah: review of analyi L. Throughou he algorihm, lengh of he hore pah never decreae. L. Afer a mo m hore pah augmenaion, he lengh of he hore augmening pah ricly increae. Shore augmening pah: improving he running ime Noe. Θ(m n) augmenaion neceary on ome nework. Try o decreae ime per augmenaion inead. Simple idea O(m n ) [Dinic 970] Dynamic ree O(m n log n) [Sleaor-Tarjan 98] Theorem. The hore augmening pah algorihm run in O(m n) ime. O(m + n) ime o find hore augmening pah via BFS. O(m) augmenaion for pah of exacly k edge. O(m n) augmenaion. Blocking-flow algorihm Two ype of augmenaion. Normal: lengh of hore pah doe no change. Special: lengh of hore pah ricly increae. Blocking flow Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, 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. May 8, 0 CS8 Lecure

5 /8/0 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, 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. 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, 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 6 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, 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. 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, 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 7 8 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, 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. 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, 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 9 0

6 /8/0 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, 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. 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, 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 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, 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. 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, 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 end of phae Blocking-flow algorihm Blocking-flow algorihm: analyi INITIALIZE(G,,, f, c) LG level-graph of Gf. P. GOTO ADVANCE(). RETREAT(v) IF (v = ) STOP. ELSE Delee v (and all inciden edge) from LG. Remove la edge (u, v) from P. GOTO ADVANCE(u). ADVANCE(v) IF (v = ) AUGMENT(P). Remove auraed edge from LG. P. GOTO ADVANCE(). IF (here exi edge (v, w) LG) Add edge (v, w) o P. GOTO ADVANCE(w). ELSE GOTO RETREAT(v). Lemma. A phae can be implemened in O(m n) ime. Iniializaion happen once per phae. A mo m augmenaion per phae. O(m) uing BFS O(mn) per phae (becaue an augmenaion delee a lea one edge from L G) A mo n rerea per phae. O(m + n) per phae (becaue a rerea delee one node from L G) A mo m n per phae. O(mn) per phae (becaue a mo n before rerea or augmenaion) Theorem. [Dinic 970] The blocking-flow algorihm run in O(mn ) ime. By lemma, O(mn) ime per phae. A mo n phae (a in hore augmen pah analyi)

7 /8/0 Chooing good augmening pah: ummary Maximum flow algorihm: heory Aumpion. Ineger capaciie beween and C. year mehod wor cae dicovered by 9 implex O(m C) Danzig mehod # augmenaion running ime augmening pah n C O(m n C) fae augmening pah m log (mc) O(m log n log (mc)) capaciy caling m log C O(m log C) improved capaciy caling m log C O(m n log C) hore augmening pah m n O(m n) improved hore augmening pah m n O(m n ) dynamic ree m n O(m n log n ) 9 augmening pah O(m C) Ford-Fulkeron 970 hore augmening pah O(m ) Dinic, Edmond-Karp 970 fae augmening pah O(m log m log( m C )) Dinic, Edmond-Karp 977 blocking flow O(m / ) Cherkaky 978 blocking flow O(m 7/ ) Galil 98 dynamic ree O(m log m) Sleaor-Tarjan 98 capaciy caling O(m log C) Gabow 997 lengh funcion O(m / log m log C) Goldberg-Rao 0 compac nework O(m / log m) Orlin?? O(m)? max-flow algorihm for pare digraph wih m edge, ineger capaciie beween and C 8 9 Biparie maching Uni capaciy imple graph Q. Which max-flow algorihm o ue for biparie maching? Generic augmening pah: O( m f * ) = O(m n). Capaciy caling: O(m log U) = O(m ). Shore augmening pah: O(m n ). Q. Sugge "more clever" algorihm are no a good a we fir hough? A. No, ju need more clever analyi! Nex. May 8, 0 CS8 Lecure 0 Uni-capaciy imple nework Def. A nework i a uni-capaciy imple nework if: Every edge capaciy i. Every node (oher han or ) ha eiher (i) a mo one enering edge or (ii) a mo one leaving edge. Propery. Le G be a imple uni-capaciy nework and le f be a 0- flow, hen Gf i a uni-capaciy imple nework. Uni-capaciy imple 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 97] In uni-capaciy imple nework, he hore augmening pah algorihm compue a maximum flow in O(m n / ) ime. L. Each phae of normal augmenaion ake O(m) ime. L. Afer a mo n / phae, f f * n /. L. Afer a mo n / addiional augmenaion, flow i opimal. We will prove a running ime of O(mn / ) 7

8 /8/0 Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. augmen Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. rerea 6 7 Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. augmen 8 9 8

9 /8/0 Uni-capaciy imple nework Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, along an edge in LG unil reach or ge uck. If reach, augmen and and updae L G. delee all edge in augmening pah from LG If ge uck, delee node from L G and go o previou node. Uni-capaciy imple nework: analyi Phae of normal augmenaion. Explicily mainain level graph L G. Sar a, 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 LEMMA. A phae of normal augmenaion ake O(m) ime. O(m) o creae level graph L G. O() per edge ince each edge ravered and deleed a mo once. O() per node ince each node deleed a mo once. 0 Uni-capaciy imple nework: analyi LEMMA. Afer a mo n / phae, f f * n /. Afer n / phae, lengh of hore augmening pah i > n /. Level graph ha more han n / level. Le h n / be layer wih min number of node: Vh n /. Uni-capaciy imple nework: analyi LEMMA. Afer a mo n / phae, f f * n /. Afer n / phae, lengh of hore augmening pah i > n /. Level graph ha more han n / level. Le h n / be layer wih min number of node: Vh n /. Le A = {v : l(v) < h} {v : l(v) = h and v ha ougoing reidual edge}. capf (A, B) Vh n / f f * n /. for flow f reidual graph Gf reidual edge A V 0 V V h Vn / V 0 V V h Vn / Maching Def. Given an undireced graph G = (V, E) a ube of edge M E i a maching if each node appear in a mo one edge in M. Biparie maching Max maching. Given a graph, find a max cardinaliy maching. May 8, 0 CS8 Lecure 9

10 /8/0 Biparie maching Def. A graph G i biparie if he node can be pariioned ino wo ube L and R uch ha every edge connec a node in L o one in R. Biparie maching Def. A graph G i biparie if he node can be pariioned ino wo ube L and R uch ha every edge connec a node in L o one in R. Biparie maching. Given a biparie graph G = (L R, E), find a max cardinaliy maching. Biparie maching. Given a biparie graph G = (L R, E), find a max cardinaliy maching. ' ' ' ' ' ' ' ' L ' R L ' R maching: -', -', -' 6 maching: -', -', -', -' 7 Biparie maching: max flow formulaion Max flow formulaion: proof of correcne Creae digraph G' = (L R {, }, E' ). Direc all edge from L o R, and aign infinie (or uni) capaciy. Add ource, and uni capaciy edge from o each node in L. Add ink, and uni capaciy edge from each node in R o. Theorem. Max cardinaliy of a maching in G = value of max flow in G'. Given a max maching M of cardinaliy k. Conider flow f ha end uni along each of k pah. f i a flow, and ha value k. G' ' ' ' ' ' ' ' ' ' ' ' ' ' G ' ' G' L R 8 9 Max flow formulaion: proof of correcne Theorem. Max cardinaliy of a maching in G = value of max flow in G'. Le f be a max flow in G' of value k. Inegraliy heorem implie k i inegral and can aume f i 0-. Conider M = e of edge from L o R wih f (e) =. - each node in L and R paricipae in a mo one edge in M - M = k: conider cu (L, R ) Perfec maching in a biparie graph Def. Given a graph G = (V, E) a ube of edge M E i a perfec maching if each node appear in exacly one edge in M. Q. When doe a biparie graph have a perfec maching? A. Hall Theorem. Le G = (L R, E) be a biparie graph wih L = R. G ha a perfec maching iff N(S) S for all ube S L. ' ' On problem e! ' ' ' ' ' ' G' ' ' G

11 /8/0 Biparie maching running ime Nonbiparie maching Theorem. The Ford-Fulkeron algorihm olve he biparie maching problem in O(m n) ime. Theorem. [Hopcrof-Karp 97] The biparie maching problem can be olved in O(m n / ) ime. Nonbiparie maching. Given an undireced graph (no necearily biparie), find a maching of maximum cardinaliy. Srucure of nonbiparie graph i more complicaed. Bu well-underood. [Tue-Berge, Edmond-Galai] Bloom algorihm: O(n ). [Edmond 96] Be known: O(m n / ). [Micali-Vazirani 980, Vazirani 99] 6 6 Edge-dijoin pah Def. Two pah are edge-dijoin if hey have no edge in common. Edge-dijoin pah Dijoin pah problem. Given a digraph G = (V, E) and wo node and, find he max number of edge-dijoin pah. 6 May 8, 0 CS8 Lecure 6 digraph G 7 6 Edge-dijoin pah Edge-dijoin pah Def. Two pah are edge-dijoin if hey have no edge in common. Max flow formulaion. Aign uni capaciy o every edge. Dijoin pah problem. Given a digraph G = (V, E) and wo node and, find he max number of edge-dijoin pah. Ex. Communicaion nework. Theorem. Max number edge-dijoin pah equal value of max flow. Suppoe here are k edge-dijoin pah P,, P k. Se f (e) = if e paricipae in ome pah P j ; ele e f (e) = 0. Since pah are edge-dijoin, f i a flow of value k. 6 digraph G edge-dijoin pah

12 /8/0 Edge-dijoin pah Max flow formulaion. Aign uni capaciy o every edge. Theorem. Max number edge-dijoin pah equal value of max flow. Suppoe max flow value i k. Inegraliy heorem implie here exi 0- flow f of value k. Conider edge (, u) wih f(, u) =. - by conervaion, here exi an edge (u, v) wih f(u, v) = - coninue unil reach, alway chooing a new edge Produce k (no necearily imple) edge-dijoin pah. can eliminae cycle o ge imple pah in O(mn) ime if deired (flow decompoiion) 68