Preflow Push Algorithm. M. Amber Hassaan

Transcription

1 Preflow Push Algorihm M. Amber Hassaan

2 Max Flow Problem Given a graph wih Source and Sink nodes we wan o compue: The maximum rae a which fluid can flow from Source o Sink The rae of flow hrough each edge of he graph Given a graph: Edges have flow capaciies Firs value is flow capaciy Second value is curren flow rae Edges behave like pipes Nodes are juncions of pipes Hisory: Ford and Fulkerson 1956 (Augmening Pahs) Dinic 1970 Edmonds and Karp 1972 a Malhora, Kumar and Maheshwari 1978 Goldberg and Tarjan 1986 (Preflow Push) 5, 2 Boykov and Kolmogorov 2006 Min-cu 4, 4 s 6, 2 8, 7 5, 5 Preflow Push Algorihm 2 b

3 Max Flow Problem Flow f(u,v) is a real-valued funcion defined for every edge (u,v) in he graph The flow needs o saisfy he following 3 properies: f(u,v) c(u,v) i.e. capaciy of edge (u,v) f(u,v) = - f(v,u) Flow coming ino node v = Flow leaving v a 5, 2 4, 4 s 6, 2 8, 7 5, 5 Preflow Push Algorihm 3 b

4 Max Flow Problem Residual Graph: A Graph ha conains edges ha can admi more flow Define a Graph G(V,E ) for G(V,E) We define residual flow r(u,v) = c(u,v) f(u,v) If r(u,v) > 0 hen (u,v) is in E Original Graph a a Residual Graph 2 5, 2 4, s 6, 2 s , 7 5, b b Preflow Push Algorihm 4

5 Preflow Push Algorihm for Maxflow problem Relaxaion algorihm: Performs local updaes repeaedly unil global consrain is saisfied Similar o Sencil compuaions Fluid flows from a higher poin o a lower poin In he beginning Source is he highes poin Sink and all oher nodes are a he lowes poin Source sends maximum flow on is ougoing edges Sending flow o ou-neighbors is called Push operaion The heigh of Source s neighbors is hen increased so ha fluid can flow ou Increasing heigh of a node is called Relabel operaion Preflow Push Algorihm 5

6 Preflow Push Algorihm for Maxflow problem A node is allowed (emporarily) o have more flow coming ino i han flow going ou i.e. a node can have excess flow Bu he edges mus respec he capaciy condiion Source can have arbirary amoun of excess flow A node is said o be acive if i has excess flow in i Preflow Push Algorihm 6

7 Preflow Push Algorihm for Maxflow problem We increase he heigh of he acive node wih a Relabel operaion If h is he minimum heigh among neighbors ha can accep flow Then heigh is relabeled o (h+1) Then we push he excess flow o he neighbors Tha are lower han he acive node and can admi flow Thus make hem acive Algorihm erminaes when here are no acive nodes lef Preflow Push Algorihm 7

8 Example A simple graph: Nodes have wo aribues: Heigh h Excess flow e Edges have pairs: Firs value is edge capaciy Second value is flow Iniialize: s has i.e. number of nodes Push 10 along (s,a) e(a) = 10 Push 12 along (s,c) e(c) = 12 s 10,0 12,0 a b c 5,0 15,0 6,0 8,0 d 3,0 17,0 Preflow Push Algorihm 8

9 Example Relabel c wih Push 6 along (c,d) e(d) = 6 Relabel c wih h=2 Push 5 along (c,a) e(a) = 5 Relabel c wih h=7 Push -1 along (s,c) s 10,10 12,12 e=10 a b c 5,0 e=12 15,0 6,0 8,0 e=10 d 3,0 17,0 Preflow Push Algorihm 9

10 Example Relabel a wih Push 15 along (a,b) e(b) = 15 s 10,10 12,11 e=15 a b c 5,5 h=7 15,0 6,6 8,0 d e=5 3,0 17,0 Preflow Push Algorihm 10

11 Example Relabel b wih Push 3 along (b,) e() = 3 Push 8 along (b,d) e(d) = 14 Relabel b wih h=2 Push -4 along (a,b) s 10,10 12,11 a b c 5,5 h=7 15,15 6,6 8,0 e=15 d e=6 3,0 17,0 e=3 e(a) = 4 Preflow Push Algorihm 11

12 Example Relabel d wih Push 14 along (d,) e() = 17 s 10,10 12,11 e=4 a b c 5,5 h=7 15,11 6,6 8,8 h=2 d e=14 3,3 17,0 e=3 Preflow Push Algorihm 12

13 Example Relabel a wih h=3 Push 4 along (a,b) e(b) = 4 s 10,10 12,11 h=3 e=4 a b c 5,5 h=7 15,11 6,6 8,8 h=2 d 3,3 17,14 e=17 Preflow Push Algorihm 13

14 Example Relabel b wih h=4 Push -4 along (a,b) e(a) = 4 s 10,10 12,11 h=3 a b c 5,5 h=7 15,15 6,6 8,8 h=4 e=4 d 3,3 17,14 e=17 Preflow Push Algorihm 14

15 Example Relabel a wih h=5 Push 4 along (a,b) e(b) = 4 s 10,10 12,11 h=5 e=4 a b c 5,5 h=7 15,11 6,6 8,8 h=4 d 3,3 17,14 e=17 Preflow Push Algorihm 15

16 Example Relabel b wih Push -4 along (a,b) e(a) = 4 s 10,10 12,11 h=5 a b c 5,5 h=7 15,15 6,6 8,8 e=4 d 3,3 17,14 e=17 Preflow Push Algorihm 16

17 Example Relabel a wih h=7 Push -4 along (s,a) e(a) = 0 s 10,10 12,11 h=7 e=4 a b c 5,5 h=7 15,11 6,6 8,8 d 3,3 17,14 e=17 Preflow Push Algorihm 17

18 Example No acive nodes lef The algorihm erminaes 10,6 h=7 15,11 a b 3,3 s 12,11 c 5,5 h=7 6,6 8,8 d 17,14 e=17 Min-cu Preflow Push Algorihm 18

19 Pseudo code 1. Worklis wl = iniializepreflowpush(); 2. while (!wl.isempy()) { 3. Node n = wl.remove(); 4. n.relabel(); 5. for (Node w in n.neighbors()) { 6. if (n can push flow o w) { 7. pushflow(n, w); 8. wl.add(w); 9. } 10. } 11. if (n has excess flow) { 12. wl.add(n); 13. } 14. } Preflow Push Algorihm 19

20 Amorphous Daa Parallelism in Preflow Push Topology: graph Operaor: local compuaion (daa driven) Acive nodes: nodes wih excess flow Neighborhoods: acive nodes and heir neighbors Ordering: unordered Parallelism: Shoos in he beginning and hen drops down Dependen on number of acive nodes Preflow Push Algorihm 20

21 Parallelism profile for Preflow Push Inpu Graph: 512x512 Grid Preflow Push Algorihm 21

22 Preflow Push: Global Relabeling The heigh of he nodes direcs he flow owards he sink: I affecs he flow of fluid globally Bu is updaed locally Which causes he fluid o flow in arbirary direcions Increases he oal number of push/relabel operaions Global Relabeling: Afer every N push/relabel operaions we compue he heigh of every node from he sink The heigh is compued by a breadh-firs scan on he Residual Graph saring a he sink The heigh is incremened a each nex level of BFS I reduces he number of push/relabel operaions significanly N is deermined heurisically Preflow Push Algorihm 22

23 Preflow Push: Global Relabeling s h=2 a b c 5 h= d e=17 Preflow Push Algorihm 23

24 Conclusion Preflow Push is an algorihm o find he max-flow in a graph I exhibis amorphous daa parallelism Parallelism is dependen on he srucure of he graph Preflow Push Algorihm 24