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Preflow Push Algorithm

Preflow Push Algorithm. M. Amber Hassaan. a. 5, 2. 4, 4. s. 6, 2. t. 8, 7. 5, 5. b. Max Flow Problem. Given a graph with “Source” and “Sink” nodes we want to compute: The maximum rate at which fluid can flow from Source to Sink The rate of flow through each edge of the graph

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Preflow Push Algorithm

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  1. Preflow Push Algorithm M. Amber Hassaan

  2. a 5, 2 4, 4 s 6, 2 t 8, 7 5, 5 b Max Flow Problem • Given a graph with “Source” and “Sink” nodes we want to compute: • The maximum rate at which fluid can flow from Source to Sink • The rate of flow through each edge of the graph • Given a graph: • Edges have flow capacities • First value is flow capacity • Second value is current flow rate • Edges behave like pipes • Nodes are junctions of pipes • History: • Ford and Fulkerson 1956 (Augmenting Paths) • Dinic 1970 • Edmonds and Karp 1972 • Malhotra, Kumar and Maheshwari 1978 • Goldberg and Tarjan 1986 (Preflow Push) • Boykov and Kolmogorov 2006 Min-cut Preflow Push Algorithm

  3. a 5, 2 4, 4 s 6, 2 t 8, 7 5, 5 b Max Flow Problem • Flow f(u,v) is a real-valued function defined for every edge (u,v) in the graph • The flow needs to satisfy the following 3 properties: • f(u,v) ≤ c(u,v) i.e. capacity of edge (u,v) • f(u,v) = - f(v,u) • Flow coming into node v = Flow leaving v Preflow Push Algorithm

  4. a a 2 4 5, 2 4, 4 3 s s 6, 2 4 t t 2 1 8, 7 5, 5 5 7 b b Max Flow Problem • Residual Graph: • A Graph that contains edges that can admit 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 then (u,v) is in E’ Original Graph Residual Graph Preflow Push Algorithm

  5. Preflow Push Algorithm for Maxflow problem • Relaxation algorithm: • Performs local updates repeatedly until global constraint is satisfied • Similar to Stencil computations • Fluid flows from a higher point to a lower point • In the beginning • “Source” is the highest point • “Sink” and all other nodes are at the lowest point • Source sends maximum flow on its outgoing edges • Sending flow to out-neighbors is called “Push” operation • The height of Source’s neighbors is then increased so that fluid can flow out • Increasing height of a node is called “Relabel” operation Preflow Push Algorithm

  6. Preflow Push Algorithm for Maxflow problem • A node is allowed (temporarily) to have more flow coming into it than flow going out • i.e. a node can have “excess flow” • But the edges must respect the capacity condition • Source can have arbitrary amount of excess flow • A node is said to be “active” if it has excess flow in it Preflow Push Algorithm

  7. Preflow Push Algorithm for Maxflow problem • We increase the height of the “active” node with a “Relabel” operation • If h is the minimum height among neighbors that can accept flow • Then height is relabeled to (h+1) • Then we “push” the excess flow to the neighbors • That are lower than the active node and can admit flow • Thus make them active • Algorithm terminates when there are no “active” nodes left Preflow Push Algorithm

  8. h=0 e=0 h=0 e=0 15,0 a b 10,0 3,0 5,0 t s 8,0 h=0 e=0 h=6 17,0 12,0 c d 6,0 h=0 e=0 h=0 e=0 Example • A simple graph: • Nodes have two attributes: • Height ‘h’ • Excess flow ‘e’ • Edges have pairs: • First value is edge capacity • Second value is flow • Initialize: • s has h=6 i.e. number of nodes • Push 10 along (s,a) • e(a) = 10 • Push 12 along (s,c) • e(c) = 12 Preflow Push Algorithm

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

  10. h=1 e=15 h=0 e=0 15,0 a b 10,10 3,0 5,5 t s 8,0 h=0 e=0 h=6 17,0 12,11 c d 6,6 h=7 e=0 h=0 e=5 Example • Relabel a with h=1 • Push 15 along (a,b) • e(b) = 15 Preflow Push Algorithm

  11. h=1 e=0 h=1 e=15 15,15 a b 10,10 3,0 5,5 t s 8,0 h=0 e=3 h=6 17,0 12,11 c d 6,6 h=7 e=0 h=0 e=6 Example • Relabel b with h=1 • Push 3 along (b,t) • e(t) = 3 • Push 8 along (b,d) • e(d) = 14 • Relabel b with h=2 • Push -4 along (a,b) • e(a) = 4 Preflow Push Algorithm

  12. h=1 e=4 h=2 e=0 15,11 a b 10,10 3,3 5,5 t s 8,8 h=0 e=3 h=6 17,0 12,11 c d 6,6 h=7 e=0 h=1 e=14 Example • Relabel d with h=1 • Push 14 along (d,t) • e(t) = 17 Preflow Push Algorithm

  13. h=3 e=4 h=2 e=0 15,11 a b 10,10 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • Relabel a with h=3 • Push 4 along (a,b) • e(b) = 4 Preflow Push Algorithm

  14. h=3 e=0 h=4 e=4 15,15 a b 10,10 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • Relabel b with h=4 • Push -4 along (a,b) • e(a) = 4 Preflow Push Algorithm

  15. h=5 e=4 h=4 e=0 15,11 a b 10,10 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • Relabel a with h=5 • Push 4 along (a,b) • e(b) = 4 Preflow Push Algorithm

  16. h=5 e=0 h=6 e=4 15,15 a b 10,10 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • Relabel b with h=6 • Push -4 along (a,b) • e(a) = 4 Preflow Push Algorithm

  17. h=7 e=4 h=6 e=0 15,11 a b 10,10 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • Relabel a with h=7 • Push -4 along (s,a) • e(a) = 0 Preflow Push Algorithm

  18. h=7 e=0 h=6 e=0 15,11 a b 10,6 3,3 5,5 t s 8,8 h=0 e=17 h=6 17,14 12,11 c d 6,6 h=7 e=0 h=1 e=0 Example • No active nodes left • The algorithm terminates Min-cut Preflow Push Algorithm

  19. Pseudo code • Worklist wl = initializePreflowPush(); • while (!wl.isEmpty()) { • Node n = wl.remove(); • n.relabel(); • for (Node w in n.neighbors()) { • if (n can push flow to w) { • pushflow(n, w); • wl.add(w); • } • } • if (n has excess flow) { • wl.add(n); • } • } Preflow Push Algorithm

  20. Amorphous Data Parallelism in Preflow Push • Topology: graph • Operator: local computation (data driven) • Active nodes: nodes with excess flow • Neighborhoods: active nodes and their neighbors • Ordering: unordered • Parallelism: Shoots in the beginning and then drops down • Dependent on number of active nodes Preflow Push Algorithm

  21. Parallelism profile for Preflow Push Input Graph: 512x512 Grid Preflow Push Algorithm

  22. Preflow Push: Global Relabeling • The height of the nodes directs the flow towards the sink: • It affects the flow of fluid globally • But is updated locally • Which causes the fluid to flow in arbitrary directions • Increases the total number of push/relabel operations • Global Relabeling: • After every N push/relabel operations we compute the height of every node from the sink • The height is computed by a breadth-first scan on the Residual Graph • starting at the sink • The height is incremented at each next level of BFS • It reduces the number of push/relabel operations significantly • N is determined heuristically Preflow Push Algorithm

  23. Preflow Push: Global Relabeling h=2 e=0 h=1 e=0 4 a b 4 11 3 6 5 t s 8 14 h=0 e=17 h=6 11 1 3 c d 6 h=2 e=0 h=1 e=0 Preflow Push Algorithm

  24. Conclusion • Preflow Push is an algorithm to find the max-flow in a graph • It exhibits amorphous data parallelism • Parallelism is dependent on the structure of the graph Preflow Push Algorithm

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