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This document discusses algorithms for solving maximum flow problems on graphs. It describes the push-relabel algorithm, which is one of the most efficient maximum flow algorithms. It was developed by Goldberg and Tarjan. The document also covers preflow basics, residual graphs, and the Edmonds-Karp algorithm. Edmonds-Karp finds the shortest augmenting path using breadth-first search and has a time complexity of O(V*E^2).
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- 1. 02/23/14
- 3. Topics Push relable method edmonds-karP algorithm 02/23/14
- 4. TERMS TO KNOW 1. 2. 3. 4. 02/23/14 Residual graph Augmented path Exes flow/ net flow Active vertex
- 5. Push Relable method • One of the most efficient algorithms Used for maximum flow problems • By Dr. Andrew Goldberg and Robert E Tarjan • Fastest maximum flow algorithm 02/23/14
- 8. Conditions • Three conditions 1. F(u, v) should be less or equal c(u, v) 2. Skew symmetry F(u, v) = -F(v, u) 3. Flow conservation 02/23/14
- 9. algorithm 9
- 15. Time complexity • Generally it has O(V2E) • It depends on its method of implementation 02/23/14
- 17. Overview • An algorithm of ford Fulkerson used for finding the shortest augmenting path • Uses breadth first search • Considering each edge has a unit weight 02/23/14
- 19. 02/23/14
- 20. Algorithm set f to 0 and Gf to 0 While Gf contains source to sink path P do 1. P is path with min no of edges from s to t 2. Augment f using P 3. Update Gf End while Return f 02/23/14
- 21. Time complexity • Runs in time O(V E2). 02/23/14
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Editor's Notes
- In optimization theory, maximum flow problems involve finding a feasible flow through a single-source, single-sink flow network that is maximum. In a flow network, flow going into a vertex must equal flow going out of a vertex (with the exception of the source node and the sink node, s and t respectively). Thus we cannot accumulate flow anywhere in the network. he Preflow algorithm relaxes this constraint while it determines the maximum flow of the network, allowing a vertex to have more flow coming into it than going out. This is called the excess flow of a vertex, or preflow. Any vertex with excess flow is known as an active vertex.
- The push-relabel algorithm finds maximum flow on a flow network. There are three conditions for a flow network: Capacity constraints: . The flow along an edge cannot exceed its capacity. Skew symmetry: . The net flow from to must be the opposite of the net flow from to (see example). Flow conservation: , unless or . The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow. he push-relabel algorithm finds maximum flow on a flow network. There are three conditions for a flow network: Capacity constraints: . The flow along an edge cannot exceed its capacity. Skew symmetry: . The net flow from to must be the opposite of the net flow from to (see example). Flow conservation: , unless or . The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow. The preflow algorithm observes the first two of the conditions of flow networks, plus a relaxed third condition for the duration of the algorithm (the algorithm finishes once all the excess is gone from the graph, at which point we have a legal flow, and the Flow conservation constraint is again observed): Non-negativity constraint: for all nodes . More flow enters a node than exits. Given a flow network with capacity from node u to node v given as , and the flow across u and v given as , source s and sink t, we want to find the maximum amount of flow you can send from s to t through the network. Some key concepts used in the algorithm are: residual edge: If available capacity we have a residual edge . residual edge (Alternate): . We can push flow back along an edge. residual graph: The set of all residual edges forms the residual graph. legal labelling: For each residual edge . We only push from u to v if . excess(u): Sum of flow to and from u. height(u): Each vertex is assigned a height value. For values , the height serves as an estimate of the distance to the sink node t. For values , the height is an estimate of the distance back to the sink s.
- Push A push from u to v means sending as much excess flow from u to v as we can. Three conditions must be met for a push to take place: is active Or . There is more flow entering than exiting the vertex. There is a residual edge from u to v, where u is higher than v. If all these conditions are met we can execute a Push: Function Push(u,v) flow = min(excess(u), c(u,v)-f(u,v)); excess(u) -= flow; // subtract the amount of flow moved from the current vertex excess(v) += flow; // add the flow to the vertex we are pushing to f(u,v) += flow; // add the amount of flow moved to the flow across the edge (u,v) f(v,u) -= flow; // subtract the flow from the edge in the other direction. Relabel When we relabel a node u we increase its height until it is 1 higher than its lowest neighbour in the residual graph. Conditions for a relabel: is active where So we have excess flow to push, but we not higher than any of our neighbours that have available capacity across their edge. Then we can execute Relabel: Function Relabel(u) height(u) = min(height(v) where r(u,v) > 0) + 1;