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19
src/algorithms/graph/flow-networks/edmonds-karp/README.md
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src/algorithms/graph/flow-networks/edmonds-karp/README.md
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# Edmonds-Karp
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The Edmonds-Karp algorithm in an implementation of the [Ford-Fulkerson method](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/ford-fulkerson) for finding the maximum flow in a network in *O(|V||E|<sup>2</sup>)* time.
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Searching for an augmenting path P in the residual graph *G<sub>f</sub>* is done by [breadth-first search](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/breadth-first-search). This is in contrast with Ford-Fulkerson where the search order is not specified.
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## References
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- [Wikipedia | Edmonds-Karp](https://en.wikipedia.org/wiki/Edmonds%E2%80%93Karp_algorithm)
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- [Youtube | Edmonds-Karp Algorithm](https://www.youtube.com/watch?v=RppuJYwlcI8)
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- [Wikibooks | Java Implementation](https://en.wikibooks.org/wiki/Algorithm_Implementation/Graphs/Maximum_flow/Edmonds-Karp)
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- [Cornell University | Edmonds-Karp](https://www.cs.cornell.edu/courses/cs4820/2012sp/handouts/edmondskarp.pdf)
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- [Wikipedia Commons | Edmonds-Karp Examples](https://commons.wikimedia.org/wiki/File:Edmonds-Karp_flow_example_0.svg)
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import GraphVertex from '../../../../../data-structures/graph/GraphVertex';
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import GraphEdge from '../../../../../data-structures/graph/GraphEdge';
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import Graph from '../../../../../data-structures/graph/Graph';
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import edmondsKarp from '../edmondsKarp';
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describe('edmondsKarp', () => {
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it('should solve the maximum flow problem', () => {
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const capacities = {
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S_A: 3, S_B: 3, A_B: 4, A_D: 3, B_C: 5, B_D: 3, C_T: 2, D_T: 4,
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};
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const vertexS = new GraphVertex('S');
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const vertexA = new GraphVertex('A');
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const vertexB = new GraphVertex('B');
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const vertexC = new GraphVertex('C');
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const vertexD = new GraphVertex('D');
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const vertexT = new GraphVertex('T');
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const edgeSA = new GraphEdge(vertexS, vertexA, 0);
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const edgeSB = new GraphEdge(vertexS, vertexB, 0);
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const edgeAB = new GraphEdge(vertexA, vertexB, 0);
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const edgeAD = new GraphEdge(vertexA, vertexD, 0);
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const edgeBC = new GraphEdge(vertexB, vertexC, 0);
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const edgeBD = new GraphEdge(vertexB, vertexD, 0);
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const edgeCT = new GraphEdge(vertexC, vertexT, 0);
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const edgeDT = new GraphEdge(vertexD, vertexT, 0);
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const graph = new Graph(true);
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graph
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.addEdge(edgeSA)
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.addEdge(edgeSB)
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.addEdge(edgeAB)
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.addEdge(edgeAD)
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.addEdge(edgeBC)
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.addEdge(edgeBD)
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.addEdge(edgeCT)
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.addEdge(edgeDT);
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// COMPUTE edmondsKarp
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const edmondsKarpObj = edmondsKarp(graph, vertexS, vertexT, capacities);
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// max flow is 6 as expected
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expect(edmondsKarpObj.maxflow).toEqual(6);
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// flow through the network is as expected
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const flowThroughEachEdge = {};
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const expectedFlow = {
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S_A: 3, S_B: 3, A_B: 0, A_D: 3, B_C: 2, B_D: 1, C_T: 2, D_T: 4,
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};
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edmondsKarpObj.graph.getAllEdges().forEach((e) => {
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flowThroughEachEdge[e.getKey()] = e.weight;
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});
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expect(JSON.stringify(flowThroughEachEdge)).toBe(JSON.stringify(expectedFlow));
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});
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});
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213
src/algorithms/graph/flow-networks/edmonds-karp/edmondsKarp.js
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src/algorithms/graph/flow-networks/edmonds-karp/edmondsKarp.js
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import GraphVertex from '../../../../data-structures/graph/GraphVertex';
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import GraphEdge from '../../../../data-structures/graph/GraphEdge';
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import Graph from '../../../../data-structures/graph/Graph';
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import breadthFirstSearch from '../../breadth-first-search/breadthFirstSearch';
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import Queue from '../../../../data-structures/queue/Queue';
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/**
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* Edmonds-Karp implementation
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*
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* @param {Graph} graph
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* @param {GraphVertex} source
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* @param {GraphVertex} sink
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* @param {object} capacities
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* @return {object}
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*
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* NB: capacities is a mapping from: GraphEdge.getKey() to number
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* & return is {graph:graph,maxflow:f} of types {Graph, Number}
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*
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*/
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export default function edmondsKarp(graph, source, sink, capacities) {
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// variable to hold the max flow
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// f gets incremented in 'augment(path)'
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let f = 0;
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// variable to hold the flow network
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// the weights of the edges of g are the flow on those edges
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const g = graph;
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// set capacities where not defined to be infinity
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const cs = capacities;
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g.getAllEdges().forEach((edge) => {
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if (!(edge.getKey() in capacities)) cs[edge.getKey()] = Number.POSITIVE_INFINITY;
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});
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// the network flow is defined as the weights of g
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// assert the edges have 0 weight initally so that flow can be set to 0
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g.getAllEdges().forEach((edge) => {
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if (edge.weight !== 0) throw Error('ASSERT: the edges of g should have 0 weight – the edges represent the flow – FordFulkerson asserts 0 inital flow in network');
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});
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// initialise residual graph
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// build is done in function buildResidualGraph
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let residualGraph;
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// FLOW HELPER functions
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const getFlow = (edgeKey) => g.edges[edgeKey].weight;
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const updateFlow = (edgeKey, value) => { g.edges[edgeKey].weight += value; };
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// RESIDUAL CAPACITY HELPER function
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const getResidualCapacity = (edgeKey) => residualGraph.edges[edgeKey].weight;
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// FUNCTION for building Residual Graph
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// -> And hence determining residual capacities
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function buildResidualGraph() {
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// the weights of the residualGraph are the residual capacities
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residualGraph = new Graph(true);
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// add to the residualGraph all the vertices in g
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Object.keys(g.getVerticesIndices()).forEach((key) => {
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residualGraph.addVertex(new GraphVertex(key));
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});
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// compute residual capacities and assign as weights to residualGraph
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g.getAllEdges().forEach((edge) => {
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// edge is an edge in g
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// get edgeKey for extracting edge capacity from capacities
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const edgeKey = edge.getKey();
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// get the residualGraph start and end vertices (from edge in g)
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const startVertex = residualGraph.getVertexByKey(edge.startVertex.getKey());
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const endVertex = residualGraph.getVertexByKey(edge.endVertex.getKey());
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// if the flow is less than the capacity:
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if (getFlow(edgeKey) < cs[edgeKey]) {
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// compute the residual capacity
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const capMinusFlow = cs[edgeKey] - getFlow(edgeKey);
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// add a forward edge to residualGraph of the difference (cap-flow)
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// -> because (cap-flow) units of flow can still be pushed down the edge
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residualGraph.addEdge(new GraphEdge(startVertex, endVertex, capMinusFlow));
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}
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// if the flow is non zero
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if (getFlow(edgeKey) > 0) {
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// add a backwards edge to residualGraph with weight of flow
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// -> so that all of the flow along edge can be undone
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residualGraph.addEdge(new GraphEdge(endVertex, startVertex, getFlow(edgeKey)));
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}
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});
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// return residualGraph;
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}
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/**
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* Augment: augments the flow network given a path of edges in residualGraph
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* @param {GraphEdge[]} edgePath
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*
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* Considers the path in ResidualCap
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*/
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function augment(edgePath) {
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// compute the bottleneck capacity of the path in residualGraph
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let bottleneck = Number.POSITIVE_INFINITY;
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edgePath.forEach((edge) => {
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// get the residual capacity of an edge
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const resCap = getResidualCapacity(edge.getKey());
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// if the residual capacity is less than bottleneck than set bottleneck
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if (resCap < bottleneck) bottleneck = resCap;
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});
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edgePath.forEach((edgeResidualGraph) => {
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// get the vertices in g corresponding to the edge in residualGraph
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const startVertex = g.getVertexByKey(edgeResidualGraph.startVertex.getKey());
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const endVertex = g.getVertexByKey(edgeResidualGraph.endVertex.getKey());
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// if it contains a forward edge, update the flow by adding bottleneck
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const forwardEdge = g.findEdge(startVertex, endVertex);
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if (forwardEdge !== null) updateFlow(forwardEdge.getKey(), bottleneck);
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// if it contains a backward edge, update the flow by adding -bottleneck
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const backwardEdge = g.findEdge(endVertex, startVertex);
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if (backwardEdge !== null) updateFlow(backwardEdge.getKey(), -bottleneck);
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});
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// increase the value of flow by bottleneck
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f += bottleneck;
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}
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// intialise the variable augmentingPath
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let augmentingPath;
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/**
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* findAugmentingPath: finds an augmenting path in residualGraph
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* Uses BFS to determine the path (Edmonds-Karp specification)
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*/
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function findAugmentingPath() {
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// intialise augmentingPath
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augmentingPath = [];
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// get the source and sink keys
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const s = source.getKey();
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const t = sink.getKey();
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// initialise a queue of vertices visited
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const vertexPathQ = new Queue();
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vertexPathQ.enqueue([s]);
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// initialise the currentPath array and seen object
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let currentPath = [];
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const seen = {};
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// initialise variable found – has sink been found
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let found = false;
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// Specify BFS traversal callbacks.
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const bfsCallbacks = {
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enterVertex: () => {
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// on entering vertex dequeue from vertex queue
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currentPath = vertexPathQ.dequeue();
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},
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allowTraversal: ({ nextVertex }) => {
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// if sink has been found
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if (found) return false;
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// if the next vertex has been seen stop traversing branch
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if (seen[nextVertex.getKey()]) return false;
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// otherwise add next vertex to the seen object (if it is not sink)
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if (nextVertex.getKey() !== t) seen[nextVertex.getKey()] = true;
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// compute thisPath from currentPath and nextVertex
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const thisPath = [];
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currentPath.forEach((v) => thisPath.push(v));
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thisPath.push(nextVertex.getKey());
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// enqueue thisPath to vertexPathQ
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vertexPathQ.enqueue(thisPath);
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// if nextVertex is the sink then found is true
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if (nextVertex.getKey() === t) found = true;
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// continue traversal if not found
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return !found;
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},
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};
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// perform breadth first search to find a vertex path from s to t
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breadthFirstSearch(residualGraph, residualGraph.getVertexByKey(s), bfsCallbacks);
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// the search path is the next of vertexPathQ to dequeue
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const vertexPath = vertexPathQ.dequeue();
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// check that vertexPath exists and ends with the sink
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if (vertexPath !== null && vertexPath[vertexPath.length - 1] === t) {
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// reset augmentingPath
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augmentingPath = [];
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for (let i = 0; i < vertexPath.length - 1; i += 1) {
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// get the start and end vertices for next edge in path in residualGraph
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const startVertex = residualGraph.getVertexByKey(vertexPath[i]);
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const endVertex = residualGraph.getVertexByKey(vertexPath[i + 1]);
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// find the edge in residualGraph
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const edge = residualGraph.findEdge(startVertex, endVertex);
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// add the edge to the augmentingPath
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augmentingPath.push(edge);
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}
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// an augmenting path has been found so return true
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return true;
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}
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// otherwise no augmenting path found so return false
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return false;
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}
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/// --- MAIN EXECUTION LOOP ---
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// build an inital residual graph
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buildResidualGraph();
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// while there is still an augmenting path in residualGraph
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while (findAugmentingPath()) {
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// augment the flow network along that graph
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augment(augmentingPath);
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// then build the updated residual graph
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buildResidualGraph();
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}
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// g is the final graph with flow values as weights on the edges.
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// f is the max flow.
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return { graph: g, maxflow: f };
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}
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67
src/algorithms/graph/flow-networks/ford-fulkerson/README.md
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src/algorithms/graph/flow-networks/ford-fulkerson/README.md
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# Ford-Fulkerson
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The Ford–Fulkerson method is a greedy algorithm that computes the maximum flow in a flow network. It is sometimes called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified.
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To see a Javascript implementation, go to the [Edmonds-Karp](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/edmonds-karp) algorithm.
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Edmonds-Karp is an implementation of Ford-Fulkerson that specifies the use of [breadth-first search](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/breadth-first-search) (BFS) to determine the search order for an augmenting path.
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### Residual Graph G<sub>f</sub>
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Construction of a residual graph G<sub>f</sub> = ( V , E<sub>f</sub> ) is dependent on a flow `f` (and the graph `G` and its flow capacity `c`). Initialised at the start of the Ford-Fulkerson algorithm, it is updated at every pass of the main loop. It maintains the residual capacity c<sub>f</sub>(e) and edge set E<sub>f</sub> that depend on the flow `f`.
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**Residual Graph Construction:**
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Start with empty E<sub>f</sub> and c<sub>f</sub>(e). For each edge `e = (u,v)` with flow `f(e)` and capacity *c(e)*:
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- if `f(e) < c(e)`: add *forward* edge e to E<sub>f</sub> with capacity *c<sub>f</sub>(e)* = `c(e) - f(e)`
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- if `f(e) > 0`: add *backward* edge *e'=(v,u)* to E<sub>f</sub> with capacity *c<sub>f</sub>(e')* = `f(e)`
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**Residual Graph Intuition:**
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- when the flow through an edge is less than its capacity, you can still *push* at most the difference `c(e) - f(e) units of flow` along that edge
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- when the flow through an edge is non-zero, you can *undo* all of the flow `f(e) units of flow` along that edge
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### Pseudocode algorithm
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**Algorithm**: Ford-Fulkerson
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**Inputs:** a network `G = ( V, E )` with flow capacity `c`, source `s`, sink `t`
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**Outputs:** the maximum value of the flow `|f|` from `s` to `t`
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```pseudocode
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FORD-FULKERSON(G,s,t,c)
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AUGMENT(f,P)
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bottleneck_cap_p = min {c_f(e) for e in P}
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f2(edge) = f(edge) for edge in E
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for edge in E
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if edge in path P
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f2(edge) = f(edge) + bottleneck_cap_p
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if reverse(edge) in path P
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f2(edge) = f(edge) - bottleneck_cap_p
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return f2
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f(edge) = 0 for edge in E
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G_f = residual graph of G (with respect to f and capacity c)
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while there exists a path P (from s to t) in G_f
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f = AUGMENT(f,P)
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Update G_f (by definition of residual graph with new f)
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return f
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```
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### Augmenting along a path
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Augmenting along a path `P` always adds `min {c_f(e) for e in P}` to the flow through the network. `P` is any path from `s` to `t` in the residual graph *G<sub>f</sub>*.
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### Demo of the algorithm working
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## References
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- [Wikipedia | Ford-Fulkerson](https://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm)
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- [Wikipedia | Maximum Flow](https://en.wikipedia.org/wiki/Maximum_flow_problem)
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- [Youtube | Max Flow Ford-Fulkerson](https://www.youtube.com/watch?v=LdOnanfc5TM)
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- [Amazon | Introduction to Algorithms (CLRS)](https://www.amazon.co.uk/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844)
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- [Wikipedia Commons | Ford-Fulkerson Demo](https://commons.wikimedia.org/wiki/File:FordFulkersonDemo.gif)
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