Problem:
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Maximum Flow

A flow network is a directed graph which has a $source$ and a $sink$. In a flow network, each edge $(u, v)$ has a capacity $c(u, v)$. Each edge receives a flow, but the amount of flow on the edge can not exceed the corresponding capacity. Find the maximum flow from the $source$ to the $sink$.

Input

A flow network is given in the following format.

$|V|\;|E|$

$u_0\;v_0\;c_0$

$u_1\;v_1\;c_1$

:

$u_{|E|-1}\;v_{|E|-1}\;c_{|E|-1}$

$|V|$, $|E|$ is the number of vertices and edges of the flow network respectively. The vertices in $G$ are named with the numbers 0, 1,..., $|V|-1$. The source is 0 and the sink is $|V|-1$.

$u_i$, $v_i$, $c_i$ represent $i$-th edge of the flow network.
A pair of $u_i$ and $v_i$ denotes that there is an edge from $u_i$ to $v_i$ and $c_i$ is the capacity of $i$-th edge.

Output

Print the maximum flow.

Constraints

$2 \leq |V| \leq 100$

$1 \leq |E| \leq 1000$

$0 \leq c_i \leq 10000$

Sample Input 1

4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

Sample Output 1

3

Solution in JAVA (judge verdict: Accepted)


import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.List;
import java.util.Scanner;

public class Main {

    public static class FordFulkerson_DFS {

        final ResidualNetwork residualG;
        final int srcVert;
        final int termVert;
        boolean[] used;

        public FordFulkerson_DFS(WeightedDigraph graph, int srcVert, int termVert) {
            this.residualG = new ResidualNetwork(graph);
            this.srcVert = srcVert;
            this.termVert = termVert;
            used = new boolean[graph.numVertex];
        }

        public int computeMaxFlow() {
            int maxFlow = 0;
            for (;;) {
                Arrays.fill(used, false);
                int flow = dfs(srcVert, termVert, Integer.MAX_VALUE);
                if (flow == 0) {
                    return maxFlow;
                }
                maxFlow += flow;
            }
        }

        private int dfs(int v, int t, int flow) {
            if (v == t) {
                return flow;
            }
            used[v] = true;
            for (WeightedDigraph.Edge edge : residualG.srcEdges[v]) {
                if (!used[edge.targetVertex] && edge.weight > 0) {
                    int minFlow = dfs(edge.targetVertex, t, Math.min(flow, edge.weight));
                    if (minFlow > 0) {
                        edge.weight -= minFlow;
                        residualG.edgeMap.get(edge).weight += minFlow;
                        return minFlow;
                    }
                }
            }
            return 0;
        }
    }

    public static class ResidualNetwork extends WeightedDigraph {

        HashMap<Edge, Edge> edgeMap;

        public ResidualNetwork(WeightedDigraph graph) {
            super(graph.numVertex);
            edgeMap = new HashMap<Edge, Edge>();
            for (int i = 0; i < graph.numVertex; i++) {
                for (WeightedDigraph.Edge edge : graph.srcEdges[i]) {
                    this.addEdge(edge.srcVertex, edge.targetVertex, edge.weight);
                }
            }
        }

        @Override
        public void addEdge(int srcVertex, int targetVertex, int weight) {
            Edge e = new WeightedDigraph.Edge(srcVertex, targetVertex, weight);
            srcEdges[srcVertex].add(e);
            Edge backE = new WeightedDigraph.Edge(targetVertex, srcVertex, 0);
            srcEdges[targetVertex].add(backE);  
            edgeMap.put(e, backE);
            edgeMap.put(backE, e);
        }
    }

    public static class WeightedDigraph {

        int numVertex;
        List<Edge>[] srcEdges;

        @SuppressWarnings("unchecked")
        public WeightedDigraph(int numV) {
            this.numVertex = numV;
            srcEdges = new List[numV];

            for (int i = 0; i < numV; i++) {
                srcEdges[i] = new ArrayList<Edge>();
            }
        }

        public void addEdge(int srcVertex, int targetVertex, int weight) {
            Edge e = new WeightedDigraph.Edge(srcVertex, targetVertex, weight);
            srcEdges[srcVertex].add(e);
        }

        public static class Edge {

            int srcVertex;
            int targetVertex;
            int weight;

            public Edge(int srcVertex, int targetVertex, int weight) {
                this.srcVertex = srcVertex;
                this.targetVertex = targetVertex;
                this.weight = weight;
            }

            @Override
            public int hashCode() {
                int hash = 3;
                hash = 59 * hash + this.srcVertex;
                hash = 59 * hash + this.targetVertex;
                return hash;
            }

            @Override
            public boolean equals(Object obj) {
                if (this == obj) {
                    return true;
                }
                if (obj == null) {
                    return false;
                }
                if (getClass() != obj.getClass()) {
                    return false;
                }
                final Edge other = (Edge) obj;
                if (this.srcVertex != other.srcVertex) {
                    return false;
                }
                if (this.targetVertex != other.targetVertex) {
                    return false;
                }
                return true;
            }

        }

    }

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int numVertex = scanner.nextInt();
        int numEdge = scanner.nextInt();
        WeightedDigraph graph = new WeightedDigraph(numVertex);
        while (numEdge-- > 0) {
            int srcV = scanner.nextInt();
            int targetV = scanner.nextInt();
            int weight = scanner.nextInt();
            graph.addEdge(srcV, targetV, weight);
        }
        
        FordFulkerson_DFS flowNet = new FordFulkerson_DFS(graph, 0, numVertex - 1);
        System.out.println(flowNet.computeMaxFlow());

    }

}