content: /*************************************************************************
   *  Compilation:  javac FordFulkerson.java
   *  Execution:    java FordFulkerson V E
   *  Dependencies: FlowNetwork.java FlowEdge.java Queue.java
   *
   *  Ford-Fulkerson algorithm for computing a max flow and 
   *  a min cut using shortest augmenthing path rule.
   *
   *********************************************************************/
  
  public class FordFulkerson {
      private boolean[] marked;     // marked[v] = true iff s->v path in residual graph
      private FlowEdge[] edgeTo;    // edgeTo[v] = last edge on shortest residual s->v path
      private double value;         // current value of max flow
    
      // max flow in flow network G from s to t
      public FordFulkerson(FlowNetwork G, int s, int t) {
          value = excess(G, t);
          if (!isFeasible(G, s, t)) {
              throw new RuntimeException("Initial flow is infeasible");
          }
  
          // while there exists an augmenting path, use it
          while (hasAugmentingPath(G, s, t)) {
  
              // compute bottleneck capacity
              double bottle = Double.POSITIVE_INFINITY;
              for (int v = t; v != s; v = edgeTo[v].other(v)) {
                  bottle = Math.min(bottle, edgeTo[v].residualCapacityTo(v));
              }
  
              // augment flow
              for (int v = t; v != s; v = edgeTo[v].other(v)) {
                  edgeTo[v].addResidualFlowTo(v, bottle); 
              }
  
              value += bottle;
          }
  
          // check optimality conditions
          assert check(G, s, t);
      }
  
      // return value of max flow
      public double value()  {
          return value;
      }
  
      // is v in the s side of the min s-t cut?
      public boolean inCut(int v)  {
          return marked[v];
      }
  
  
      // return an augmenting path if one exists, otherwise return null
      private boolean hasAugmentingPath(FlowNetwork G, int s, int t) {
          edgeTo = new FlowEdge[G.V()];
          marked = new boolean[G.V()];
  
          // breadth-first search
          Queue<Integer> q = new Queue<Integer>();
          q.enqueue(s);
          marked[s] = true;
          while (!q.isEmpty()) {
              int v = q.dequeue();
  
              for (FlowEdge e : G.adj(v)) {
                  int w = e.other(v);
  
                  // if residual capacity from v to w
                  if (e.residualCapacityTo(w) > 0) {
                      if (!marked[w]) {
                          edgeTo[w] = e;
                          marked[w] = true;
                          q.enqueue(w);
                      }
                  }
              }
          }
  
          // is there an augmenting path?
          return marked[t];
      }
  
  
  
      // return excess flow at vertex v
      private double excess(FlowNetwork G, int v) {
          double excess = 0.0;
          for (FlowEdge e : G.adj(v)) {
              if (v == e.from()) excess -= e.flow();
              else               excess += e.flow();
          }
          return excess;
      }
  
      // return excess flow at vertex v
      private boolean isFeasible(FlowNetwork G, int s, int t) {
          double EPSILON = 1E-11;
  
          // check that capacity constraints are satisfied
          for (int v = 0; v < G.V(); v++) {
              for (FlowEdge e : G.adj(v)) {
                  if (e.flow() < 0 || e.flow() > e.capacity()) {
                      System.err.println("Edge does not satisfy capacity constraints: " + e);
                      return false;
                  }
              }
          }
  
          // check that net flow into a vertex equals zero, except at source and sink
          if (Math.abs(value + excess(G, s)) > EPSILON) {
              System.err.println("Excess at source = " + excess(G, s));
              System.err.println("Max flow         = " + value);
              return false;
          }
          if (Math.abs(value - excess(G, t)) > EPSILON) {
              System.err.println("Excess at sink   = " + excess(G, t));
              System.err.println("Max flow         = " + value);
              return false;
          }
          for (int v = 0; v < G.V(); v++) {
              if (v == s || v == t) continue;
              else if (Math.abs(excess(G, v)) > EPSILON) {
                  System.err.println("Net flow out of " + v + " doesn't equal zero");
                  return false;
              }
          }
          return true;
      }
  
  
  
      // check optimality conditions
      private boolean check(FlowNetwork G, int s, int t) {
  
          // check that flow is feasible
          if (!isFeasible(G, s, t)) {
              System.err.println("Flow is infeasible");
              return false;
          }
  
          // check that s is on the source side of min cut and that t is not on source side
          if (!inCut(s)) {
              System.err.println("source " + s + " is not on source side of min cut");
              return false;
          }
          if (inCut(t)) {
              System.err.println("sink " + t + " is on source side of min cut");
              return false;
          }
  
          // check that value of min cut = value of max flow
          double mincutValue = 0.0;
          for (int v = 0; v < G.V(); v++) {
              for (FlowEdge e : G.adj(v)) {
                  if ((v == e.from()) && inCut(e.from()) && !inCut(e.to()))
                      mincutValue += e.capacity();
              }
          }
  
          double EPSILON = 1E-11;
          if (Math.abs(mincutValue - value) > EPSILON) {
              System.err.println("Max flow value = " + value + ", min cut value = " + mincutValue);
              return false;
          }
  
          return true;
      }
  
  
      // test client that creates random network, solves max flow, and prints results
      public static void main(String[] args) {
  
          // create flow network with V vertices and E edges
          int V = Integer.parseInt(args[0]);
          int E = Integer.parseInt(args[1]);
          int s = 0, t = V-1;
          FlowNetwork G = new FlowNetwork(V, E);
          StdOut.println(G);
  
          // compute maximum flow and minimum cut
          FordFulkerson maxflow = new FordFulkerson(G, s, t);
          StdOut.println("Max flow from " + s + " to " + t);
          for (int v = 0; v < G.V(); v++) {
              for (FlowEdge e : G.adj(v)) {
                  if ((v == e.from()) && e.flow() > 0)
                      StdOut.println("   " + e);
              }
          }
  
          // print min-cut
          StdOut.print("Min cut: ");
          for (int v = 0; v < G.V(); v++) {
              if (maxflow.inCut(v)) StdOut.print(v + " ");
          }
          StdOut.println();
  
          StdOut.println("Max flow value = " +  maxflow.value());
      }
  
  }
  