package edu.princeton.cs.algs4;
import edu.princeton.cs.algs4.ch13.Stack;
import edu.princeton.cs.algs4.ch42.DirectedEdge;
import edu.princeton.cs.algs4.ch43.EdgeWeightedDigraph;
import edu.princeton.cs.algs4.ch43.EdgeWeightedDirectedCycle;
import edu.princeton.cs.introcs.*;
/*************************************************************************
* Compilation: javac FloydWarshall.java
* Execution: java FloydWarshall V E
* Dependencies: AdjMatrixEdgeWeightedDigraph.java
*
* Floyd-Warshall all-pairs shortest path algorithm.
*
* % java FloydWarshall 100 500
*
* Should check for negative cycles during triple loop; otherwise
* intermediate numbers can get exponentially large.
* Reference: "The Floyd-Warshall algorithm on graphs with negative cycles"
* by Stefan Hougardy
*
*************************************************************************/
/**
* The <tt>FloydWarshall</tt> class represents a data type for solving the
* all-pairs shortest paths problem in edge-weighted digraphs with
* no negative cycles.
* The edge weights can be positive, negative, or zero.
* This class finds either a shortest path between every pair of vertices
* or a negative cycle.
* <p>
* This implementation uses the Floyd-Warshall algorithm.
* The constructor takes time proportional to <em>V</em><sup>3</sup> in the
* worst case, where <em>V</em> is the number of vertices.
* Afterwards, the <tt>dist()</tt>, <tt>hasPath()</tt>, and <tt>hasNegativeCycle()</tt>
* methods take constant time; the <tt>path()</tt> and <tt>negativeCycle()</tt>
* method takes time proportional to the number of edges returned.
* <p>
* For additional documentation, see <a href="/algs4/44sp">Section 4.4</a> of
* <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/ public class FloydWarshall {
private boolean hasNegativeCycle; // is there a negative cycle?
private double[][] distTo; // distTo[v][w] = length of shortest v->w path
private DirectedEdge[][] edgeTo; // edgeTo[v][w] = last edge on shortest v->w path
/**
* Computes a shortest paths tree from each vertex to to every other vertex in
* the edge-weighted digraph <tt>G</tt>. If no such shortest path exists for
* some pair of vertices, it computes a negative cycle.
* @param G the edge-weighted digraph
*/
public FloydWarshall(AdjMatrixEdgeWeightedDigraph G) {
int V = G.V();
distTo = new double[V][V];
edgeTo = new DirectedEdge[V][V];
// initialize distances to infinity
for (int v = 0; v < V; v++) {
for (int w = 0; w < V; w++) {
distTo[v][w] = Double.POSITIVE_INFINITY;
}
}
// initialize distances using edge-weighted digraph's
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
distTo[e.from()][e.to()] = e.weight();
edgeTo[e.from()][e.to()] = e;
}
// in case of self-loops
if (distTo[v][v] >= 0.0) {
distTo[v][v] = 0.0;
edgeTo[v][v] = null;
}
}
// Floyd-Warshall updates
for (int i = 0; i < V; i++) {
// compute shortest paths using only 0, 1, ..., i as intermediate vertices
for (int v = 0; v < V; v++) {
if (edgeTo[v][i] == null) continue; // optimization
for (int w = 0; w < V; w++) {
if (distTo[v][w] > distTo[v][i] + distTo[i][w]) {
distTo[v][w] = distTo[v][i] + distTo[i][w];
edgeTo[v][w] = edgeTo[i][w];
}
}
// check for negative cycle
if (distTo[v][v] < 0.0) {
hasNegativeCycle = true;
return;
}
}
}
}
/**
* Is there a negative cycle?
* @return <tt>true</tt> if there is a negative cycle, and <tt>false</tt> otherwise
*/
public boolean hasNegativeCycle() {
return hasNegativeCycle;
}
/**
* Returns a negative cycle, or <tt>null</tt> if there is no such cycle.
* @return a negative cycle as an iterable of edges,
* or <tt>null</tt> if there is no such cycle
*/
public Iterable<DirectedEdge> negativeCycle() {
for (int v = 0; v < distTo.length; v++) {
// negative cycle in v's predecessor graph
if (distTo[v][v] < 0.0) {
int V = edgeTo.length;
EdgeWeightedDigraph spt = new EdgeWeightedDigraph(V);
for (int w = 0; w < V; w++)
if (edgeTo[v][w] != null)
spt.addEdge(edgeTo[v][w]);
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(spt);
assert finder.hasCycle();
return finder.cycle();
}
}
return null;
}
/**
* Is there a path from the vertex <tt>s</tt> to vertex <tt>t</tt>?
* @param s the source vertex
* @param t the destination vertex
* @return <tt>true</tt> if there is a path from vertex <tt>s</tt>
* to vertex <tt>t</tt>, and <tt>false</tt> otherwise
*/
public boolean hasPath(int s, int t) {
return distTo[s][t] < Double.POSITIVE_INFINITY;
}
/**
* Returns the length of a shortest path from vertex <tt>s</tt> to vertex <tt>t</tt>.
* @param s the source vertex
* @param t the destination vertex
* @return the length of a shortest path from vertex <tt>s</tt> to vertex <tt>t</tt>;
* <tt>Double.POSITIVE_INFINITY</tt> if no such path
* @throws UnsupportedOperationException if there is a negative cost cycle
*/
public double dist(int s, int t) {
if (hasNegativeCycle())
throw new UnsupportedOperationException("Negative cost cycle exists");
return distTo[s][t];
}
/**
* Returns a shortest path from vertex <tt>s</tt> to vertex <tt>t</tt>.
* @param s the source vertex
* @param t the destination vertex
* @return a shortest path from vertex <tt>s</tt> to vertex <tt>t</tt>
* as an iterable of edges, and <tt>null</tt> if no such path
* @throws UnsupportedOperationException if there is a negative cost cycle
*/
public Iterable<DirectedEdge> path(int s, int t) {
if (hasNegativeCycle())
throw new UnsupportedOperationException("Negative cost cycle exists");
if (!hasPath(s, t)) return null;
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[s][t]; e != null; e = edgeTo[s][e.from()]) {
path.push(e);
}
return path;
}
// check optimality conditions
private boolean check(EdgeWeightedDigraph G, int s) {
// no negative cycle
if (!hasNegativeCycle()) {
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
for (int i = 0; i < G.V(); i++) {
if (distTo[i][w] > distTo[i][v] + e.weight()) {
System.err.println("edge " + e + " is eligible");
return false;
}
}
}
}
}
return true;
}
/**
* Unit tests the <tt>FloydWarshall</tt> data type.
*/
public static void main(String[] args) {
// random graph with V vertices and E edges, parallel edges allowed
int V = Integer.parseInt(args[0]);
int E = Integer.parseInt(args[1]);
AdjMatrixEdgeWeightedDigraph G = new AdjMatrixEdgeWeightedDigraph(V);
for (int i = 0; i < E; i++) {
int v = (int) (V * Math.random());
int w = (int) (V * Math.random());
double weight = Math.round(100 * (Math.random() - 0.15)) / 100.0;
if (v == w) G.addEdge(new DirectedEdge(v, w, Math.abs(weight)));
else G.addEdge(new DirectedEdge(v, w, weight));
}
StdOut.println(G);
// run Floyd-Warshall algorithm
FloydWarshall spt = new FloydWarshall(G);
// print all-pairs shortest path distances
StdOut.printf(" ");
for (int v = 0; v < G.V(); v++) {
StdOut.printf("%6d ", v);
}
StdOut.println();
for (int v = 0; v < G.V(); v++) {
StdOut.printf("%3d: ", v);
for (int w = 0; w < G.V(); w++) {
if (spt.hasPath(v, w)) StdOut.printf("%6.2f ", spt.dist(v, w));
else StdOut.printf(" Inf ");
}
StdOut.println();
}
// print negative cycle
if (spt.hasNegativeCycle()) {
StdOut.println("Negative cost cycle:");
for (DirectedEdge e : spt.negativeCycle())
StdOut.println(e);
StdOut.println();
}
// print all-pairs shortest paths
else {
for (int v = 0; v < G.V(); v++) {
for (int w = 0; w < G.V(); w++) {
if (spt.hasPath(v, w)) {
StdOut.printf("%d to %d (%5.2f) ", v, w, spt.dist(v, w));
for (DirectedEdge e : spt.path(v, w))
StdOut.print(e + " ");
StdOut.println();
}
else {
StdOut.printf("%d to %d no path\n", v, w);
}
}
}
}
}
}
/*************************************************************************
* Copyright 2002-2012, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4-package.jar, which accompanies the textbook
*
* Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
* Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
* http://algs4.cs.princeton.edu
*
*
* algs4-package.jar is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* algs4-package.jar is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with algs4-package.jar. If not, see http://www.gnu.org/licenses.
*************************************************************************/