Find the single source longest path in a graph by tweaking the SSSP algorithms i
ID: 3861118 • Letter: F
Question
Find the single source longest path in a graph by tweaking the SSSP algorithms in a Bellman-Ford implementation. JAVA
BellmanFordSP.java
/******************************************************************************
* Compilation: javac BellmanFordSP.java
* Execution: java BellmanFordSP filename.txt s
* Dependencies: EdgeWeightedDigraph.java DirectedEdge.java Queue.java
* EdgeWeightedDirectedCycle.java
* Data files: http://algs4.cs.princeton.edu/44sp/tinyEWDn.txt
* http://algs4.cs.princeton.edu/44sp/mediumEWDnc.txt
*
* Bellman-Ford shortest path algorithm. Computes the shortest path tree in
* edge-weighted digraph G from vertex s, or finds a negative cost cycle
* reachable from s.
*
* % java BellmanFordSP tinyEWDn.txt 0
* 0 to 0 ( 0.00)
* 0 to 1 ( 0.93) 0->2 0.26 2->7 0.34 7->3 0.39 3->6 0.52 6->4 -1.25 4->5 0.35 5->1 0.32
* 0 to 2 ( 0.26) 0->2 0.26
* 0 to 3 ( 0.99) 0->2 0.26 2->7 0.34 7->3 0.39
* 0 to 4 ( 0.26) 0->2 0.26 2->7 0.34 7->3 0.39 3->6 0.52 6->4 -1.25
* 0 to 5 ( 0.61) 0->2 0.26 2->7 0.34 7->3 0.39 3->6 0.52 6->4 -1.25 4->5 0.35
* 0 to 6 ( 1.51) 0->2 0.26 2->7 0.34 7->3 0.39 3->6 0.52
* 0 to 7 ( 0.60) 0->2 0.26 2->7 0.34
*
* % java BellmanFordSP tinyEWDnc.txt 0
* 4->5 0.35
* 5->4 -0.66
*
*
******************************************************************************/
//package edu.princeton.cs.algs4;
/**
* The {@code BellmanFordSP} class represents a data type for solving the
* single-source 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 from the source vertex s
* to every other vertex or a negative cycle reachable from the source vertex.
*
* This implementation uses the Bellman-Ford-Moore algorithm.
* The constructor takes time proportional to V (V + E)
* in the worst case, where V is the number of vertices and E
* is the number of edges.
* Afterwards, the {@code distTo()}, {@code hasPathTo()}, and {@code hasNegativeCycle()}
* methods take constant time; the {@code pathTo()} and {@code negativeCycle()}
* method takes time proportional to the number of edges returned.
*
* For additional documentation,
* see Section 4.4 of
* Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class BellmanFordSP {
private double[] distTo; // distTo[v] = distance of shortest s->v path
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on shortest s->v path
private boolean[] onQueue; // onQueue[v] = is v currently on the queue?
private Queue queue; // queue of vertices to relax
private int cost; // number of calls to relax()
private Iterable cycle; // negative cycle (or null if no such cycle)
/**
* Computes a shortest paths tree from {@code s} to every other vertex in
* the edge-weighted digraph {@code G}.
* @param G the acyclic digraph
* @param s the source vertex
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public BellmanFordSP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
onQueue = new boolean[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// Bellman-Ford algorithm
queue = new Queue();
queue.enqueue(s);
onQueue[s] = true;
while (!queue.isEmpty() && !hasNegativeCycle()) {
int v = queue.dequeue();
onQueue[v] = false;
relax(G, v);
}
assert check(G, s);
}
// relax vertex v and put other endpoints on queue if changed
private void relax(EdgeWeightedDigraph G, int v) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[w] > distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
if (!onQueue[w]) {
queue.enqueue(w);
onQueue[w] = true;
}
}
if (cost++ % G.V() == 0) {
findNegativeCycle();
if (hasNegativeCycle()) return; // found a negative cycle
}
}
}
/**
* Is there a negative cycle reachable from the source vertex {@code s}?
* @return {@code true} if there is a negative cycle reachable from the
* source vertex {@code s}, and {@code false} otherwise
*/
public boolean hasNegativeCycle() {
return cycle != null;
}
/**
* Returns a negative cycle reachable from the source vertex {@code s}, or {@code null}
* if there is no such cycle.
* @return a negative cycle reachable from the soruce vertex {@code s}
* as an iterable of edges, and {@code null} if there is no such cycle
*/
public Iterable negativeCycle() {
return cycle;
}
// by finding a cycle in predecessor graph
private void findNegativeCycle() {
int V = edgeTo.length;
EdgeWeightedDigraph spt = new EdgeWeightedDigraph(V);
for (int v = 0; v < V; v++)
if (edgeTo[v] != null)
spt.addEdge(edgeTo[v]);
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(spt);
cycle = finder.cycle();
}
/**
* Returns the length of a shortest path from the source vertex {@code s} to vertex {@code v}.
* @param v the destination vertex
* @return the length of a shortest path from the source vertex {@code s} to vertex {@code v};
* {@code Double.POSITIVE_INFINITY} if no such path
* @throws UnsupportedOperationException if there is a negative cost cycle reachable
* from the source vertex {@code s}
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public double distTo(int v) {
validateVertex(v);
if (hasNegativeCycle())
throw new UnsupportedOperationException("Negative cost cycle exists");
return distTo[v];
}
/**
* Is there a path from the source {@code s} to vertex {@code v}?
* @param v the destination vertex
* @return {@code true} if there is a path from the source vertex
* {@code s} to vertex {@code v}, and {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public boolean hasPathTo(int v) {
validateVertex(v);
return distTo[v] < Double.POSITIVE_INFINITY;
}
/**
* Returns a shortest path from the source {@code s} to vertex {@code v}.
* @param v the destination vertex
* @return a shortest path from the source {@code s} to vertex {@code v}
* as an iterable of edges, and {@code null} if no such path
* @throws UnsupportedOperationException if there is a negative cost cycle reachable
* from the source vertex {@code s}
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public Iterable pathTo(int v) {
validateVertex(v);
if (hasNegativeCycle())
throw new UnsupportedOperationException("Negative cost cycle exists");
if (!hasPathTo(v)) return null;
Stack path = new Stack();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
path.push(e);
}
return path;
}
// check optimality conditions: either
// (i) there exists a negative cycle reacheable from s
// or
// (ii) for all edges e = v->w: distTo[w] <= distTo[v] + e.weight()
// (ii') for all edges e = v->w on the SPT: distTo[w] == distTo[v] + e.weight()
private boolean check(EdgeWeightedDigraph G, int s) {
// has a negative cycle
if (hasNegativeCycle()) {
double weight = 0.0;
for (DirectedEdge e : negativeCycle()) {
weight += e.weight();
}
if (weight >= 0.0) {
System.err.println("error: weight of negative cycle = " + weight);
return false;
}
}
// no negative cycle reachable from source
else {
// check that distTo[v] and edgeTo[v] are consistent
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distanceTo[s] and edgeTo[s] inconsistent");
return false;
}
for (int v = 0; v < G.V(); v++) {
if (v == s) continue;
if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
System.err.println("distTo[] and edgeTo[] inconsistent");
return false;
}
}
// check that all edges e = v->w satisfy distTo[w] <= distTo[v] + e.weight()
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[v] + e.weight() < distTo[w]) {
System.err.println("edge " + e + " not relaxed");
return false;
}
}
}
// check that all edges e = v->w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (int w = 0; w < G.V(); w++) {
if (edgeTo[w] == null) continue;
DirectedEdge e = edgeTo[w];
int v = e.from();
if (w != e.to()) return false;
if (distTo[v] + e.weight() != distTo[w]) {
System.err.println("edge " + e + " on shortest path not tight");
return false;
}
}
}
StdOut.println("Satisfies optimality conditions");
StdOut.println();
return true;
}
// throw an IllegalArgumentException unless {@code 0 <= v < V}
private void validateVertex(int v) {
int V = distTo.length;
if (v < 0 || v >= V)
throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
}
/**
* Unit tests the {@code BellmanFordSP} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
int s = Integer.parseInt(args[1]);
EdgeWeightedDigraph G = new EdgeWeightedDigraph(in);
BellmanFordSP sp = new BellmanFordSP(G, s);
// print negative cycle
if (sp.hasNegativeCycle()) {
for (DirectedEdge e : sp.negativeCycle())
StdOut.println(e);
}
// print shortest paths
else {
for (int v = 0; v < G.V(); v++) {
if (sp.hasPathTo(v)) {
StdOut.printf("%d to %d (%5.2f) ", s, v, sp.distTo(v));
for (DirectedEdge e : sp.pathTo(v)) {
StdOut.print(e + " ");
}
StdOut.println();
}
else {
StdOut.printf("%d to %d no path ", s, v);
}
}
}
}
}
/******************************************************************************
* Copyright 2002-2016, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4.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.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.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.jar. If not, see http://www.gnu.org/licenses.
******************************************************************************/
tinyEWD.txt
8
15
4 5 0.35
5 4 0.35
4 7 0.37
5 7 0.28
7 5 0.28
5 1 0.32
0 4 0.38
0 2 0.26
7 3 0.39
1 3 0.29
2 7 0.34
6 2 0.40
3 6 0.52
6 0 0.58
6 4 0.93
Explanation / Answer
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public class BellmanFord {
public Integer[][] singleSourceShortestPath(Integer[][] weight,
int source) throws Exception
{
//auxiliary constants
final int SIZE = weight.length;
final int EVE = -1;//to indicate no predecessor
final int INFINITY = Integer.MAX_VALUE;
//declare and initialize pred to EVE and minDist to INFINITY
Integer[] pred = new Integer[SIZE];
Integer[] minDist = new Integer[SIZE];
Arrays.fill(pred, EVE);
Arrays.fill(minDist, INFINITY);
//set minDist[source] = 0 because source is 0 distance from itself.
minDist[source] = 0;
//relax the edge set V-1 times to find all shortest paths
for (int i = 1; i < minDist.length - 1; i++) {
for (int v = 0; v < SIZE; v++) {
for (int x : adjacency(weight, v)) {
if (minDist[x] > minDist[v] + weight[v][x]) {
minDist[x] = minDist[v] + weight[v][x];
pred[x] = v;
}
}
}
}
//detect cycles if any
for (int v = 0; v < SIZE; v++) {
for (int x : adjacency(weight, v)) {
if (minDist[x] > minDist[v] + weight[v][x]) {
throw new Exception("Negative cycle found");
}
}
}
Integer[][] result = {pred, minDist};
return result;
}
/******************************************************************
* Retrieve all the neighbors of vertex v.
*****************************************************************/
private List<Integer> adjacency(Integer[][] G, int v) {
List<Integer> result = new ArrayList<Integer>();
for (int x = 0; x < G.length; x++) {
if (G[v][x] != null) {
result.add(x);
}
}
return result;
}
}
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