/******************************************************************************
* Compilation: javac DijkstraSP.java
* Execution: java DijkstraSP input.txt s
* Dependencies: EdgeWeightedDigraph.java IndexMinPQ.java Stack.java DirectedEdge.java
* Data files: http://algs4.cs.princeton.edu/44sp/tinyEWD.txt
* http://algs4.cs.princeton.edu/44sp/mediumEWD.txt
* http://algs4.cs.princeton.edu/44sp/largeEWD.txt
*
* Dijkstra's algorithm. Computes the shortest path tree.
* Assumes all weights are nonnegative.
*
* % java DijkstraSP tinyEWD.txt 0
* 0 to 0 (0.00)
* 0 to 1 (1.05) 0->4 0.38 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 (0.38) 0->4 0.38
* 0 to 5 (0.73) 0->4 0.38 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 DijkstraSP mediumEWD.txt 0
* 0 to 0 (0.00)
* 0 to 1 (0.71) 0->44 0.06 44->93 0.07 ... 107->1 0.07
* 0 to 2 (0.65) 0->44 0.06 44->231 0.10 ... 42->2 0.11
* 0 to 3 (0.46) 0->97 0.08 97->248 0.09 ... 45->3 0.12
* 0 to 4 (0.42) 0->44 0.06 44->93 0.07 ... 77->4 0.11
* ...
*
******************************************************************************/
package edu.princeton.cs.algs4;
/**
* The <tt>DijkstraSP</tt> class represents a data type for solving the
* single-source shortest paths problem in edge-weighted digraphs
* where the edge weights are nonnegative.
* <p>
* This implementation uses Dijkstra's algorithm with a binary heap.
* The constructor takes time proportional to <em>E</em> log <em>V</em>,
* where <em>V</em> is the number of vertices and <em>E</em> is the number of edges.
* Afterwards, the <tt>distTo()</tt> and <tt>hasPathTo()</tt> methods take
* constant time and the <tt>pathTo()</tt> method takes time proportional to the
* number of edges in the shortest path returned.
* <p>
* For additional documentation,
* see <a href="http://algs4.cs.princeton.edu/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 DijkstraSP {
private double[] distTo; // distTo[v] = distance of shortest s->v path
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on shortest s->v path
private IndexMinPQ<Double> pq; // priority queue of vertices
/**
* Computes a shortest-paths tree from the source vertex <tt>s</tt> to every other
* vertex in the edge-weighted digraph <tt>G</tt>.
*
* @param G the edge-weighted digraph
* @param s the source vertex
* @throws IllegalArgumentException if an edge weight is negative
* @throws IllegalArgumentException unless 0 ≤ <tt>s</tt> ≤ <tt>V</tt> - 1
*/
public DijkstraSP(EdgeWeightedDigraph G, int s) {
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0)
throw new IllegalArgumentException("edge " + e + " has negative weight");
}
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// relax vertices in order of distance from s
pq = new IndexMinPQ<Double>(G.V());
pq.insert(s, distTo[s]);
while (!pq.isEmpty()) {
int v = pq.delMin();
for (DirectedEdge e : G.adj(v))
relax(e);
}
// check optimality conditions
assert check(G, s);
}
// relax edge e and update pq if changed
private void relax(DirectedEdge e) {
int v = e.from(), w = e.to();
if (distTo[w] > distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
if (pq.contains(w)) pq.decreaseKey(w, distTo[w]);
else pq.insert(w, distTo[w]);
}
}
/**
* Returns the length of a shortest path from the source vertex <tt>s</tt> to vertex <tt>v</tt>.
* @param v the destination vertex
* @return the length of a shortest path from the source vertex <tt>s</tt> to vertex <tt>v</tt>;
* <tt>Double.POSITIVE_INFINITY</tt> if no such path
*/
public double distTo(int v) {
return distTo[v];
}
/**
* Returns true if there is a path from the source vertex <tt>s</tt> to vertex <tt>v</tt>.
*
* @param v the destination vertex
* @return <tt>true</tt> if there is a path from the source vertex
* <tt>s</tt> to vertex <tt>v</tt>; <tt>false</tt> otherwise
*/
public boolean hasPathTo(int v) {
return distTo[v] < Double.POSITIVE_INFINITY;
}
/**
* Returns a shortest path from the source vertex <tt>s</tt> to vertex <tt>v</tt>.
*
* @param v the destination vertex
* @return a shortest path from the source vertex <tt>s</tt> to vertex <tt>v</tt>
* as an iterable of edges, and <tt>null</tt> if no such path
*/
public Iterable<DirectedEdge> pathTo(int v) {
if (!hasPathTo(v)) return null;
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
path.push(e);
}
return path;
}
// check optimality conditions:
// (i) for all edges e: distTo[e.to()] <= distTo[e.from()] + e.weight()
// (ii) for all edge e on the SPT: distTo[e.to()] == distTo[e.from()] + e.weight()
private boolean check(EdgeWeightedDigraph G, int s) {
// check that edge weights are nonnegative
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0) {
System.err.println("negative edge weight detected");
return false;
}
}
// check that distTo[v] and edgeTo[v] are consistent
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distTo[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;
}
}
return true;
}
/**
* Unit tests the <tt>DijkstraSP</tt> data type.
*/
public static void main(String[] args) {
In in = new In(args[0]);
EdgeWeightedDigraph G = new EdgeWeightedDigraph(in);
int s = Integer.parseInt(args[1]);
// compute shortest paths
DijkstraSP sp = new DijkstraSP(G, s);
// print shortest path
for (int t = 0; t < G.V(); t++) {
if (sp.hasPathTo(t)) {
StdOut.printf("%d to %d (%.2f) ", s, t, sp.distTo(t));
for (DirectedEdge e : sp.pathTo(t)) {
StdOut.print(e + " ");
}
StdOut.println();
}
else {
StdOut.printf("%d to %d no path\n", s, t);
}
}
}
}
