/******************************************************************************
* Compilation: javac PrimMST.java
* Execution: java PrimMST filename.txt
* Dependencies: EdgeWeightedGraph.java Edge.java Queue.java
* IndexMinPQ.java UF.java In.java StdOut.java
* Data files: https://algs4.cs.princeton.edu/43mst/tinyEWG.txt
* https://algs4.cs.princeton.edu/43mst/mediumEWG.txt
* https://algs4.cs.princeton.edu/43mst/largeEWG.txt
*
* Compute a minimum spanning forest using Prim's algorithm.
*
* % java PrimMST tinyEWG.txt
* 1-7 0.19000
* 0-2 0.26000
* 2-3 0.17000
* 4-5 0.35000
* 5-7 0.28000
* 6-2 0.40000
* 0-7 0.16000
* 1.81000
*
* % java PrimMST mediumEWG.txt
* 1-72 0.06506
* 2-86 0.05980
* 3-67 0.09725
* 4-55 0.06425
* 5-102 0.03834
* 6-129 0.05363
* 7-157 0.00516
* ...
* 10.46351
*
* % java PrimMST largeEWG.txt
* ...
* 647.66307
*
******************************************************************************/
/**
* The {@code PrimMST} class represents a data type for computing a
* minimum spanning tree in an edge-weighted graph.
* The edge weights can be positive, zero, or negative and need not
* be distinct. If the graph is not connected, it computes a minimum
* spanning forest, which is the union of minimum spanning trees
* in each connected component. The {@code weight()} method returns the
* weight of a minimum spanning tree and the {@code edges()} method
* returns its edges.
*
* This implementation uses Prim's algorithm with an indexed
* binary heap.
* The constructor takes Θ(E log V) time in
* the worst case, where V is the number of
* vertices and E is the number of edges.
* Each instance method takes Θ(1) time.
* It uses Θ(V) extra space (not including the
* edge-weighted graph).
*
* This {@code weight()} method correctly computes the weight of the MST
* if all arithmetic performed is without floating-point rounding error
* or arithmetic overflow.
* This is the case if all edge weights are non-negative integers
* and the weight of the MST does not exceed 252.
*
* For additional documentation,
* see Section 4.3 of
* Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
* For alternate implementations, see {@link LazyPrimMST}, {@link KruskalMST},
* and {@link BoruvkaMST}.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class PrimMST {
private static final double FLOATING_POINT_EPSILON = 1.0E-12;
private Edge[] edgeTo; // edgeTo[v] = shortest edge from tree vertex to non-tree vertex
private double[] distTo; // distTo[v] = weight of shortest such edge
private boolean[] marked; // marked[v] = true if v on tree, false otherwise
private IndexMinPQ pq;
/**
* Compute a minimum spanning tree (or forest) of an edge-weighted graph.
* @param G the edge-weighted graph
*/
public PrimMST(EdgeWeightedGraph G) {
edgeTo = new Edge[G.V()];
distTo = new double[G.V()];
marked = new boolean[G.V()];
pq = new IndexMinPQ(G.V());
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
for (int v = 0; v < G.V(); v++) // run from each vertex to find
if (!marked[v]) prim(G, v); // minimum spanning forest
// check optimality conditions
assert check(G);
}
// run Prim's algorithm in graph G, starting from vertex s
private void prim(EdgeWeightedGraph G, int s) {
distTo[s] = 0.0;
pq.insert(s, distTo[s]);
while (!pq.isEmpty()) {
int v = pq.delMin();
scan(G, v);
}
}
// scan vertex v
private void scan(EdgeWeightedGraph G, int v) {
marked[v] = true;
for (Edge e : G.adj(v)) {
int w = e.other(v);
if (marked[w]) continue; // v-w is obsolete edge
if (e.weight() < distTo[w]) {
distTo[w] = e.weight();
edgeTo[w] = e;
if (pq.contains(w)) pq.decreaseKey(w, distTo[w]);
else pq.insert(w, distTo[w]);
}
}
}
/**
* Returns the edges in a minimum spanning tree (or forest).
* @return the edges in a minimum spanning tree (or forest) as
* an iterable of edges
*/
public Iterable edges() {
Queue mst = new Queue();
for (int v = 0; v < edgeTo.length; v++) {
Edge e = edgeTo[v];
if (e != null) {
mst.enqueue(e);
}
}
return mst;
}
/**
* Returns the sum of the edge weights in a minimum spanning tree (or forest).
* @return the sum of the edge weights in a minimum spanning tree (or forest)
*/
public double weight() {
double weight = 0.0;
for (Edge e : edges())
weight += e.weight();
return weight;
}
// check optimality conditions (takes time proportional to E V lg* V)
private boolean check(EdgeWeightedGraph G) {
// check weight
double totalWeight = 0.0;
for (Edge e : edges()) {
totalWeight += e.weight();
}
if (Math.abs(totalWeight - weight()) > FLOATING_POINT_EPSILON) {
System.err.printf("Weight of edges does not equal weight(): %f vs. %f\n", totalWeight, weight());
return false;
}
// check that it is acyclic
UF uf = new UF(G.V());
for (Edge e : edges()) {
int v = e.either(), w = e.other(v);
if (uf.find(v) == uf.find(w)) {
System.err.println("Not a forest");
return false;
}
uf.union(v, w);
}
// check that it is a spanning forest
for (Edge e : G.edges()) {
int v = e.either(), w = e.other(v);
if (uf.find(v) != uf.find(w)) {
System.err.println("Not a spanning forest");
return false;
}
}
// check that it is a minimal spanning forest (cut optimality conditions)
for (Edge e : edges()) {
// all edges in MST except e
uf = new UF(G.V());
for (Edge f : edges()) {
int x = f.either(), y = f.other(x);
if (f != e) uf.union(x, y);
}
// check that e is min weight edge in crossing cut
for (Edge f : G.edges()) {
int x = f.either(), y = f.other(x);
if (uf.find(x) != uf.find(y)) {
if (f.weight() < e.weight()) {
System.err.println("Edge " + f + " violates cut optimality conditions");
return false;
}
}
}
}
return true;
}
/**
* Unit tests the {@code PrimMST} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
EdgeWeightedGraph G = new EdgeWeightedGraph(in);
PrimMST mst = new PrimMST(G);
for (Edge e : mst.edges()) {
StdOut.println(e);
}
StdOut.printf("%.5f\n", mst.weight());
}
}