import edu.princeton.cs.introcs.In; import edu.princeton.cs.introcs.StdOut; /************************************************************************* * Compilation: javac LazyPrimMST.java * Execution: java LazyPrimMST filename.txt * Dependencies: EdgeWeightedGraph.java Edge.java Queue.java * MinPQ.java UF.java In.java StdOut.java * Data files: http://algs4.cs.princeton.edu/43mst/tinyEWG.txt * http://algs4.cs.princeton.edu/43mst/mediumEWG.txt * http://algs4.cs.princeton.edu/43mst/largeEWG.txt * * Compute a minimum spanning forest using a lazy version of Prim's * algorithm. * * % java LazyPrimMST tinyEWG.txt * 0-7 0.16000 * 1-7 0.19000 * 0-2 0.26000 * 2-3 0.17000 * 5-7 0.28000 * 4-5 0.35000 * 6-2 0.40000 * 1.81000 * * % java LazyPrimMST mediumEWG.txt * 0-225 0.02383 * 49-225 0.03314 * 44-49 0.02107 * 44-204 0.01774 * 49-97 0.03121 * 202-204 0.04207 * 176-202 0.04299 * 176-191 0.02089 * 68-176 0.04396 * 58-68 0.04795 * 10.46351 * * % java LazyPrimMST largeEWG.txt * ... * 647.66307 * *************************************************************************/ /** * The LazyPrimMST 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 weight() method returns the * weight of a minimum spanning tree and the edges() method * returns its edges. * * This implementation uses a lazy version of Prim's algorithm * with a binary heap of edges. * The constructor takes time proportional to E log E * and extra space (not including the graph) proportional to E , * where V is the number of vertices and E is the number of edges. * Afterwards, the weight() method takes constant time * and the edges() method takes time proportional to V . * * For additional documentation, see Section 4.4 of * Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne. * For alternate implementations, see {@link PrimMST}, {@link KruskalMST}, * and {@link BoruvkaMST}. * * @author Robert Sedgewick * @author Kevin Wayne */ public class LazyPrimMST { private double weight; // total weight of MST private Queue mst; // edges in the MST private boolean[] marked; // marked[v] = true if v on tree private MinPQ pq; // edges with one endpoint in tree /** * Compute a minimum spanning tree (or forest) of an edge-weighted graph. * @param G the edge-weighted graph */ public LazyPrimMST(EdgeWeightedGraph G) { mst = new Queue(); pq = new MinPQ(); marked = new boolean[G.V()]; for (int v = 0; v < G.V(); v++) // run Prim from all vertices to if (!marked[v]) prim(G, v); // get a minimum spanning forest // check optimality conditions assert check(G); } // run Prim's algorithm private void prim(EdgeWeightedGraph G, int s) { scan(G, s); while (!pq.isEmpty()) { // better to stop when mst has V-1 edges Edge e = pq.delMin(); // smallest edge on pq int v = e.either(), w = e.other(v); // two endpoints assert marked[v] || marked[w]; if (marked[v] && marked[w]) continue; // lazy, both v and w already scanned mst.enqueue(e); // add e to MST weight += e.weight(); if (!marked[v]) scan(G, v); // v becomes part of tree if (!marked[w]) scan(G, w); // w becomes part of tree } } // add all edges e incident to v onto pq if the other endpoint has not yet been scanned private void scan(EdgeWeightedGraph G, int v) { assert !marked[v]; marked[v] = true; for (Edge e : G.adj(v)) if (!marked[e.other(v)]) pq.insert(e); } /** * 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() { 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() { 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(); } double EPSILON = 1E-12; if (Math.abs(totalWeight - weight()) > 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.connected(v, 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.connected(v, 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 : mst) { 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.connected(x, y)) { if (f.weight() < e.weight()) { System.err.println("Edge " + f + " violates cut optimality conditions"); return false; } } } } return true; } /** * Unit tests the LazyPrimMST data type. */ public static void main(String[] args) { In in = new In(args[0]); EdgeWeightedGraph G = new EdgeWeightedGraph(in); LazyPrimMST mst = new LazyPrimMST(G); for (Edge e : mst.edges()) { StdOut.println(e); } StdOut.printf("%.5f\n", mst.weight()); } }