package edu.princeton.cs.algs4.ch55;
import edu.princeton.cs.algs4.ch13.Bag;
import edu.princeton.cs.algs4.ch15.UF;
import edu.princeton.cs.algs4.ch43.Edge;
import edu.princeton.cs.algs4.ch43.EdgeWeightedGraph;
import edu.princeton.cs.introcs.*;
/*************************************************************************
* Compilation: javac BoruvkaMST.java
* Execution: java BoruvkaMST filename.txt
* Dependencies: EdgeWeightedGraph.java Edge.java Bag.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 Boruvka's algorithm.
*
* % java BoruvkaMST tinyEWG.txt
* 0-2 0.26000
* 6-2 0.40000
* 5-7 0.28000
* 4-5 0.35000
* 2-3 0.17000
* 1-7 0.19000
* 0-7 0.16000
* 1.81000
*
*************************************************************************/
/**
* The <tt>BoruvkaMST</tt> class represents a data type for computing a
* <em>minimum spanning tree</em> 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 <em>minimum
* spanning forest</em>, which is the union of minimum spanning trees
* in each connected component. The <tt>weight()</tt> method returns the
* weight of a minimum spanning tree and the <tt>edges()</tt> method
* returns its edges.
* <p>
* This implementation uses <em>Boruvka's algorithm</em> and the union-find
* data type.
* The constructor takes time proportional to <em>E</em> log <em>V</em>
* and extra space (not including the graph) proportional to <em>V</em>,
* where <em>V</em> is the number of vertices and <em>E</em> is the number of edges.
* Afterwards, the <tt>weight()</tt> method takes constant time
* and the <tt>edges()</tt> method takes time proportional to <em>V</em>.
* <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.
* For alternate implementations, see {@link edu.princeton.cs.algs4.ch43.LazyPrimMST}, {@link edu.princeton.cs.algs4.ch43.PrimMST},
* and {@link edu.princeton.cs.algs4.ch43.KruskalMST}.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class BoruvkaMST {
private Bag<Edge> mst = new Bag<Edge>(); // edges in MST
private double weight; // weight of MST
/**
* Compute a minimum spanning tree (or forest) of an edge-weighted graph.
* @param G the edge-weighted graph
*/
public BoruvkaMST(EdgeWeightedGraph G) {
UF uf = new UF(G.V());
// repeat at most log V times or until we have V-1 edges
for (int t = 1; t < G.V() && mst.size() < G.V() - 1; t = t + t) {
// foreach tree in forest, find closest edge
// if edge weights are equal, ties are broken in favor of first edge in G.edges()
Edge[] closest = new Edge[G.V()];
for (Edge e : G.edges()) {
int v = e.either(), w = e.other(v);
int i = uf.find(v), j = uf.find(w);
if (i == j) continue; // same tree
if (closest[i] == null || less(e, closest[i])) closest[i] = e;
if (closest[j] == null || less(e, closest[j])) closest[j] = e;
}
// add newly discovered edges to MST
for (int i = 0; i < G.V(); i++) {
Edge e = closest[i];
if (e != null) {
int v = e.either(), w = e.other(v);
// don't add the same edge twice
if (!uf.connected(v, w)) {
mst.add(e);
weight += e.weight();
uf.union(v, w);
}
}
}
}
// check optimality conditions
assert check(G);
}
/**
* 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<Edge> 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;
}
// is the weight of edge e strictly less than that of edge f?
private static boolean less(Edge e, Edge f) {
return e.weight() < f.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 <tt>BoruvkaMST</tt> data type.
*/
public static void main(String[] args) {
In in = new In(args[0]);
EdgeWeightedGraph G = new EdgeWeightedGraph(in);
BoruvkaMST mst = new BoruvkaMST(G);
for (Edge e : mst.edges()) {
StdOut.println(e);
}
StdOut.printf("%.5f\n", mst.weight());
}
}
/*************************************************************************
* 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.
*************************************************************************/