/******************************************************************************
* Compilation: javac KruskalMST.java
* Execution: java KruskalMST filename.txt
* Dependencies: EdgeWeightedGraph.java Edge.java Queue.java MinPQ.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 Kruskal's algorithm.
*
* % java KruskalMST tinyEWG.txt
* 0-7 0.16000
* 2-3 0.17000
* 1-7 0.19000
* 0-2 0.26000
* 5-7 0.28000
* 4-5 0.35000
* 6-2 0.40000
* 1.81000
*
* % java KruskalMST mediumEWG.txt
* 168-231 0.00268
* 151-208 0.00391
* 7-157 0.00516
* 122-205 0.00647
* 8-152 0.00702
* 156-219 0.00745
* 28-198 0.00775
* 38-126 0.00845
* 10-123 0.00886
* ...
* 10.46351
*
******************************************************************************/
import java.util.Arrays;
/**
* The {@code KruskalMST} 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 Kruskal's algorithm and the
* union-find data type.
* The constructor takes Θ(E log E) time in
* the worst case.
* Each instance method takes Θ(1) time.
* It uses Θ(E) extra space (not including the 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 PrimMST},
* and {@link BoruvkaMST}.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class KruskalMST {
private static final double FLOATING_POINT_EPSILON = 1.0E-12;
private double weight; // weight of MST
private Queue mst = new Queue(); // edges in MST
/**
* Compute a minimum spanning tree (or forest) of an edge-weighted graph.
* @param G the edge-weighted graph
*/
public KruskalMST(EdgeWeightedGraph G) {
// create array of edges, sorted by weight
Edge[] edges = new Edge[G.E()];
int t = 0;
for (Edge e: G.edges()) {
edges[t++] = e;
}
Arrays.sort(edges);
// run greedy algorithm
UF uf = new UF(G.V());
for (int i = 0; i < G.E() && mst.size() < G.V() - 1; i++) {
Edge e = edges[i];
int v = e.either();
int w = e.other(v);
// v-w does not create a cycle
if (uf.find(v) != uf.find(w)) {
uf.union(v, w); // merge v and w components
mst.enqueue(e); // add edge e to mst
weight += e.weight();
}
}
// 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 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 total weight
double total = 0.0;
for (Edge e : edges()) {
total += e.weight();
}
if (Math.abs(total - weight()) > FLOATING_POINT_EPSILON) {
System.err.printf("Weight of edges does not equal weight(): %f vs. %f\n", total, 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 : 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.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 KruskalMST} 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);
KruskalMST mst = new KruskalMST(G);
for (Edge e : mst.edges()) {
StdOut.println(e);
}
StdOut.printf("%.5f\n", mst.weight());
}
}