/******************************************************************************
* 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: 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 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 {@code 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 {@code weight()} method returns the
* weight of a minimum spanning tree and the {@code edges()} method
* returns its edges.
*
* This implementation uses a lazy version of Prim's algorithm
* with a binary heap of edges.
* The constructor takes Θ(E log E) 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(E) extra space in the worst case
* (not including the edge-weighted graph).
*
* For additional documentation,
* see Section 4.3 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 static final double FLOATING_POINT_EPSILON = 1.0E-12;
private double weight; // total weight of MST
private Queue mst; // edges in the MST
private boolean[] marked; // marked[v] = true iff 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();
}
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 : 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 LazyPrimMST} 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);
LazyPrimMST mst = new LazyPrimMST(G);
for (Edge e : mst.edges()) {
StdOut.println(e);
}
StdOut.printf("%.5f\n", mst.weight());
}
}