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);
}
// 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.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 : 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.connected(x, y)) {
if (f.weight() < e.weight()) {
System.err.println("Edge " + f + " violates cut optimality conditions");
return false;
}
}
}
}
return true;
}
/**
* Reads in a an integer {@code n} and a sequence of pairs of integers
* (between {@code 0} and {@code n-1}) from standard input, where each integer
* in the pair represents some site;
* if the sites are in different components, merge the two components
* and print the pair to standard output.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
int n = StdIn.readInt();
UF uf = new UF(n);
while (!StdIn.isEmpty()) {
int p = StdIn.readInt();
int q = StdIn.readInt();
if (uf.connected(p, q)) continue;
uf.union(p, q);
StdOut.println(p + " " + q);
}
StdOut.println(uf.count() + " components");
}
/**
* Computes a minimum cut of the edge-weighted graph. Precisely, it computes
* the lightest of the cuts-of-the-phase which yields the desired minimum
* cut.
*
* @param G the edge-weighted graph
* @param a the starting vertex
*/
private void minCut(EdgeWeightedGraph G, int a) {
UF uf = new UF(G.V());
boolean[] marked = new boolean[G.V()];
cut = new boolean[G.V()];
CutPhase cp = new CutPhase(0.0, a, a);
for (int v = G.V(); v > 1; v--) {
cp = minCutPhase(G, marked, cp);
if (cp.weight < weight) {
weight = cp.weight;
makeCut(cp.t, uf);
}
G = contractEdge(G, cp.s, cp.t);
marked[cp.t] = true;
uf.union(cp.s, cp.t);
}
}
private Queue<Edge> mst = new Queue<Edge>(); // 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) {
// more efficient to build heap by passing array of edges
MinPQ<Edge> pq = new MinPQ<Edge>();
for (Edge e : G.edges()) {
pq.insert(e);
}
// run greedy algorithm
UF uf = new UF(G.V());
while (!pq.isEmpty() && mst.size() < G.V() - 1) {
Edge e = pq.delMin();
int v = e.either();
int w = e.other(v);
if (!uf.connected(v, w)) { // v-w does not create a cycle
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);
}
/**
* Makes a cut for the current edge-weighted graph by partitioning its set
* of vertices into two nonempty subsets. The vertices connected to the
* vertex {@code t} belong to the first subset. Other vertices not connected
* to {@code t} belong to the second subset.
*
* @param t the vertex {@code t}
* @param uf the union-find data type
*/
private void makeCut(int t, UF uf) {
for (int v = 0; v < cut.length; v++) {
cut[v] = uf.connected(v, t);
}
}
