// 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();
}
double EPSILON = 1E-12;
if (Math.abs(total - weight()) > 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.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;
}
public KruskalMST(EdgeWeightedGraph G) {
mst = new Queue<Edge>();
MaxPQ<Edge> pq = new MaxPQ<Edge>();
for (Edge e : G.edges()) pq.insert(e); // 将所有的边进入优先队列，按照权重升序排列
UF uf = new UF(G.V()); // 构建union-find对象
while (!pq.isEmpty() && mst.size() < G.V() - 1) {
Edge e = pq.delMax(); // 从pq得到权重最小的边和他的顶点
int v = e.either(), w = e.other(v);
if (uf.connected(v, w)) continue; // 忽略失效的边，两个顶点已经在生成树中了
uf.union(v, w); // 否则将两者合并，在一个树中,顶点
mst.enqueue(e); // 将边添加到最小生成树中
}
}
KruskalMST(EdgeWeightedGraph G) {
MinPQ<Edge> pq = new MinPQ<Edge>(); // we dont need to pass a new comparator let it
// follow the natural ordering of edges
double temp_wt = 0;
for (Edge e : G.edges()) pq.insert(e);
UF set = 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 (!set.connected(v, w)) {
set.union(v, w);
mst.add(e);
temp_wt += e.weight();
}
}
this.weight = temp_wt;
}
public KruskalMST(EdgeWeightedGraph g) {
mst = new Queue<Edge>();
MinPQ<Edge> pq = new MinPQ<Edge>();
for (Edge e : g.edges()) {
pq.insert(e);
}
UnionFind uf = new UnionFind(g.vertices());
while (!pq.isEmpty() && mst.size() < g.vertices() - 1) {
Edge e = pq.delMin();
int v = e.either(), w = e.other(v);
if (uf.connected(v, w)) continue; // -- would form a cycle
uf.union(v, w);
mst.enqueue(e);
}
}
public static void main(String[] args) {
EdgeWeightedGraph ewg = new EdgeWeightedGraph(Integer.parseInt(args[0]));
ewg.addEdge(new Edge(0, 1, 10));
ewg.addEdge(new Edge(1, 2, 11));
ewg.addEdge(new Edge(2, 3, 12));
ewg.addEdge(new Edge(3, 4, 13));
ewg.addEdge(new Edge(4, 5, 14));
ewg.addEdge(new Edge(5, 6, 21));
ewg.addEdge(new Edge(6, 7, 22));
ewg.addEdge(new Edge(7, 8, 23));
ewg.addEdge(new Edge(8, 9, 24));
ewg.addEdge(new Edge(9, 0, 25));
int count = 0;
for (Edge e : ewg.edges()) {
StdOut.println(count + " => " + e);
count++;
}
}
// Kruskal's algorithm
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);
}
/**
* Dado un grafo no dirigido y con pesos, ejecuta el algoritmo de Kruskal para la obtención del su
* árbol de cobertura mínima.
*
* @param G Grafo no dirigido con pesos al cual se le deesea obtener su arbol de cobertura mínima.
*/
public void minimumSpanningTree(EdgeWeightedGraph G) // By Kruskal's algorithm
{
LinkedList<Edge> mst = new LinkedList<>();
MinPQ<Edge> pq = new MinPQ<>(G.E());
for (Edge edge : G.edges()) // Bag<Edge> != Comparator<Edge> :c
pq.insert(edge);
UF uf = new UF(G.V());
while (!pq.isEmpty() && mst.size() < G.V() - 1) {
Edge edge = pq.delMin();
int v = edge.either(), w = edge.other(v);
if (!uf.connected(v, w)) {
uf.union(v, w);
mst.add(edge);
}
}
System.out.println("");
for (Edge edge : mst) System.out.println(edge);
}