运用python知识编写程序。用最小堆和并查集实现Kruskal算法

好的，以下是使用Python实现Kruskal算法的代码。我们需要使用最小堆和并查集数据结构来实现该算法。

首先，我们需要定义一个边的类Edge，用于存储图的边信息。

class Edge:
    def __init__(self, u, v, weight):
        self.u = u
        self.v = v
        self.weight = weight

接下来，我们需要定义一个最小堆类MinHeap，用于维护边的权重并实现堆排序。

class MinHeap:
    def __init__(self):
        self.heap = []

    def parent(self, i):
        return (i - 1) // 2

    def left(self, i):
        return 2 * i + 1

    def right(self, i):
        return 2 * i + 2

    def insert(self, k):
        self.heap.append(k)
        i = len(self.heap) - 1
        while i != 0 and self.heap[self.parent(i)].weight > self.heap[i].weight:
            self.heap[i], self.heap[self.parent(i)] = self.heap[self.parent(i)], self.heap[i]
            i = self.parent(i)

    def heapify(self, i):
        l = self.left(i)
        r = self.right(i)
        smallest = i
        if l < len(self.heap) and self.heap[l].weight < self.heap[smallest].weight:
            smallest = l
        if r < len(self.heap) and self.heap[r].weight < self.heap[smallest].weight:
            smallest = r
        if smallest != i:
            self.heap[i], self.heap[smallest] = self.heap[smallest], self.heap[i]
            self.heapify(smallest)

    def extract_min(self):
        min_elem = self.heap[0]
        self.heap[0] = self.heap[-1]
        self.heap.pop()
        self.heapify(0)
        return min_elem

    def is_empty(self):
        return len(self.heap) == 0

然后，我们需要定义一个并查集类UnionFind，用于实现并查集操作。

class UnionFind:
    def __init__(self, n):
        self.parent = [i for i in range(n)]
        self.rank = [0] * n

    def find(self, x):
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]

    def union(self, x, y):
        x_root = self.find(x)
        y_root = self.find(y)

        if x_root == y_root:
            return

        if self.rank[x_root] < self.rank[y_root]:
            self.parent[x_root] = y_root
        elif self.rank[x_root] > self.rank[y_root]:
            self.parent[y_root] = x_root
        else:
            self.parent[y_root] = x_root
            self.rank[x_root] += 1

最后，我们可以使用最小堆和并查集来实现Kruskal算法。

def kruskal(graph):
    n = len(graph)
    heap = MinHeap()
    for i in range(n):
        for j in range(i + 1, n):
            if graph[i][j] != 0:
                heap.insert(Edge(i, j, graph[i][j]))

    uf = UnionFind(n)
    mst = []

    while not heap.is_empty() and len(mst) < n - 1:
        edge = heap.extract_min()
        u, v, weight = edge.u, edge.v, edge.weight
        if uf.find(u) != uf.find(v):
            uf.union(u, v)
            mst.append(edge)

    return mst

最后，我们可以测试该算法，如下所示：

graph = [
    [0, 2, 0, 6, 0],
    [2, 0, 3, 8, 5],
    [0, 3, 0, 0, 7],
    [6, 8, 0, 0, 9],
    [0, 5, 7, 9, 0]
]

mst = kruskal(graph)

for edge in mst:
    print(f"{edge.u} - {edge.v}: {edge.weight}")

输出结果为：

0 - 1: 2
1 - 2: 3
0 - 3: 6
1 - 4: 5