添加边以满足给定条件的最小数量

问题描述

给定一个无向图 $G = (V, E)$，以及一些额外的约束条件，如：图 $G$ 必须是连通的， 结点 $a$ 和结点 $b$ 必须连通，等等。现在，我们需要根据这些约束条件添加最少的边，使得原图变成满足条件的图。

解决方案

这个问题可以被看作是一种最小生成树问题。我们可以先将图 $G$ 变成无向带权图，然后用 Kruskal 算法求出 $G$ 的最小生成树 $T$，随后根据给定的约束条件，一次添加边，构成满足条件的图。

如果给出的约束条件可以被拆分为多个独立的子问题，我们可以分别解决每个子问题，然后将它们的解合并起来得到最终的解。

实现细节

把无向图变成无向带权图

为了将无向图变成无向带权图，我们可以将所有没有权值的边（也就是原图中没有被标记的边），加上一个默认权值。这个默认权值可以是一个以图中边权为基础的常数，也可以是一个和边出现频率紧密相关的值。

使用 Kruskal 算法获取最小生成树

Kruskal 算法是一种用于计算最小生成树的贪心算法。它按照边的权值递增的顺序来选取边，同时保证选中的边不会形成环。

在实现 Kruskal 算法时，我们可以使用归并集合（disjoint-set）数据结构，将图中的结点分组到不同的集合，并在遍历边时判断当前边的两个结点是否属于同一集合，如果不属于同一集合，就可以将创造一条新的连通路径。

处理给定的条件约束

对于一些满足特定条件的约束，比如连通性，我们可以使用深度优先搜索（DFS）来判断它们是否满足。对于一些其它的约束，比如结点之间必须连通，我们也可以使用 DFS 来判断两个结点之间是否连通。

合并多个子问题

如果给出的约束条件可以被拆分为多个独立的子问题，我们需要将每个子问题分别解决，然后将它们的解合并起来才能得到最终的解。比如，我们需要同时考虑图的连通性和结点 $a$ 和结点 $b$ 之间的连通性，我们可以先将图变成连通图，然后再加上结点 $a$ 和结点 $b$ 之间的边。如果图已经连通，我们就可以直接加上结点 $a$ 和结点 $b$ 之间的边。

示例代码

以下是 Python 实现的示例代码：

from collections import defaultdict
from typing import List

class Graph:
    def __init__(self):
        self.edges = defaultdict(list)

    def add_edge(self, u, v, weight=1):
        self.edges[u].append((v, weight))
        self.edges[v].append((u, weight))

    def add_edges(self, edges):
        for edge in edges:
            self.add_edge(*edge)

    def find_parent(self, node, parent):
        if parent[node] == node:
            return node
        return self.find_parent(parent[node], parent)

    def union(self, node1, node2, parent, rank):
        parent1 = self.find_parent(node1, parent)
        parent2 = self.find_parent(node2, parent)

        if rank[parent1] > rank[parent2]:
            parent[parent2] = parent1
        elif rank[parent1] < rank[parent2]:
            parent[parent1] = parent2
        else:
            parent[parent1] = parent2
            rank[parent2] += 1

    def get_mst(self):
        parent = {}
        rank = {}

        # Initialize disjoint set
        for node in self.edges:
            parent[node] = node
            rank[node] = 0

        # Sort edges by increasing weight
        edges = []
        for node in self.edges:
            for neighbor, weight in self.edges[node]:
                edges.append((node, neighbor, weight))
        edges.sort(key=lambda x: x[2])

        # Kruskal's algorithm
        mst_edges = []
        for u, v, weight in edges:
            if self.find_parent(u, parent) != self.find_parent(v, parent):
                mst_edges.append((u, v))
                self.union(u, v, parent, rank)

        return Graph().add_edges(mst_edges)

    def is_connected(self):
        visited = {node: False for node in self.edges}
        stack = []
        stack.append(list(self.edges.keys())[0])
        visited[stack[-1]] = True

        # DFS traversal
        while len(stack) > 0:
            node = stack.pop()
            for neighbor, weight in self.edges[node]:
                if not visited[neighbor]:
                    visited[neighbor] = True
                    stack.append(neighbor)

        return all(visited.values())

    def add_edges_to_connect_all_nodes(self):
        if self.is_connected():
            return Graph()

        mst = self.get_mst()
        components = []
        components_nodes = {}
        for node in mst.edges:
            component_found = False
            for i in range(len(components)):
                if node in components[i]:
                    components[i].extend(mst.edges[node])
                    components_nodes[i].append(node)
                    component_found = True
                    break
            if not component_found:
                components.append(mst.edges[node])
                components_nodes[len(components)-1] = [node]

        for i in range(len(components)):
            edges = components[i]
            nodes = components_nodes[i]
            visited = {node: False for node in nodes}
            stack = []
            stack.append(nodes[0])
            visited[stack[-1]] = True

            # DFS traversal
            while len(stack) > 0:
                node = stack.pop()
                for neighbor, weight in edges:
                    if node == neighbor and not visited[neighbor]:
                        visited[neighbor] = True
                        stack.append(neighbor)

            for node in nodes:
                if not visited[node]:
                    min_weight = float("inf")
                    min_edge = None
                    for neighbor, weight in self.edges[node]:
                        if neighbor in nodes and weight < min_weight:
                            min_weight = weight
                            min_edge = (node, neighbor)
                    if min_edge:
                        edges.append(min_edge)

        new_graph = Graph()
        for edges in components:
            new_graph.add_edges(edges)
        return new_graph

    def add_edge_to_connect_nodes(self, a, b):
        if a == b:
            return Graph()

        mst = self.get_mst()
        parent = {}
        rank = {}

        # Initialize disjoint set
        for node in mst.edges:
            parent[node] = node
            rank[node] = 0

        # Find paths from a and b to the roots of their components
        a_path = [(a, 0)]
        while a_path[-1][0] != parent[a_path[-1][0]]:
            a_path.append((parent[a_path[-1][0]], a_path[-1][1]+1))

        b_path = [(b, 0)]
        while b_path[-1][0] != parent[b_path[-1][0]]:
            b_path.append((parent[b_path[-1][0]], b_path[-1][1]+1))

        # Find the lca of a and b
        lca = None
        if len(a_path) <= len(b_path):
            for i in range(len(a_path)):
                if a_path[i][0] == b_path[i][0]:
                    lca = a_path[i][0]
                    break
        else:
            for i in range(len(b_path)):
                if b_path[i][0] == a_path[i][0]:
                    lca = b_path[i][0]
                    break

        # Find the minimum weight edge that connects a and b
        min_weight = float("inf")
        min_edge = None
        for a_neighbor, a_weight in self.edges[a]:
            if self.find_parent(a_neighbor, parent) != self.find_parent(a, parent):
                for b_neighbor, b_weight in self.edges[b]:
                    if self.find_parent(b_neighbor, parent) != self.find_parent(b, parent) and \
                        (a_neighbor, b_neighbor) not in mst.edges and (b_neighbor, a_neighbor) not in mst.edges:
                        weight = a_weight + b_weight
                        if weight < min_weight:
                            min_weight = weight
                            min_edge = (a_neighbor, b_neighbor)

        # Check if adding the minimum edge will create cycle
        cycle_found = False
        if min_edge:
            mst_edges = [(u, v) for u, v in mst.edges]
            mst_edges.append(min_edge)
            for u, v in mst_edges:
                if self.find_parent(u, parent) == self.find_parent(v, parent):
                    cycle_found = True
                    break

        # Add the new edge if it doesn't create cycle
        if min_edge and not cycle_found:
            mst.add_edge(*min_edge, weight=min_weight)

        return mst

# Example usage
g = Graph()
g.add_edges([(1, 2), (2, 3), (4, 5)])
g = g.add_edges_to_connect_all_nodes()
g = g.add_edge_to_connect_nodes(2, 5)
print(g.edges)

这个示例代码演示了如何使用 Kruskal 算法计算最小生成树，以及如何根据给定的约束条件添加边使得图满足条件。