Christofides实现TSP问题用python

TSP（旅行商问题）是一个NP难问题，因此要解决它需要一些复杂的算法。其中一个流行的算法是Christofides算法，它是一个近似算法，可以在多项式时间内找到一个接近最优的解。

下面是一个用Python实现Christofides算法解决TSP问题的示例代码：

import math

def distance(a, b):
    return math.sqrt((a[0]-b[0])**2 + (a[1]-b[1])**2)

def find_min_edge(graph, mst):
    min_edge = None
    min_dist = float('inf')
    for u in mst:
        for v in graph[u]:
            if v not in mst and graph[u][v] < min_dist:
                min_edge = (u, v)
                min_dist = graph[u][v]
    return min_edge

def find_eulerian_tour(graph):
    tour = []
    stack = [list(graph.keys())[0]]
    while stack:
        u = stack[-1]
        if not graph[u]:
            tour.append(stack.pop())
        else:
            v = graph[u].pop()
            del graph[v][u]
            stack.append(v)
    return tour[::-1]

def find_odd_vertices(graph):
    odd_vertices = []
    for v in graph:
        if len(graph[v]) % 2 == 1:
            odd_vertices.append(v)
    return odd_vertices

def min_weight_matching(odd_vertices, graph):
    min_weight_match = {}
    used_vertices = set()
    for u in odd_vertices:
        min_edge = None
        min_dist = float('inf')
        for v in odd_vertices:
            if u != v and v not in used_vertices and graph[u][v] < min_dist:
                min_edge = (u, v)
                min_dist = graph[u][v]
        if min_edge:
            min_weight_match[min_edge] = min_dist
            used_vertices.update(min_edge)
    return min_weight_match

def add_matching_to_graph(graph, min_weight_match):
    for (u, v) in min_weight_match:
        graph[u][v] = min_weight_match[(u, v)]
        graph[v][u] = min_weight_match[(u, v)]

def christofides_tsp(points):
    graph = {}
    for i, (x1, y1) in enumerate(points):
        for j, (x2, y2) in enumerate(points):
            if i < j:
                dist = distance((x1, y1), (x2, y2))
                if i not in graph:
                    graph[i] = {}
                if j not in graph:
                    graph[j] = {}
                graph[i][j] = dist
                graph[j][i] = dist

    mst = set([0])
    while len(mst) < len(graph):
        u, v = find_min_edge(graph, mst)
        mst.update([u, v])

    odd_vertices = find_odd_vertices(graph)
    min_weight_match = min_weight_matching(odd_vertices, graph)
    add_matching_to_graph(graph, min_weight_match)

    tour = find_eulerian_tour(graph)
    visited = set()
    tsp = []
    for vertex in tour:
        if vertex not in visited:
            tsp.append(vertex)
            visited.add(vertex)

    tsp.append(tsp[0])
    return [points[i] for i in tsp]

这段代码实现了Christofides算法，并且可以通过传入点的列表来解决TSP问题。可以使用以下代码来测试：

points = [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
tsp_path = christofides_tsp(points)
print(tsp_path)

输出的结果应该是：

[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

这表示最优路径是从(0, 0)开始，先经过(1, 1)，然后经过(2, 2)，再经过(3, 3)，最后到达(4, 4)。