必经点最短路径问题 matlab代码

以下是基于Dijkstra算法的必经点最短路径问题的Matlab代码实现：

function [path, dist] = shortest_path_with_waypoints(graph, start, finish, waypoints)
% 输入：
% graph：有向图的邻接矩阵，graph(i,j)表示i到j的边权重，若不存在i到j的边，则graph(i,j)=Inf
% start：起点
% finish：终点
% waypoints：必经点数组，例如waypoints=[2,4,5]表示必经点为2、4、5
% 输出：
% path：最短路径
% dist：最短路径的长度

n = size(graph, 1);  % 图的大小
m = length(waypoints);  % 必经点数量

% 将起点、终点和必经点映射为图中的编号
waypoints = [start, waypoints, finish];
start = 1;
finish = n - 1;
for i = 2:m+1
    waypoints(i) = waypoints(i) - 1;
end

% 计算所有点之间的最短路径
all_paths = zeros(n,n);
for i = 1:n
    [all_paths(i,:), ~] = dijkstra(graph, i);
end

% 生成新的有向图，其中每个必经点都被分成两个节点
new_n = n + m;
new_graph = Inf(new_n);
for i = 1:n
    for j = 1:n
        if graph(i,j) ~= Inf
            for k = 1:m+1
                if any(waypoints == i) && any(waypoints == j)
                    new_i = find(waypoints == i);
                    new_j = find(waypoints == j);
                    new_graph(new_i, new_j) = all_paths(i,j);
                elseif any(waypoints == i)
                    new_i = find(waypoints == i);
                    new_j = j + m;
                    new_graph(new_i, new_j) = all_paths(i,j);
                elseif any(waypoints == j)
                    new_i = i;
                    new_j = find(waypoints == j);
                    new_graph(new_i, new_j) = all_paths(i,j);
                else
                    new_graph(i,j) = graph(i,j);
                end
            end
        end
    end
end

% 从起点到终点计算最短路径
[new_dist, path] = dijkstra(new_graph, start);
path = path - 1;
dist = new_dist(finish);

% 修正路径，恢复必经点
for i = length(path):-1:2
    if path(i) > m
        path(i) = path(i) - m;
    end
end
path = path(2:end-1);

其中，dijkstra函数实现了Dijkstra算法，可以使用以下代码添加到Matlab文件中：

function [distances, path] = dijkstra(graph, start)
% 输入：
% graph：有向图的邻接矩阵，graph(i,j)表示i到j的边权重，若不存在i到j的边，则graph(i,j)=Inf
% start：起点
% 输出：
% distances：起点到其他点的最短距离
% path：起点到其他点的最短路径

n = size(graph, 1);  % 图的大小

% 初始化distances和path数组
distances = Inf(1, n);
distances(start) = 0;
path = repmat({[]}, 1, n);

% 初始化visited数组，表示已访问的节点
visited = false(1, n);

while any(~visited)
    % 找到未访问节点中距离start最近的节点
    [~, current] = min(distances(~visited));
    current = current(1);
    visited(current) = true;

    % 更新与当前节点相邻的节点的距离
    neighbors = find(graph(current, :));
    for neighbor = neighbors
        if ~visited(neighbor)
            new_distance = distances(current) + graph(current, neighbor);
            if new_distance < distances(neighbor)
                distances(neighbor) = new_distance;
                path{neighbor} = [path{current}, current];
            end
        end
    end
end

% 生成起点到其他节点的路径
for i = 1:n
    path{i} = [path{i}, i];
end

使用示例：

% 创建有向图的邻接矩阵，例如：
% graph = [Inf, 2, Inf, 1, Inf;
%          Inf, Inf, 3, Inf, 5;
%          Inf, Inf, Inf, Inf, 6;
%          Inf, Inf, 1, Inf, Inf;
%          Inf, Inf, Inf, 4, Inf];
% 表示1到2的边权重为2，1到4的边权重为1，2到3的边权重为3，等等。
graph = ...

% 计算起点为1，终点为5，必经点为2、4的最短路径
[start, finish, waypoints] = [1, 5, [2, 4]];
[path, dist] = shortest_path_with_waypoints(graph, start, finish, waypoints);

注意：以上代码中的有向图邻接矩阵中，若不存在i到j的边，则必须将graph(i,j)赋值为Inf，否则会引发错误。在实际应用中，可以根据具体情况进行修改。