数模 04图论模型 最短路径 Dijkstra算法

Dijkstra算法

迪杰斯特拉算法是由荷兰计算机科学家狄克斯特拉于1959 年提出的，因此又叫狄克斯特拉算法。是从一个顶点到其余各顶点的最短路径算法，解决的是有向图中最短路径问题。迪杰斯特拉算法主要特点是以起始点为中心向外层层扩展，直到扩展到终点为止。

此算法求的是两点间的最短路径，注意，得先画出带权邻接矩阵：

    [0     2     8     1   Inf   Inf   Inf   Inf   Inf   Inf   Inf;
     2     0     6   Inf     1   Inf   Inf   Inf   Inf   Inf   Inf;
     8     6     0     7     5     1     2   Inf   Inf   Inf   Inf;
     1   Inf     7     0   Inf   Inf     9   Inf   Inf   Inf   Inf;
   Inf     1     5   Inf     0     3   Inf     2     9   Inf   Inf;
   Inf   Inf     1   Inf     3     0     4   Inf     6   Inf   Inf;
   Inf   Inf     2     9   Inf     4     0   Inf     3     1   Inf;
   Inf   Inf   Inf   Inf     2   Inf   Inf     0     7   Inf     9;
   Inf   Inf   Inf   Inf     9     6     3     7     0     1     2;
   Inf   Inf   Inf   Inf   Inf   Inf     1   Inf     1     0     4;
   Inf   Inf   Inf   Inf   Inf   Inf   Inf     9     2     4     0;]


    [0     8   Inf   Inf   Inf   Inf     7     8   Inf   Inf   Inf;
   Inf     0     3   Inf   Inf   Inf   Inf   Inf   Inf   Inf   Inf;
   Inf   Inf     0     5     6   Inf     5   Inf   Inf   Inf   Inf;
   Inf   Inf   Inf     0     1   Inf   Inf   Inf   Inf   Inf    12;
   Inf   Inf     6   Inf     0     2   Inf   Inf   Inf   Inf    10;
   Inf   Inf   Inf   Inf     2     0     9   Inf     3   Inf   Inf;
   Inf   Inf   Inf   Inf   Inf     9     0   Inf   Inf   Inf   Inf;
     8   Inf   Inf   Inf   Inf   Inf   Inf     0     9   Inf   Inf;
   Inf   Inf   Inf   Inf     7   Inf   Inf     9     0     2   Inf;
   Inf   Inf   Inf   Inf   Inf   Inf   Inf   Inf     2     0     2;
   Inf   Inf   Inf   Inf    10   Inf   Inf   Inf   Inf   Inf     0;];

 

至于具体算法在数据结构课程中已经了解到了，下面直接给出matlab实操过程。

weight=    [0     2     8     1   Inf   Inf   Inf   Inf   Inf   Inf   Inf;
            2     0     6   Inf     1   Inf   Inf   Inf   Inf   Inf   Inf;
            8     6     0     7     5     1     2   Inf   Inf   Inf   Inf;
            1   Inf     7     0   Inf   Inf     9   Inf   Inf   Inf   Inf;
          Inf     1     5   Inf     0     3   Inf     2     9   Inf   Inf;
          Inf   Inf     1   Inf     3     0     4   Inf     6   Inf   Inf;
          Inf   Inf     2     9   Inf     4     0   Inf     3     1   Inf;
          Inf   Inf   Inf   Inf     2   Inf   Inf     0     7   Inf     9;
          Inf   Inf   Inf   Inf     9     6     3     7     0     1     2;
          Inf   Inf   Inf   Inf   Inf   Inf     1   Inf     1     0     4;
          Inf   Inf   Inf   Inf   Inf   Inf   Inf     9     2     4     0;];
[dis, path]=dijkstra(weight,1, 11)

在主程序中的weight即为需要自行填写的带权邻接矩阵，dijkstra(weight,1, 11)表示求的是1到11的最短路径

function [min,path]=dijkstra(w,start,terminal)
n=size(w,1); label(start)=0; f(start)=start;
for i=1:n
   if i~=start
       label(i)=inf;
end, end
s(1)=start; u=start;
while length(s)<n
   for i=1:n
      ins=0;
      for j=1:length(s)
         if i==s(j)
            ins=1;
         end,  
      end
      if ins==0
         v=i;
         if label(v)>(label(u)+w(u,v))
            label(v)=(label(u)+w(u,v)); 
         f(v)=u;
         end, 
      end, 
   end   
v1=0;
   k=inf;
   for i=1:n
         ins=0;
         for j=1:length(s)
            if i==s(j)
               ins=1;
            end, 
         end
         if ins==0
            v=i;
            if k>label(v)
               k=label(v);  v1=v;
            end,  
         end,  
   end
   s(length(s)+1)=v1;  
   u=v1;
end
min=label(terminal); path(1)=terminal;
i=1; 
while path(i)~=start
      path(i+1)=f(path(i));
      i=i+1 ;
end
path(i)=start;
L=length(path);
path=path(L:-1:1);

然后在命令行输入主程序保存的文件名，这里我保存的是tulun1，输出：

 

 dis为最短路径长度，path表示最短路径经过的点。