最小生成树-Kruskal算法

比较成熟的算法，百度百科有，不赘述

Kruskal算法适用于边稀疏的情形，而Prim算法适用于边稠密的情形

主要是各种代码收集，最主要的还是c++的

3.1 伪代码

3.2 C

3.3 matlab

3.4 pascal

3.5 c++

3.6 java代码实现

3.7 Mathematica实现（如下）

(*Kruskal算法,点从1开始,时间不最优，但空间最小*)
Clear[kruskal]
kruskal[dian_, tuer_] := Module[
  {dianf = {}, son = {}, biannum = Length[tuer], result = {}, s1, s2, 
   f1, f2, bflag = False, tu},
  son = Table[1, {i, 1, dian}];
  dianf = Table[i, {i, 1, dian}];
  tu = Sort[tuer, #1[[3]] < #2[[3]] &];
  (*Print[tu];*)
  Do[
   s1 = tu[[i, 1]]; s2 = tu[[i, 2]];(*取边，点*);
   f1 = dianf[[s1]]; f2 = dianf[[s2]];
   While[f1 != s1, s1 = f1; f1 = dianf[[s1]]];
   While[f2 != s2, s2 = f2; f2 = dianf[[s2]]];(*取父节点*)
   If[
    s1 < s2,
    dianf[[s2]] = s1;
    son[[s1]] += son[[s2]]; AppendTo[result, tu[[i]]],
    If[s1 > s2, dianf[[s1]] = s2; son[[s2]] += son[[s1]]; 
     AppendTo[result, tu[[i]]]]
    ];
   If[son[[1]] == dian, bflag = True; Break];
   , {i, 1, biannum}];
  If[bflag, Return[result], Print["图不完整"]]
  ]

(*Kruskal算法,点从1开始,时间最小*)
Clear[getfa, kruskal2, dianf]
dianf = {};(*dianji需要实现定义*)
getfa[s_] := 
 Module[ {}, 
  If[dianf[[s]] == s, Return [s], dianf[[s]] = getfa[dianf[[s]]]; 
   Return [dianf[[s]]]]];
kruskal2[dian_, tuer_] := Module[
  {son = {}, biannum = Length[tuer], result = {}, s1, s2, f1, f2, 
   bflag = False, tu},
  son = Table[1, {i, 1, dian}];
  dianf = Table[i, {i, 1, dian}];
  tu = Sort[tuer, #1[[3]] < #2[[3]] &];
  (*Print[tu];*)
  Do[
   s1 = tu[[i, 1]]; s2 = tu[[i, 2]];(*取边，点*);
   f1 = getfa[s1]; f2 = getfa[s2];(*取父节点*)
   (*Print[s1," ",f1," ",s2," ",f2];*)
   If[
    f1 < f2,
    dianf[[f2]] = f1; dianf[[s2]] = f1; son[[f1]] += son[[f2]]; 
    AppendTo[result, tu[[i]]],
    If[
     f1 > f2,
     dianf[[f1]] = f2; dianf[[s1]] = f2; son[[f2]] += son[[f1]]; 
     AppendTo[result, tu[[i]]]]
    ];
   If[son[[1]] == dian, bflag = True; Break];
   , {i, 1, biannum}];
  If[bflag, Return[result], Print["图不完整"]]
  ]

示例：
tuer = {{1, 2, 1}, {1, 9, 2}, {1, 8, 1}, {2, 9, 1}, {2, 3, 1}, {9, 3, 
   3}, {3, 4, 1}, {9, 4, 4}, {4, 5, 5}, {5, 9, 4}, {5, 6, 2}, {6, 9, 
   2}, {6, 7, 3}, {7, 9, 5}, {7, 8, 5}, {8, 9, 4}}

(*example 1*)
In[139]:= kruskal[9, tuer]

Out[139]= {{3, 4, 1}, {2, 3, 1}, {2, 9, 1}, {1, 8, 1}, {1, 2, 1}, {6, 
  9, 2}, {5, 6, 2}, {6, 7, 3}}

(*example 2*)
In[136]:= kruskal2[9, tuer]

Out[136]= {{3, 4, 1}, {2, 3, 1}, {2, 9, 1}, {1, 8, 1}, {1, 2, 1}, {6, 
  9, 2}, {5, 6, 2}, {6, 7, 3}}