最短路-Floyd算法和Dijkstra算法

两者在负权问题上不是很好，最好只处理正值

Dijkstra算法的话，为了方便，我认为从i到i点不可达；百部百科解释挺好，那个堆优化挺好的

Floyd算法百部百科也不错，都是老算法了，哪都有资料

博客园这位筒子的写得很好 http://www.cnblogs.com/biyeymyhjob/archive/2012/07/31/2615833.html

Mathematica下的代码【我也觉得mathematica写很诡异。。。】

(*Dijkstra算法,其思想和Prim有点像,输出的是每个点的前向节点*)
Dijkstra[tu_, dian_, point_] := Module[
  {dis = tu[[point]], pre = Table[point, {i, 1, dian}],
   diannum = dian, visin = Table[False, {i, 1, dian}], k, lval,
   mindis, flag = True},
  visin[[point]] = True;
  k = point;
  While[diannum != 0,
   mindis = Infinity;
   Do[
    If[visin[[i]] == False && dis[[i]] < mindis, mindis = dis[[i]];
     k = i]
    , {i, 1, dian}];

   If[mindis == Infinity, flag = True; Break[]];
   diannum--;
   visin[[k]] = True;

   Do[
    If[visin[[j]], Continue[]];
    lval = tu[[k]][[j]] + dis[[k]];
    If[lval < dis[[j]], dis[[j]] = lval; pre[[j]] = k]
    , {j, 1, dian}]
   ];
  pre
  ]

Dijkstra

Input:
tuer = {{INF, 7, INF, 5, INF, INF, INF}, {7, INF, 8, 9, 7, INF, INF}, {INF, 8, INF, INF, 5, INF, INF}, {5, 9, INF, INF, 15, 6, INF}, {INF, 7, 5, 15, INF, 8, 9}, {INF, INF, INF, 6, 8, INF, 11}, {INF, INF, INF, INF, 9, 11, INF}}

Dijkstra[tuer, 7, 1]

Out:
 {1, 1, 2, 1, 2, 4, 6}

示例

(*Floyd算法,打印输出了长度，函数输出前节点*)
Floyd[tu_, dian_] := Module[
  {path = Table[-1, {i, 1, dian}, {j, 1, dian}], Dis = tu, lenn},
  Do[
   Do[
    Do[
     lenn = Dis[[i, k]] + Dis[[k, j]];
     If[Dis[[i, j]] > lenn, path[[i, j]] = k; Dis[[i, j]] = lenn]
     , {k, 1, dian}]
    , {j, 1, dian}]
   , {i, 1, dian}];
  Print[Dis];
  path
  ]

Floyd

Input:
tuer = {{INF, 7, INF, 5, INF, INF, INF}, {7, INF, 8, 9, 7, INF, INF}, {INF, 8, INF, INF, 5, INF, INF}, {5, 9, INF, INF, 15, 6, INF}, {INF, 7, 5, 15, INF, 8, 9}, {INF, INF, INF, 6, 8, INF, 11}, {INF, INF, INF, INF, 9, 11, INF}}
Floyd[tuer, 7]

During evaluation of Input：
 {{0,7,15,5,14,11,22},{7,0,8,9,7,15,16},{15,8,0,17,5,13,14},{5,9,17,0,14,6,17},{14,7,5,14,0,8,9},{11,15,13,6,8,0,11},{22,16,14,17,9,11,0}}

Out：
{{-1, -1, 2, -1, 2, 4, 6}, {-1, -1, -1, -1, -1, 4,
  5}, {2, -1, -1, 2, -1, 5, 5}, {-1, -1, 2, -1, 6, -1, 6}, {2, -1, -1,
   6, -1, -1, -1}, {4, 4, 5, -1, -1, -1, -1}, {6, 5, 5,
  6, -1, -1, -1}}

示例