【理論知識的，可以參考】

漫畫：圖的最短路徑問題

最短路徑演算法

該演算法得到的是單源最短路徑，即起點到任意目標點的距離

【lua實現】

 1 local Dijkstra = {}
 2 Dijkstra.__index = Dijkstra
 3 
 4 function Dijkstra.new(g)
 5     local obj = {}
 6     setmetatable(obj, Dijkstra)
 7 
 8     obj:ctor(g)
 9     return obj
10 end
11 
12 function Dijkstra:ctor(g)
13     self.graph = g
 
14     self.startVertex = nil
15     self.visited = {}
16     self.visitFrom = {}
17     self.distFromStart = {}
18 end
19 
20 function Dijkstra:IsReachable(endVertex)
21     return self.distFromStart[endVertex].dist < 9999
22 end
23 
24 function Dijkstra:VisitAllReachable(startVertex)
25     self.startVertex = startVertex
 
26     self.visitFrom = {}
27     self.distFromStart = {}
28 
29     for i=1,self.graph:GetVertexCount() do
30         local v = self.graph:GetVertex(i)
31         self.distFromStart[v] = {
32             dist = 9999,
33             used = false,
34         }
35     end
36 
37     local distInfo = self.distFromStart[startVertex]
 
38     distInfo.dist = 0
39     distInfo.used = true
40 
41     local v = startVertex
42     while true do
43         local adjList = self.graph:GetAdjacent(v)
44         for i=1,#adjList do
45             local adjVInfo = adjList[i]
46             local oldDist = self.distFromStart[adjVInfo.v].dist
47             local newDist = self.distFromStart[v].dist + adjVInfo.w
48             if newDist < oldDist then
49                 self.distFromStart[adjVInfo.v].dist = newDist
50                 self.visitFrom[adjVInfo.v] = v
51             end
52         end
53 
54         local minDistInfo = nil
55         local minV = nil
56         for i=1,self.graph:GetVertexCount() do
57             local v = self.graph:GetVertex(i)
58             local distInfo = self.distFromStart[v]
59             if not distInfo.used then
60                 if nil == minDistInfo or distInfo.dist < minDistInfo.dist then
61                     minDistInfo = distInfo
62                     minV = v
63                 end
64             end
65         end
66         if nil == minDistInfo then --所有都遍歷過了
67             break
68         end
69 
70         minDistInfo.used = true
71         v = minV
72     end
73 end
74 
75 function Dijkstra:GetDist(endVertex)
76     return self.distFromStart[endVertex].dist
77 end
78 
79 function Dijkstra:GetPath(endVertex)
80     if not self:IsReachable(endVertex) then return nil end
81 
82     local pathQueue = {}
83     local v = endVertex
84     while v ~= self.startVertex do
85         table.insert(pathQueue, 1, v)
86         v = self.visitFrom[v]
87     end
88     table.insert(pathQueue, 1, v)
89     return pathQueue
90 end

計算下面這張圖的最短路徑：

 1 function CreateGraph()
 2     local g = WeightedGraph.new()
 3     g:AddVertex("A")
 4     g:AddVertex("B")
 5     g:AddVertex("C")
 6     g:AddVertex("D")
 7     g:AddVertex("E")
 8     g:AddVertex("F")
 9     g:AddVertex("G")
10 
11     g:AddEdge("A", "B", 5)
12     g:AddEdge("A", "C", 2)
13     g:AddEdge("B", "D", 1)
14     g:AddEdge("B", "E", 6)
15     g:AddEdge("C", "D", 6)
16     g:AddEdge("C", "F", 8)
17     g:AddEdge("D", "E", 1)
18     g:AddEdge("D", "F", 2)
19     g:AddEdge("E", "G", 7)
20     g:AddEdge("F", "G", 3)
21 
22     --tostring(g)
23     return g
24 end
25 
26 function Test1()
27     local g = CreateGraph()
28     local d = Dijkstra.new(g)
29     d:VisitAllReachable("A")
30     assert(d:IsReachable("B"))
31     assert(d:IsReachable("G"))
32 
33     assert(5 == d:GetDist("B"))
34     assert(2 == d:GetDist("C"))
35     assert(6 == d:GetDist("D"))
36     assert(7 == d:GetDist("E"))
37     assert(8 == d:GetDist("F"))
38     assert(11 == d:GetDist("G"))
39 end
40 Test1()