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Dijkstra 算法是一种用于计算带权有向图中单源最短路径(SSSP:Single-Source Shortest Path)的算法,由计算机科学家 Edsger Dijkstra 于 1956 年构思并于 1959 年发表。其解决的问题是:给定图 G 和源顶点 v,找到从 v 至图中所有顶点的最短路径。
Dijkstra 算法采用贪心算法(Greedy Algorithm)范式进行设计。在最短路径问题中,对于带权有向图 G = (V, E),Dijkstra 算法的初始实现版本未使用最小优先队列实现,其时间复杂度为 O(V2),基于 Fibonacci heap 的最小优先队列实现版本,其时间复杂度为 O(E + VlogV)。
Bellman-Ford 算法和 Dijkstra 算法同为解决单源最短路径的算法。对于带权有向图 G = (V, E),Dijkstra 算法要求图 G 中边的权值均为非负,而 Bellman-Ford 算法能适应一般的情况(即存在负权边的情况)。一个实现的很好的 Dijkstra 算法比 Bellman-Ford 算法的运行时间 O(V*E) 要低。
Dijkstra 算法描述:
伪码实现如下:
1 function Dijkstra(Graph, source): 2 3 dist[source] ← 0 // Distance from source to source 4 prev[source] ← undefined // Previous node in optimal path initialization 5 6 for each vertex v in Graph: // Initialization 7 if v ≠ source // Where v has not yet been removed from Q (unvisited nodes) 8 dist[v] ← infinity // Unknown distance function from source to v 9 prev[v] ← undefined // Previous node in optimal path from source 10 end if 11 add v to Q // All nodes initially in Q (unvisited nodes) 12 end for 13 14 while Q is not empty: 15 u ← vertex in Q with min dist[u] // Source node in first case 16 remove u from Q 17 18 for each neighbor v of u: // where v has not yet been removed from Q. 19 alt ← dist[u] + length(u, v) 20 if alt < dist[v]: // A shorter path to v has been found 21 dist[v] ← alt 22 prev[v] ← u 23 end if 24 end for 25 end while 26 27 return dist[], prev[] 28 29 end function
例如,下面是一个包含 9 个顶点的图,每条边分别标识了距离。
源顶点 source = 0,初始时,
将 0 包含至 sptSet 中;
更新 0 至其邻接节点的距离;
选择不在 sptSet 中的 Min Distance 的顶点,为顶点 1,则将 1 包含至 sptSet;
更新 1 至其邻接节点的距离;
选择不在 sptSet 中的 Min Distance 的顶点,为顶点 7,则将 7 包含至 sptSet;
更新 7 至其邻接节点的距离;
选择不在 sptSet 中的 Min Distance 的顶点,为顶点 6,则将 6 包含至 sptSet;
更新 6 至其邻接节点的距离;
以此类推,直到遍历结束。
最终结果为源顶点 0 至所有顶点的距离:
Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14
C#代码实现:
1 using System; 2 using System.Collections.Generic; 3 using System.Linq; 4 5 namespace GraphAlgorithmTesting 6 { 7 class Program 8 { 9 static void Main(string[] args) 10 { 11 int[,] graph = new int[9, 9] 12 { 13 {0, 4, 0, 0, 0, 0, 0, 8, 0}, 14 {4, 0, 8, 0, 0, 0, 0, 11, 0}, 15 {0, 8, 0, 7, 0, 4, 0, 0, 2}, 16 {0, 0, 7, 0, 9, 14, 0, 0, 0}, 17 {0, 0, 0, 9, 0, 10, 0, 0, 0}, 18 {0, 0, 4, 0, 10, 0, 2, 0, 0}, 19 {0, 0, 0, 14, 0, 2, 0, 1, 6}, 20 {8, 11, 0, 0, 0, 0, 1, 0, 7}, 21 {0, 0, 2, 0, 0, 0, 6, 7, 0} 22 }; 23 24 Graph g = new Graph(graph.GetLength(0)); 25 for (int i = 0; i < graph.GetLength(0); i++) 26 { 27 for (int j = 0; j < graph.GetLength(1); j++) 28 { 29 if (graph[i, j] > 0) 30 g.AddEdge(i, j, graph[i, j]); 31 } 32 } 33 34 int[] dist = g.Dijkstra(0); 35 Console.WriteLine("Vertex\t\tDistance from Source"); 36 for (int i = 0; i < dist.Length; i++) 37 { 38 Console.WriteLine("{0}\t\t{1}", i, dist[i]); 39 } 40 41 Console.ReadKey(); 42 } 43 44 class Edge 45 { 46 public Edge(int begin, int end, int distance) 47 { 48 this.Begin = begin; 49 this.End = end; 50 this.Distance = distance; 51 } 52 53 public int Begin { get; private set; } 54 public int End { get; private set; } 55 public int Distance { get; private set; } 56 } 57 58 class Graph 59 { 60 private Dictionary<int, List<Edge>> _adjacentEdges 61 = new Dictionary<int, List<Edge>>(); 62 63 public Graph(int vertexCount) 64 { 65 this.VertexCount = vertexCount; 66 } 67 68 public int VertexCount { get; private set; } 69 70 public void AddEdge(int begin, int end, int distance) 71 { 72 if (!_adjacentEdges.ContainsKey(begin)) 73 { 74 var edges = new List<Edge>(); 75 _adjacentEdges.Add(begin, edges); 76 } 77 78 _adjacentEdges[begin].Add(new Edge(begin, end, distance)); 79 } 80 81 public int[] Dijkstra(int source) 82 { 83 // dist[i] will hold the shortest distance from source to i 84 int[] distSet = new int[VertexCount]; 85 86 // sptSet[i] will true if vertex i is included in shortest 87 // path tree or shortest distance from source to i is finalized 88 bool[] sptSet = new bool[VertexCount]; 89 90 // initialize all distances as INFINITE and stpSet[] as false 91 for (int i = 0; i < VertexCount; i++) 92 { 93 distSet[i] = int.MaxValue; 94 sptSet[i] = false; 95 } 96 97 // distance of source vertex from itself is always 0 98 distSet[source] = 0; 99 100 // find shortest path for all vertices 101 for (int i = 0; i < VertexCount - 1; i++) 102 { 103 // pick the minimum distance vertex from the set of vertices not 104 // yet processed. u is always equal to source in first iteration. 105 int u = CalculateMinDistance(distSet, sptSet); 106 107 // mark the picked vertex as processed 108 sptSet[u] = true; 109 110 // update dist value of the adjacent vertices of the picked vertex. 111 for (int v = 0; v < VertexCount; v++) 112 { 113 // update dist[v] only if is not in sptSet, there is an edge from 114 // u to v, and total weight of path from source to v through u is 115 // smaller than current value of dist[v] 116 if (!sptSet[v] 117 && distSet[u] != int.MaxValue 118 && _adjacentEdges[u].Exists(e => e.End == v)) 119 { 120 int d = _adjacentEdges[u].Single(e => e.End == v).Distance; 121 if (distSet[u] + d < distSet[v]) 122 { 123 distSet[v] = distSet[u] + d; 124 } 125 } 126 } 127 } 128 129 return distSet; 130 } 131 132 /// <summary> 133 /// A utility function to find the vertex with minimum distance value, 134 /// from the set of vertices not yet included in shortest path tree 135 /// </summary> 136 private int CalculateMinDistance(int[] distSet, bool[] sptSet) 137 { 138 int minDistance = int.MaxValue; 139 int minDistanceIndex = -1; 140 141 for (int v = 0; v < VertexCount; v++) 142 { 143 if (!sptSet[v] && distSet[v] <= minDistance) 144 { 145 minDistance = distSet[v]; 146 minDistanceIndex = v; 147 } 148 } 149 150 return minDistanceIndex; 151 } 152 } 153 } 154 }
本篇文章《Dijkstra 单源最短路径算法》由 Dennis Gao 发表自博客园,未经作者本人同意禁止任何形式的转载,任何自动或人为的爬虫转载行为均为耍流氓。
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原文地址:http://www.cnblogs.com/gaochundong/p/dijkstra_algorithm.html