首页 > 代码库 > Prim 最小生成树算法
Prim 最小生成树算法
Prim 算法是一种解决最小生成树问题（Minimum Spanning Tree）的算法。和 Kruskal 算法类似，Prim 算法的设计也是基于贪心算法（Greedy algorithm）。
Prim 算法的思想很简单，一棵生成树必须连接所有的顶点，而要保持最小权重则每次选择邻接的边时要选择较小权重的边。Prim 算法看起来非常类似于单源最短路径 Dijkstra 算法，从源点出发，寻找当前的最短路径，每次比较当前可达邻接顶点中最小的一个边加入到生成树中。
例如，下面这张连通的无向图 G，包含 9 个顶点和 14 条边，所以期待的最小生成树应包含 (9 - 1) = 8 条边。
创建 mstSet 包含到所有顶点的距离，初始为 INF，源点 0 的距离为 0，{0, INF, INF, INF, INF, INF, INF, INF, INF}。
选择当前最短距离的顶点，即还是顶点 0，将 0 加入 MST，此时邻接顶点为 1 和 7。
选择当前最小距离的顶点 1，将 1 加入 MST，此时邻接顶点为 2。
选择 2 和 7 中最小距离的顶点为 7，将 7 加入 MST，此时邻接顶点为 6 和 8。
选择 2, 6, 8 中最小距离的顶点为 6，将 6 加入 MST，此时邻接顶点为 5。
重复上面步骤直到遍历完所有顶点为止，会得到如下 MST。
C# 实现 Prim 算法如下。Prim 算法可以达到 O(ElogV) 的运行时间，如果采用斐波那契堆实现，运行时间可以减少到 O(E + VlogV)，如果 V 远小于 E 的话，将是对算法较大的改进。
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 Graph g = new Graph(9); 12 g.AddEdge(0, 1, 4); 13 g.AddEdge(0, 7, 8); 14 g.AddEdge(1, 2, 8); 15 g.AddEdge(1, 7, 11); 16 g.AddEdge(2, 3, 7); 17 g.AddEdge(2, 5, 4); 18 g.AddEdge(3, 4, 9); 19 g.AddEdge(3, 5, 14); 20 g.AddEdge(5, 4, 10); 21 g.AddEdge(6, 5, 2); 22 g.AddEdge(7, 6, 1); 23 g.AddEdge(7, 8, 7); 24 g.AddEdge(8, 2, 2); 25 g.AddEdge(8, 6, 6); 26 27 // sorry, this is an undirect graph, 28 // so, you know that this is not a good idea. 29 List<Edge> edges = g.Edges 30 .Select(e => new Edge(e.End, e.Begin, e.Weight)) 31 .ToList(); 32 foreach (var edge in edges) 33 { 34 g.AddEdge(edge.Begin, edge.End, edge.Weight); 35 } 36 37 Console.WriteLine(); 38 Console.WriteLine("Graph Vertex Count : {0}", g.VertexCount); 39 Console.WriteLine("Graph Edge Count : {0}", g.EdgeCount); 40 Console.WriteLine(); 41 42 List<Edge> mst = g.Prim(); 43 Console.WriteLine("MST Edges:"); 44 foreach (var edge in mst.OrderBy(e => e.Weight)) 45 { 46 Console.WriteLine("\t{0}", edge); 47 } 48 49 Console.ReadKey(); 50 } 51 52 class Edge 53 { 54 public Edge(int begin, int end, int weight) 55 { 56 this.Begin = begin; 57 this.End = end; 58 this.Weight = weight; 59 } 60 61 public int Begin { get; private set; } 62 public int End { get; private set; } 63 public int Weight { get; private set; } 64 65 public override string ToString() 66 { 67 return string.Format( 68 "Begin[{0}], End[{1}], Weight[{2}]", 69 Begin, End, Weight); 70 } 71 } 72 73 class Graph 74 { 75 private Dictionary<int, List<Edge>> _adjacentEdges 76 = new Dictionary<int, List<Edge>>(); 77 78 public Graph(int vertexCount) 79 { 80 this.VertexCount = vertexCount; 81 } 82 83 public int VertexCount { get; private set; } 84 85 public IEnumerable<int> Vertices { get { return _adjacentEdges.Keys; } } 86 87 public IEnumerable<Edge> Edges 88 { 89 get { return _adjacentEdges.Values.SelectMany(e => e); } 90 } 91 92 public int EdgeCount { get { return this.Edges.Count(); } } 93 94 public void AddEdge(int begin, int end, int weight) 95 { 96 if (!_adjacentEdges.ContainsKey(begin)) 97 { 98 var edges = new List<Edge>(); 99 _adjacentEdges.Add(begin, edges);100 }101 102 _adjacentEdges[begin].Add(new Edge(begin, end, weight));103 }104 105 public List<Edge> Prim()106 {107 // Array to store constructed MST108 int[] parent = new int[VertexCount];109 110 // Key values used to pick minimum weight edge in cut111 int[] keySet = new int[VertexCount];112 113 // To represent set of vertices not yet included in MST114 bool[] mstSet = new bool[VertexCount];115 116 // Initialize all keys as INFINITE117 for (int i = 0; i < VertexCount; i++)118 {119 keySet[i] = int.MaxValue;120 mstSet[i] = false;121 }122 123 // Always include first 1st vertex in MST.124 // Make key 0 so that this vertex is picked as first vertex125 keySet[0] = 0;126 parent[0] = -1; // First node is always root of MST 127 128 // The MST will have V vertices129 for (int i = 0; i < VertexCount - 1; i++)130 {131 // Pick thd minimum key vertex from the set of vertices132 // not yet included in MST133 int u = CalculateMinDistance(keySet, mstSet);134 135 // Add the picked vertex to the MST Set136 mstSet[u] = true;137 138 // Update key value and parent index of the adjacent vertices of139 // the picked vertex. Consider only those vertices which are not yet140 // included in MST141 for (int v = 0; v < VertexCount; v++)142 {143 // graph[u, v] is non zero only for adjacent vertices of m144 // mstSet[v] is false for vertices not yet included in MST145 // Update the key only if graph[u, v] is smaller than key[v]146 if (!mstSet[v]147 && _adjacentEdges.ContainsKey(u)148 && _adjacentEdges[u].Exists(e => e.End == v))149 {150 int d = _adjacentEdges[u].Single(e => e.End == v).Weight;151 if (d < keySet[v])152 {153 keySet[v] = d;154 parent[v] = u;155 }156 }157 }158 }159 160 // get all MST edges161 List<Edge> mst = new List<Edge>();162 for (int i = 1; i < VertexCount; i++)163 mst.Add(_adjacentEdges[parent[i]].Single(e => e.End == i));164 165 return mst;166 }167 168 private int CalculateMinDistance(int[] keySet, bool[] mstSet)169 {170 int minDistance = int.MaxValue;171 int minDistanceIndex = -1;172 173 for (int v = 0; v < VertexCount; v++)174 {175 if (!mstSet[v] && keySet[v] <= minDistance)176 {177 minDistance = keySet[v];178 minDistanceIndex = v;179 }180 }181 182 return minDistanceIndex;183 }184 }185 }186 }
输出结果如下：
参考资料
- Connectivity in a directed graph
- Strongly Connected Components
- Tarjan‘s Algorithm to find Strongly Connected Components
本篇文章《Prim 最小生成树算法》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。
Prim 最小生成树算法
声明：以上内容来自用户投稿及互联网公开渠道收集整理发布，本网站不拥有所有权，未作人工编辑处理，也不承担相关法律责任，若内容有误或涉及侵权可进行投诉： 投诉/举报 工作人员会在5个工作日内联系你，一经查实，本站将立刻删除涉嫌侵权内容。