背景 以前我写过一些图文来介绍有关数据结构与算法的知识:
本次,向大家介绍图论中构造最小生成树的 Kruskal 算法。
技术分析 Kruskal 算法:
Krusal算法 例子:
例子 该例子演示了一个含有6个结点,10条边的连通网,通过 Kruskal 算法逐步演化为含有6个结点,5条边的连通子网的过程,即构造最小生成树的过程。
代码实现 Step1 构造边表结点的结构 EdgeNode
。
public class EdgeNode { /// <summary> /// 获取边终点在顶点数组中的位置 /// </summary> public int Index { get; } /// <summary> /// 获取边上的权值 /// </summary> public double Weight { get; } /// <summary> /// 获取或设置下一个邻接点 /// </summary> public EdgeNode Next { get; set ; } /// <summary> /// 初始化EdgeNode类的新实例 /// </summary> /// <param name="index">边终点在顶点数组中的位置</param> /// <param name="weight">边上的权值</param> /// <param name="next">下一个邻接点</param> public EdgeNode (int index, double weight = 0.0 , EdgeNode next = null) { if (index < 0 ) throw new ArgumentOutOfRangeException(); Index = index; Weight = weight; Next = next; } }
Step2 构造顶点表结点的结构 VertexNode
。
public class VertexNode { /// <summary> /// 获取或设置顶点的名字 /// </summary> public string VertexName { get; set ; } /// <summary> /// 获取或设置顶点是否被访问 /// </summary> public bool Visited { get; set ; } /// <summary> /// 获取或设置顶点的第一个邻接点 /// </summary> public EdgeNode FirstNode { get; set ; } /// <summary> /// 初始化VertexNode类的新实例 /// </summary> /// <param name="vName">顶点的名字</param> /// <param name="firstNode">顶点的第一个邻接点</param> public VertexNode (string vName, EdgeNode firstNode = null) { VertexName = vName; Visited = false ; FirstNode = firstNode; } }
Step3 构造利用邻接表存储图的结构AdGraph
。
通过 AdGraph
的索引器可以为顶点表赋值,通过 AddEdge
方法可以为边表赋值。
public class AdGraph { private readonly VertexNode[] _vertexList; //结点表 /// <summary> /// 获取图的结点数 /// </summary> public int VertexCount { get; } /// <summary> /// 初始化AdGraph类的新实例 /// </summary> /// <param name="vCount">图中结点的个数</param> public AdGraph (int vCount) { if (vCount <= 0 ) throw new ArgumentOutOfRangeException(); VertexCount = vCount; _vertexList = new VertexNode[vCount]; } /// <summary> /// 获取或设置图中各结点的名称 /// </summary> /// <param name="index">结点名称从零开始的索引</param> /// <returns>指定索引处结点的名称</returns> public string this [int index] { get { if (index < 0 || index > VertexCount - 1 ) throw new ArgumentOutOfRangeException(); return _vertexList[index] == null ? "NULL" : _vertexList[index].VertexName; } set { if (index < 0 || index > VertexCount - 1 ) throw new ArgumentOutOfRangeException(); if (_vertexList[index] == null) _vertexList[index] = new VertexNode(value); else _vertexList[index].VertexName = value; } } /// <summary> /// 得到结点在结点表中的位置 /// </summary> /// <param name="vertexName">结点的名称</param> /// <returns>结点的位置</returns> private int GetIndex (string vertexName) { int i; for (i = 0 ; i < VertexCount; i++) { if (_vertexList[i] != null && _vertexList[i].VertexName == vertexName) break ; } return i == VertexCount ? -1 : i; } /// <summary> /// 给图加边 /// </summary> /// <param name="startVertexName">起始结点的名字</param> /// <param name="endVertexName">终止结点的名字</param> /// <param name="weight">边上的权值</param> public void AddEdge (string startVertexName, string endVertexName , double weight = 0.0 ) { int i = GetIndex(startVertexName); int j = GetIndex(endVertexName); if (i == -1 || j == -1 ) throw new Exception("图中不存在该边." ); EdgeNode temp = _vertexList[i].FirstNode; if (temp == null) { _vertexList[i].FirstNode = new EdgeNode(j, weight); } else { while (temp.Next != null) temp = temp.Next; temp.Next = new EdgeNode(j, weight); } } }
上面例子对应的邻接表如下所示:
邻接表 Step4 构造最小生成树结点的结构 SpanTreeNode
。
public class SpanTreeNode { /// <summary> /// 获取或设置结点本身的名称 /// </summary> public string SelfName { get; } /// <summary> /// 获取或设置结点双亲的名称 /// </summary> public string ParentName { get; } /// <summary> /// 获取或设置边的权值 /// </summary> public double Weight { get; set ; } /// <summary> /// 构造SpanTreeNode实例 /// </summary> /// <param name="selfName">结点本身的名称</param> /// <param name="parentName">结点双亲的名称</param> /// <param name="weight">边的权值</param> public SpanTreeNode (string selfName, string parentName, double weight) { if (string .IsNullOrEmpty(selfName) || string .IsNullOrEmpty(parentName)) throw new ArgumentNullException(); SelfName = selfName; ParentName = parentName; Weight = weight; } }
Step5 构造边的结构 Edge
。
internal class Edge { /// <summary> /// 起点编号 /// </summary> public int Begin { get;} /// <summary> /// 终点编号 /// </summary> public int End { get; } /// <summary> /// 权值 /// </summary> public double Weight { get; } /// <summary> /// 创建一个 Edge 类的新实例 /// </summary> /// <param name="begin">起点编号</param> /// <param name="end">终点编号</param> /// <param name="weight">权值</param> public Edge (int begin, int end, double weight = 0.0 ) { Begin = begin; End = end; Weight = weight; } }
Step6 获取边集合的方法 GetEdges
。
private Edge[] GetEdges() { for (int i = 0 ; i < VertexCount; i++) _vertexList[i].Visited = false ; List <Edge> result = new List <Edge>(); for (int i = 0 ; i < VertexCount; i++) { _vertexList[i].Visited = true ; EdgeNode p = _vertexList[i].FirstNode; while (p != null ) { if (_vertexList[p.Index].Visited == false ) { Edge edge = new Edge(i, p.Index, p.Weight); result.Add(edge); } p = p.Next; } } return result.OrderBy(a => a.Weight).ToArray(); }
上面例子对应的边的集合如下所示:
边 Step7 获取最小生成树的 Kruskal 算法。
private int Find (int [] parent, int f) { while (parent[f] > 0 ) f = parent[f]; return f; }/// <summary> /// 克鲁斯卡尔算法 最小生成树 /// </summary> /// <returns></returns> public SpanTreeNode[] MiniSpanTree() { int [] parent = new int [VertexCount]; for (int i = 0 ; i < VertexCount; i++) { parent[i] = 0 ; } SpanTreeNode[] tree = new SpanTreeNode[VertexCount]; int count = 0 ; Edge[] edges = GetEdges(); for (int i = 0 ; i < edges.Length; i++) { int begin = edges[i].Begin; int end = edges[i].End; int n = Find(parent, begin); int m = Find(parent, end); if (n != m) { if (i == 0 ) { tree[count] = new SpanTreeNode(_vertexList[begin].VertexName, "NULL" , 0.0 ); count++; } parent[n] = m; tree[count] = new SpanTreeNode(_vertexList[end].VertexName, _vertexList[begin].VertexName, edges[i].Weight); count++; } } return tree; }
总结 到此为止代码部分就全部介绍完了,我们来看一下上面例子的应用。
static void Main (string [] args) { AdGraph alg = new AdGraph(6 ); alg[0 ] = "V0" ; alg[1 ] = "V1" ; alg[2 ] = "V2" ; alg[3 ] = "V3" ; alg[4 ] = "V4" ; alg[5 ] = "V5" ; alg.AddEdge("V0" , "V1" , 6 ); alg.AddEdge("V0" , "V2" , 1 ); alg.AddEdge("V0" , "V3" , 5 ); alg.AddEdge("V1" , "V0" , 6 ); alg.AddEdge("V1" , "V2" , 5 ); alg.AddEdge("V1" , "V4" , 3 ); alg.AddEdge("V2" , "V0" , 1 ); alg.AddEdge("V2" , "V1" , 5 ); alg.AddEdge("V2" , "V3" , 7 ); alg.AddEdge("V2" , "V4" , 5 ); alg.AddEdge("V2" , "V5" , 4 ); alg.AddEdge("V3" , "V0" , 5 ); alg.AddEdge("V3" , "V2" , 7 ); alg.AddEdge("V3" , "V5" , 2 ); alg.AddEdge("V4" , "V1" , 3 ); alg.AddEdge("V4" , "V2" , 5 ); alg.AddEdge("V4" , "V5" , 6 ); alg.AddEdge("V5" , "V2" , 4 ); alg.AddEdge("V5" , "V3" , 2 ); alg.AddEdge("V5" , "V4" , 6 ); SpanTreeNode[] tree = alg.MiniSpanTree(); double sum = 0 ; for (int i = 0 ; i < tree.Length; i++) { string str = "(" + tree[i].ParentName + "," + tree[i].SelfName + ") Weight:" + tree[i].Weight; Console.WriteLine(str); sum += tree[i].Weight; } Console.WriteLine(sum); }
结果如下:
结果 我们再通过一个例子来演示如何应用:
地图 上面是一幅纽约市附近的地图,对应的数据存储在 graph.txt 文件中。
数据 读入该文件,构造好 AdGraph
结构后,调用我们写好的 Kruskal 算法,得到的结果如下:
结果 是不是很有趣,今天就到这里吧!马上要放假了,我们的招新活动也即将开启,希望大家关注呦!
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