第6章 图

3. 克鲁斯卡尔(Kruskal) 算法

 克鲁斯卡尔算法的基本思想是：对一个有n个顶点的无向连通图，将图中的边按权值大小依次选取，若选取的边使生成树不形成回路，则把它加入到树中；若形成回路，则将它舍弃。如此进行下去，直到树中包含有n-1条边为止。(当整个图为连通图时为n-1条边)

根据邻接矩阵存储结构实现Kruskal算法：（邻接链表的实现在前面的博客）

        public void Kruskal()
        {
            bool[,] markers = new bool[NodeNum, NodeNum];
            Dictionary<AdjListNode<T>, VexListNode<T>> dic = new Dictionary<AdjListNode<T>, VexListNode<T>>();

            while (true)
            {
                AdjListNode<T> adjNode = null;
                VexListNode<T> vexNode = null;

                AdjListNode<T> currentAdjNode = null;
                VexListNode<T> currentVexNode = null;
                for (int j = 0; j < NodeNum; j++)
                {
                    currentVexNode = vexList[j];
                    currentAdjNode = currentVexNode.FirstAdj;

                    while (currentAdjNode != null)
                    {
                        if (markers[j, currentAdjNode.AdjVexIndex] == false)
                        {
                            if (adjNode == null || currentAdjNode.Weight < adjNode.Weight)
                            {
                                vexNode = currentVexNode;
                                adjNode = currentAdjNode;
                            }
                        }

                        currentAdjNode = currentAdjNode.Next;
                    }
                }

                if (adjNode == null)
                    break;

                dic.Add(adjNode, vexNode);

                markers[IsNode(vexNode.Node), adjNode.AdjVexIndex] = true;
                markers[adjNode.AdjVexIndex, IsNode(vexNode.Node)] = true;


                bool ifCycle = true;

                while (ifCycle)
                {
                    ifCycle = false;

                    for (int j = 0; j < NodeNum; j++)
                    {
                        for (int k = 0; k < NodeNum; k++)
                        {
                            if (markers[j, k] == true)
                            {
                                for (int m = 0; m < NodeNum; m++)
                                {
                                    if (m == k)
                                        continue;
                                    if (markers[k, m] == true && markers[j, m] == false)
                                    {
                                        markers[j, m] = true;
                                        markers[m, j] = true;
                                        ifCycle = true;
                                    }
                                }
                            }
                        }
                    }
                }
            }

            foreach (var i in dic.Keys)
            {
                Console.WriteLine(dic[i].Node.Value.ToString() + " -> " + i.Weight + " -> " + vexList[i.AdjVexIndex].Node.Value.ToString());
            }
        }

调用代码:

            GraphAdjList<int> adjList = new GraphAdjList<int>(100);

            //Inial graph object
            GraphNode<int> node1 = new GraphNode<int>(1);
            GraphNode<int> node2 = new GraphNode<int>(2);
            GraphNode<int> node3 = new GraphNode<int>(3);
            GraphNode<int> node4 = new GraphNode<int>(4);
            GraphNode<int> node5 = new GraphNode<int>(5);
            GraphNode<int> node6 = new GraphNode<int>(6);
            GraphNode<int> node7 = new GraphNode<int>(7);
            GraphNode<int> node8 = new GraphNode<int>(8);

            adjList.SetNode(node1);
            adjList.SetNode(node2);
            adjList.SetNode(node3);
            adjList.SetNode(node4);
            adjList.SetNode(node5);
            adjList.SetNode(node6);
            adjList.SetNode(node7);
            adjList.SetNode(node8);

            adjList.SetEdge(0, node2, node1, 2);
            adjList.SetEdge(1,node1, node3, 4);
            adjList.SetEdge(2,node1, node4, 4);
            adjList.SetEdge(3, node1, node5, 3);
            adjList.SetEdge(4, node2, node3, 3);
            adjList.SetEdge(5, node2, node4, 1);
            adjList.SetEdge(6, node2, node5, 2);
            adjList.SetEdge(7, node3, node4, 4);
            adjList.SetEdge(8, node3, node5, 4);
            adjList.SetEdge(9, node5, node4, 2);
            adjList.SetEdge(10, node6, node7, 3);
            adjList.SetEdge(11, node8, node6, 2);
            adjList.SetEdge(12, node7, node8, 3);

            adjList.Kruskal();
            System.Console.ReadKey();

输出为：

2 -> 1 -> 4
2 -> 2 -> 5
2 -> 2 -> 1
8 -> 2 -> 6
2 -> 3 -> 3
6 -> 3 -> 7