数据结构入门学习系列-10(霍夫曼树)

什么是哈夫曼树呢？

哈夫曼树是一种带权路径长度最短的二叉树，也称为最优二叉树。下面用一幅图来说明。

它们的带权路径长度分别为：

图a： WPL=5*2+7*2+2*2+13*2=54

图b： WPL=5*3+2*3+7*2+13*1=48

可见，图b的带权路径长度较小，我们可以证明图b就是哈夫曼树(也称为最优二叉树)。

一般可以按下面步骤构建：

1，将所有左，右子树都为空的作为根节点。

2，在森林中选出两棵根节点的权值最小的树作为一棵新树的左，右子树，且置新树的附加根节点的权值为其左，右子树上根节点的权值之和。注意，左子树的权值应小于右子树的权值。

3，从森林中删除这两棵树，同时把新树加入到森林中。

4，重复2，3步骤，直到森林中只有一棵树为止，此树便是哈夫曼树。

下面是构建哈夫曼树的图解过程：

利用哈夫曼树求得的用于通信的二进制编码称为哈夫曼编码。树中从根到每个叶子节点都有一条路径，对路径上的各分支约定指向左子树的分支表示”0”码，指向右子树的分支表示“1”码，取每条路径上的“0”或“1”的序列作为各个叶子节点对应的字符编码，即是哈夫曼编码。

就拿上图例子来说：

A，B，C，D对应的哈夫曼编码分别为：111，10，110，0

用图说明如下：

记住，设计电文总长最短的二进制前缀编码，就是以n个字符出现的频率作为权构造一棵哈夫曼树，由哈夫曼树求得的编码就是哈夫曼编码。

算法实现：

namespace HuffTree.CSharp{    class Program    {        static void Main(string[] args)        {            //四个叶子节点            int leafNum = 4;            //赫夫曼树的节点总数            int totalNodes = 2 * leafNum - 1;            //各叶子节点的权值            int[] weight = new int[] { 5,7,2,13};            //各叶子节点的值            string[] alphabet = new string[] { "A","B","C","D"};            //初始化赫夫曼树            HuffmanTree[] huffmanTree = new HuffmanTree[totalNodes].Select(p => new HuffmanTree() { }).ToArray();            //构建赫夫曼树            HuffmanTreeBLL.Create(huffmanTree,leafNum,weight);            //赫夫曼编码            string[] huffmanCode = HuffmanTreeBLL.Coding(huffmanTree,leafNum);            //打印结果            PrintResult(alphabet,huffmanTree,huffmanCode,leafNum);            Console.ReadKey();        }        /// <summary>        /// 打印结果        /// </summary>        /// <param name="alphabet"></param>        /// <param name="huffmanTree"></param>        /// <param name="huffmanCode"></param>        /// <param name="leafNum"></param>        private static void PrintResult(string[] alphabet,HuffmanTree[] huffmanTree,string[] huffmanCode,int leafNum)        {            if (alphabet.Count() < 1 || huffmanTree.Count() < 1 || huffmanCode.Count() < 1) return;            for (int i = 0; i < leafNum; i++)            {                Console.WriteLine("字符：{0},权重值:{1},赫夫曼编码：{2}",alphabet[i],huffmanTree[i].Weight,huffmanCode[i]);            }        }    }}namespace DS.BLL{    /// <summary>    /// 描述:赫夫曼树操作类    /// 作者:鲁宁    /// 时间:2013/9/17 18:14:33    /// </summary>    public class HuffmanTreeBLL    {        /// <summary>        /// 构建赫夫曼树        /// 思路：一步一步向上搭建        /// </summary>        /// <param name="huffmanTree">待操作的赫夫曼树</param>        /// <param name="leafNum">叶节点数量</param>        /// <param name="weight">节点权重值</param>        /// <returns>构建好的赫夫曼树</returns>        public static HuffmanTree[] Create(HuffmanTree[] huffmanTree, int leafNum, int[] weight)        {            //获取赫夫曼树结点总数            int totalNodes = 2 * leafNum - 1;            InitLeafNode(huffmanTree,leafNum,weight);                        //构造赫夫曼树(4个节点只需要3步就可以完成构建)            for (int i = leafNum; i < totalNodes; i++)            {                 //获取权重最小的两个叶子节点的下标                int minIndex1 = -1;                int minIndex2 = -1;                SelectNode(huffmanTree,i,ref minIndex1,ref minIndex2);                huffmanTree[minIndex1].Parent = i;                huffmanTree[minIndex2].Parent = i;                huffmanTree[i].Left = minIndex1;                huffmanTree[i].Right = minIndex2;                huffmanTree[i].Weight = huffmanTree[minIndex1].Weight + huffmanTree[minIndex2].Weight;            }            return huffmanTree;        }        /// <summary>        /// 赫夫曼编码        /// 思路：左子树为0，右子树为1，对应的编码后的规则是：从根节点到子节点        /// </summary>        /// <param name="huffmanTree">待操作的赫夫曼树</param>        /// <param name="leafNum">叶子节点的数量</param>        /// <returns>赫夫曼编码</returns>        public static string[] Coding(HuffmanTree[] huffmanTree, int leafNum)        {             string[] huffmanCode= new string[leafNum];            //当前节点下标            int current = 0;            //父节点下标            int parent = 0;            for (int i = 0; i < leafNum; i++)            {                string codeTemp = string.Empty;                current = i;                                //第一次获取最左节点                parent = huffmanTree[current].Parent;                while (parent != 0)                {                    if (huffmanTree[parent].Left == current) codeTemp += "0";                    else codeTemp += "1";                    current = parent;                    parent = huffmanTree[parent].Parent;                }                huffmanCode[i] = new string(codeTemp.Reverse().ToArray());            }            return huffmanCode;        }        /// <summary>        /// 初始化叶节点        /// </summary>        /// <param name="huffmanTree"></param>        /// <param name="leafNum"></param>        /// <param name="weight"></param>        private static void InitLeafNode(HuffmanTree[] huffmanTree, int leafNum, int[] weight)        {            if (huffmanTree == null || leafNum<1 || weight.Count()<1) return;            for (int i = 0; i < leafNum; i++)            {                huffmanTree[i].Weight = weight[i];            }        }        /// <summary>        /// 获取叶子节点中权重最小的两个节点        /// </summary>        /// <param name="huffmanTree">待操作的赫夫曼</param>        /// <param name="searchNode">要查找的节点数</param>        /// <param name="minIndex1"></param>        /// <param name="minIndex2"></param>        private static void SelectNode(HuffmanTree[] huffmanTree, int searchNode, ref int minIndex1, ref int minIndex2)        {            HuffmanTree minNode1 = null;            HuffmanTree minNode2 = null;            for (int i = 0; i < searchNode; i++)            {                //只查找独根树叶子节点                if (huffmanTree[i].Parent == 0)                {                    //如果为null，则表示当前节叶子节点最小                    if (minNode1 == null)                    {                        minIndex1 = i;                        minNode1= huffmanTree[i];                        continue;                    }                    if (minNode2 == null)                    {                        minIndex2 = i;                        minNode2= huffmanTree[i];                        //交换位置，确保minIndex1为最小                        if (minNode1.Weight >= minNode2.Weight)                        {                             //节点交换                            var temp = minNode1;                            minNode1 = minNode2;                            minNode2 = temp;                            //交换下标                            var tempIndex = minIndex1;                            minIndex1 = minIndex2;                            minIndex2 = tempIndex;                            continue;                        }                    }                    if (minNode1 != null && minNode2 != null)                    {                        if (huffmanTree[i].Weight < minNode1.Weight) //注意，不能是“<=”                        {                            //将min1临时转存给min2                            minNode2 = minNode1;                            minNode1 = huffmanTree[i];                            //记录在数组中的下标                            minIndex2 = minIndex1;                            minIndex1 = i;                        }                        else                        {                            if (huffmanTree[i].Weight < minNode2.Weight)                            {                                 minNode2= huffmanTree[i];                                minIndex2 = i;                            }                        }                    }                }            }        }    }    /// <summary>    /// 赫夫曼树存储结构    /// </summary>    public class HuffmanTree    {        public int Weight { get; set; } //权值        public int Parent { get; set; } //父节点        public int Left { get; set; } //左孩子节点        public int Right { get; set; } //右孩子节点    }}