数据结构和算法系列16 哈夫曼树

数据结构和算法系列16 哈夫曼树

阅读目录

一，什么是哈夫曼树
二，如何构建哈夫曼树
四，算法实现

这一篇要总结的是树中的最后一种，即哈夫曼树，我想从以下几点对其进行总结：

1，什么是哈夫曼树？

2，如何构建哈夫曼树？

3，哈夫曼编码？

4，算法实现？

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一，什么是哈夫曼树

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)。

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二，如何构建哈夫曼树

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：





三，哈夫曼编码

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。

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四，算法实现

C#版：

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
}
}

程序输出结果为：





转载：http://www.cnblogs.com/syblogs/articles/2020145.html