算法系列15天速成——第十三天 树操作【下】

今天说下最后一种树，大家可否知道，文件压缩程序里面的核心结构，核心算法是什么？或许你知道，他就运用了赫夫曼树。

听说赫夫曼胜过了他的导师，被认为”青出于蓝而胜于蓝“，这句话也是我比较欣赏的，嘻嘻。

一 概念

了解”赫夫曼树“之前，几个必须要知道的专业名词可要熟练记住啊。

1： 结点的权

“权”就相当于“重要度”，我们形象的用一个具体的数字来表示，然后通过数字的大小来决定谁重要，谁不重要。

2： 路径

树中从“一个结点"到“另一个结点“之间的分支。

3： 路径长度

一个路径上的分支数量。

4： 树的路径长度

从树的根节点到每个节点的路径长度之和。

5： 节点的带权路径路劲长度

其实也就是该节点到根结点的路径长度*该节点的权。

6: 树的带权路径长度

树中各个叶节点的路径长度*该叶节点的权的和，常用WPL(Weight Path Length)表示。

二： 构建赫夫曼树

上面说了那么多，肯定是为下面做铺垫，这里说赫夫曼树，肯定是要说赫夫曼树咋好咋好，赫夫曼树是一种最优二叉树,

因为他的WPL是最短的，何以见得？我们可以上图说话。

View Code

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace HuffmanTree
{
class Program
{
static void Main(string[] args)
{
//有四个叶节点
int leafNum = 4;

//赫夫曼树中的节点总数
int huffmanNodes = 2 * leafNum - 1;

//各节点的权值
int[] weight = { 5, 7, 2, 13 };

string[] alphabet = { "A", "B", "C", "D" };

string testCode = "DBDBDABDCDADBDADBDADACDBDBD";

//赫夫曼树用数组来保存，每个赫夫曼都作为一个实体存在
HuffmanTree[] huffman = new HuffmanTree[huffmanNodes].Select(i => new HuffmanTree() { }).ToArray();

HuffmanTreeManager manager = new HuffmanTreeManager();

manager.CreateTree(huffman, leafNum, weight);

string[] huffmanCode = manager.HuffmanCoding(huffman, leafNum);

for (int i = 0; i < leafNum; i++)
{
Console.WriteLine("字符：{0}，权重:{1},编码为:{2}", alphabet[i], huffman[i].weight, huffmanCode[i]);
}

Console.WriteLine("原始的字符串为：" + testCode);

string encode = manager.Encode(huffmanCode, alphabet, testCode);

Console.WriteLine("被编码的字符串为：" + encode);

string decode = manager.Decode(huffman, huffmanNodes, alphabet, encode);

Console.WriteLine("解码后的字符串为：" + decode);
}
}

#region 赫夫曼树结构
/// <summary>
/// 赫夫曼树结构
/// </summary>
public class HuffmanTree
{
public int weight { get; set; }

public int parent { get; set; }

public int left { get; set; }

public int right { get; set; }
}
#endregion

/// <summary>
/// 赫夫曼树的操作类
/// </summary>
public class HuffmanTreeManager
{
#region 赫夫曼树的创建
/// <summary>
/// 赫夫曼树的创建
/// </summary>
/// <param name="huffman">赫夫曼树</param>
/// <param name="leafNum">叶子节点</param>
/// <param name="weight">节点权重</param>
public HuffmanTree[] CreateTree(HuffmanTree[] huffman, int leafNum, int[] weight)
{
//赫夫曼树的节点总数
int huffmanNode = 2 * leafNum - 1;

//初始化节点，赋予叶子节点值
for (int i = 0; i < huffmanNode; i++)
{
if (i < leafNum)
{
huffman[i].weight = weight[i];
}
}

//这里面也要注意，4个节点，其实只要3步就可以构造赫夫曼树
for (int i = leafNum; i < huffmanNode; i++)
{
int minIndex1;
int minIndex2;
SelectNode(huffman, i, out minIndex1, out minIndex2);

//最后得出minIndex1和minindex2中实体的weight最小
huffman[minIndex1].parent = i;
huffman[minIndex2].parent = i;

huffman[i].left = minIndex1;
huffman[i].right = minIndex2;

huffman[i].weight = huffman[minIndex1].weight + huffman[minIndex2].weight;
}

return huffman;
}
#endregion

#region 选出叶子节点中最小的二个节点
/// <summary>
/// 选出叶子节点中最小的二个节点
/// </summary>
/// <param name="huffman"></param>
/// <param name="searchNodes">要查找的结点数</param>
/// <param name="minIndex1"></param>
/// <param name="minIndex2"></param>
public void SelectNode(HuffmanTree[] huffman, int searchNodes, out int minIndex1, out int minIndex2)
{
HuffmanTree minNode1 = null;

HuffmanTree minNode2 = null;

//最小节点在赫夫曼树中的下标
minIndex1 = minIndex2 = 0;

//查找范围
for (int i = 0; i < searchNodes; i++)
{
///只有独根树才能进入查找范围
if (huffman[i].parent == 0)
{
//如果为null，则认为当前实体为最小
if (minNode1 == null)
{
minIndex1 = i;

minNode1 = huffman[i];

continue;
}

//如果为null，则认为当前实体为最小
if (minNode2 == null)
{
minIndex2 = i;

minNode2 = huffman[i];

//交换一个位置，保证minIndex1为最小，为后面判断做准备
if (minNode1.weight > minNode2.weight)
{
//节点交换
var temp = minNode1;
minNode1 = minNode2;
minNode2 = temp;

//下标交换
var tempIndex = minIndex1;
minIndex1 = minIndex2;
minIndex2 = tempIndex;

continue;
}
}
if (minNode1 != null && minNode2 != null)
{
if (huffman[i].weight <= minNode1.weight)
{
//将min1临时转存给min2
minNode2 = minNode1;
minNode1 = huffman[i];

//记录在数组中的下标
minIndex2 = minIndex1;
minIndex1 = i;
}
else
{
if (huffman[i].weight < minNode2.weight)
{
minNode2 = huffman[i];

minIndex2 = i;
}
}
}
}
}
}
#endregion

#region 赫夫曼编码
/// <summary>
/// 赫夫曼编码
/// </summary>
/// <param name="huffman"></param>
/// <param name="leafNum"></param>
/// <param name="huffmanCode"></param>
public string[] HuffmanCoding(HuffmanTree[] huffman, int leafNum)
{
int current = 0;

int parent = 0;

string[] huffmanCode = new string[leafNum];

//四个叶子节点的循环
for (int i = 0; i < leafNum; i++)
{
//单个字符的编码串
string codeTemp = string.Empty;

current = i;

//第一次获取最左节点
parent = huffman[current].parent;

while (parent != 0)
{
//如果父节点的左子树等于当前节点就标记为0
if (current == huffman[parent].left)
codeTemp += "0";
else
codeTemp += "1";

current = parent;
parent = huffman[parent].parent;
}

huffmanCode[i] = new string(codeTemp.Reverse().ToArray());
}
return huffmanCode;
}
#endregion

#region 对指定字符进行压缩
/// <summary>
/// 对指定字符进行压缩
/// </summary>
/// <param name="huffmanCode"></param>
/// <param name="alphabet"></param>
/// <param name="test"></param>
public string Encode(string[] huffmanCode, string[] alphabet, string test)
{
//返回的0,1代码
string encodeStr = string.Empty;

//对每个字符进行编码
for (int i = 0; i < test.Length; i++)
{
//在模版里面查找
for (int j = 0; j < alphabet.Length; j++)
{
if (test[i].ToString() == alphabet[j])
{
encodeStr += huffmanCode[j];
}
}
}

return encodeStr;
}
#endregion

#region 对指定的二进制进行解压
/// <summary>
/// 对指定的二进制进行解压
/// </summary>
/// <param name="huffman"></param>
/// <param name="leafNum"></param>
/// <param name="alphabet"></param>
/// <param name="test"></param>
/// <returns></returns>
public string Decode(HuffmanTree[] huffman, int huffmanNodes, string[] alphabet, string test)
{
string decodeStr = string.Empty;

//所有要解码的字符
for (int i = 0; i < test.Length; )
{
int j = 0;
//赫夫曼树结构模板(用于循环的解码单个字符)
for (j = huffmanNodes - 1; (huffman[j].left != 0 || huffman[j].right != 0); )
{
if (test[i].ToString() == "0")
{
j = huffman[j].left;
}
if (test[i].ToString() == "1")
{
j = huffman[j].right;
}
i++;
}
decodeStr += alphabet[j];
}
return decodeStr;
}

#endregion
}
}