哈夫曼树和哈夫曼编码

给定n个权值作为n个叶子结点，构造一棵二叉树，若带权路径长度达到最小，称这样的二叉树为最优二叉树，也称为哈夫曼树(Huffman tree)。

哈夫曼树的构造

  哈夫曼树的构造

假设有n个权值，则构造出的哈夫曼树有n个叶子结点。 n个权值分别设为 w1、w2、…、wn，则哈夫曼树的构造规则为：

(1) 将w1、w2、…，wn看成是有n 棵树的森林(每棵树仅有一个结点)；

(2) 在森林中选出两个根结点的权值最小的树合并，作为一棵新树的左、右子树，且新树的根结点权值为其左、右子树根结点权值之和；

(3)从森林中删除选取的两棵树，并将新树加入森林；

(4)重复(2)、(3)步，直到森林中只剩一棵树为止，该树即为所求得的哈夫曼树。

程序部分如下：

#include <iostream>
#include <stdlib.h>
using namespace std;
const int MaxValue = 10000; //初始设定的权值最大值
const int MaxBit = 4; //初始设定的最大编码位数
const int MaxN = 10; //初始设定的最大结点个数
struct HaffNode //哈夫曼树的结点结构
{
	int weight; //权值
	int flag; //标记
	int parent; //双亲结点下标
	int leftChild; //左孩子下标
	int rightChild; //右孩子下标
};
struct Code //存放哈夫曼编码的数据元素结构
{
	int bit[MaxN]; //数组
	int start; //编码的起始下标
	int weight; //字符的权值
};
void Haffman(int weight[], int n, HaffNode haffTree[])
//建立叶结点个数为n权值为weight的哈夫曼树haffTree
{
	int j, m1, m2, x1, x2;
	//哈夫曼树haffTree初始化。n个叶结点的哈夫曼树共有2n-1个结点
	for(int i = 0; i < 2 * n - 1 ; i++)
	{
		if(i < n)
			haffTree[i].weight = weight[i];
		else 
			haffTree[i].weight = 0;
		haffTree[i].parent = 0;
		haffTree[i].flag = 0;
		haffTree[i].leftChild = -1;
		haffTree[i].rightChild = -1;
	}
	//构造哈夫曼树haffTree的n-1个非叶结点
	for(int i = 0;i < n-1;i++)
	{
		m1 = m2 = MaxValue;
		x1 = x2 = 0;
		for(j = 0; j < n+i;j++)//找出剩下的两个比较小的节点
		{
			if (haffTree[j].weight < m1 && haffTree[j].flag == 0)
			{
				m2 = m1;
				x2 = x1;
				m1 = haffTree[j].weight;
				x1 = j;
			}
			else
				if(haffTree[j].weight < m2 && haffTree[j].flag == 0)
				{
					m2 = haffTree[j].weight;
					x2 = j;
				}
		}
		//将找出的两棵权值最小的子树合并为一棵子树
		haffTree[x1].parent = n+i;
		haffTree[x2].parent = n+i;
		haffTree[x1].flag = 1;
		haffTree[x2].flag = 1;
		haffTree[n+i].weight = haffTree[x1].weight+haffTree[x2].weight;
		haffTree[n+i].leftChild = x1;
		haffTree[n+i].rightChild = x2;
	}
}
void HaffmanCode(HaffNode haffTree[], int n, Code haffCode[])
//由n个结点的哈夫曼树haffTree构造哈夫曼编码haffCode
{
	Code *cd = new Code;
	int child, parent;
	//求n个叶结点的哈夫曼编码
	for(int i = 0; i < n; i++)
	{
		cd->start = n-1; //不等长编码的最后一位为n-1
		cd->weight = haffTree[i].weight; //取得编码对应权值的字符
		child = i;
		parent = haffTree[child].parent;
		//由叶结点向上直到根结点
		while(parent != 0)
		{
			if(haffTree[parent].leftChild == child)
				cd->bit[cd->start] = 0; //左孩子结点编码0
			else
				cd->bit[cd->start] = 1;//右孩子结点编码1
			cd->start--;
			child = parent;
			parent = haffTree[child].parent;
		}
		//保存叶结点的编码和不等长编码的起始位
		for(int j = cd->start+1; j < n; j++)
			haffCode[i].bit[j] = cd->bit[j];
		haffCode[i].start = cd->start;
		haffCode[i].weight = cd->weight; //保存编码对应的权值
	}
}
int main()
{
	int i, j, n = 4;
	int weight[] = {1,3,5,7};
	HaffNode *myHaffTree = new HaffNode[2*n+1];
	Code *myHaffCode = new Code[n];
	if(n > MaxN)
	{
		cout << "定义的n越界，修改MaxN! " << endl;
		exit(0);
	}
	Haffman(weight, n, myHaffTree);
	HaffmanCode(myHaffTree, n, myHaffCode);
	//输出每个叶结点的哈夫曼编码
	for(i = 0; i < n; i++)
	{
		cout << "Weight = " << myHaffCode[i].weight << " Code = ";
		for(j = myHaffCode[i].start+1; j < n; j++)
			cout << myHaffCode[i].bit[j];
		cout << endl;
	}
	system("pause");
	return 0;
}