理论文字来源于网络
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)。
1、路径和路径长度
在一棵树中,从一个结点往下可以达到的孩子或子孙结点之间的通路,称为路径。通路中分支的数目称为路径长度。若规定根结点的层数为1,则从根结点到第L层结点的路径长度为L-1。
2、结点的权及带权路径长度
若将树中结点赋给一个有着某种含义的数值,则这个数值称为该结点的权。结点的带权路径长度为:从根结点到该结点之间的路径长度与该结点的权的乘积。
3、树的带权路径长度
树的带权路径长度规定为所有叶子结点的带权路径长度之和,记为WPL
哈夫曼树的构造
哈夫曼树的构造
(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
通俗点说(原创)
就是权值为a,b,c,d,e,f,g,h;的几个数,先从里面选两个最小的数a,b(假定的),组成一个节点ab,然后再从
下面是哈夫曼树的基本操作ab,c,d,e,f,g,h中再选两个最小的数组成节点,一个节点的左儿子值为0,有儿子为1;然后每个数向上到头决定自己的值;
#include
using namespace std;
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
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++) {
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;
}
}
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;
//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;
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;
}
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
}
}
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;
}
}
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;
}
}