哈夫曼算法原理




Wikipedia上面说的非常清楚了，这里我就不再赘述，直接贴过来了。




1952年, David A. Huffman提出了一个不同的算法，这个算法能够为不论什么的可能性提供出一个理想的树。香农-范诺编码（Shanno-Fano）是从树的根节点到叶子节点所进行的的编码，哈夫曼编码算法却是从相反的方向，暨从叶子节点到根节点的方向编码的。

  2. 为每一个符号建立一个叶子节点，并加上其对应的发生频率
  3. 当有一个以上的节点存在时，进行下列循环:
  2. 把这些节点作为带权值的二叉树的根节点，左右子树为空
  3. 选择两棵根结点权值最小的树作为左右子树构造一棵新的二叉树，且至新的二叉树的根结点的权值为其左右子树上根结点的权值之和。
  4. 把权值最小的两个根节点移除
  5. 将新的二叉树添�队列中.
  1. 最后剩下的节点暨为根节点，此时二叉树已经完毕。





演示样例




 




符号

A

B

C

D

E

计数

15

7

6

6

5

概率

0.38461538

0.17948718

0.15384615

0.15384615

0.12820513


在这样的情况下，D，E的最低频率和分配分别为0和1，分组结合概率的0.28205128。如今最低的一双是B和C，所以他们就分配0和1组合结合概率的0.33333333在一起。这使得BC和DE所以0和1的前面加上他们的代码和它们结合的概率最低。然后离开仅仅是一个和BCDE，当中有前缀分别为0和1，然后结合。这使我们与一个单一的节点，我们的算法是完整的。


可得A代码的代码长度是1比特，其余字符是3比特。


字符

A

B

C

D

E

代码

0

100

101

110

111


    Entropy：









Pseudo-code




 1:  begin
 2:     count frequencies of single characters (source units)
 3:     output(frequencies using Fibonacci Codes of degree 2)
 4:     sort them to non-decreasing sequence
 5:     create a leaf node (character, frequency c, left son = NULL, right son = NULL) 
 6:      of the tree for each character and put nodes into queue F
 7:     while (|F|>=2) do 
 8:      begin
 9:        pop the first two nodes (u1, u2) with the lowest 
10:         frequencies from sorted queue
11:        create a node evaluated with sum of the chosen units, 
12:         successors are chosen units (eps, c(u1)+c(u2), u1, u2)
13:        insert new node into queue
14:      end
15:     node evaluate with way from root to leaf node (left son 1, right son 0)
16:     create output from coded intput characters
17:  end









哈夫曼算法实现






实现的时候我们用vector<bool>记录每一个char的编码；用map<char,vector<bool>>表示整个字典；


就得到了以下的代码（以下有两个，一对一错）：




先放出来这个错的，以示警戒：





/************************************************************************/
/*  File Name: Huffman.cpp
*   @Function: Lossless Compression
    @Author: Sophia Zhang
    @Create Time: 2012-9-26 10:40
    @Last Modify: 2012-9-26 11:10
*/
/************************************************************************/

#include"iostream"
#include "queue"
#include "map"
#include "string"
#include "iterator"
#include "vector"
#include "algorithm"
using namespace std;

#define NChar 8 //suppose use at most 8 bits to describe all symbols
#define Nsymbols 1<<NChar //can describe 256 symbols totally (include a-z, A-Z)
typedef vector<bool> Huff_code;//8 bit code of one char
map<char,Huff_code> Huff_Dic; //huffman coding dictionary

class HTree
{
public :
  HTree* left;
  HTree* right;
  char ch;
  int weight;

  HTree(){left = right = NULL; weight=0;}
  HTree(HTree* l,HTree* r,int w,char c){left = l; right = r;  weight=w; ch=c;}
  ~HTree(){delete left; delete right;}
  int Getweight(){return weight?weight:left->weight+right->weight;}
  bool Isleaf(){return !left && !right; }
  bool operator < (const HTree tr) const
  {
    return tr.weight < weight;
  }
};

HTree* BuildTree(int *frequency)
{
  priority_queue<HTree*> QTree;

  //1st level add characters
  for (int i=0;i<Nsymbols;i++)
  {
    if(frequency[i])
      QTree.push(new HTree(NULL,NULL,frequency[i],(char)i));      
  }

  //build
  while (QTree.size()>1)
  {
    HTree* lc  = QTree.top();
    QTree.pop();
    HTree* rc = QTree.top();
    QTree.pop();

    HTree* parent = new HTree(lc,rc,parent->Getweight(),(char)256);
    QTree.push(parent);
  }
  //return tree root
  return QTree.top();
}

void Huffman_Coding(HTree* root, Huff_code& curcode)
{
  if(root->Isleaf())
  {
    Huff_Dic[root->ch] = curcode;
    return;
  }
  Huff_code& lcode = curcode;
  Huff_code& rcode = curcode;
  lcode.push_back(false);
  rcode.push_back(true);

  Huffman_Coding(root->left,lcode);
  Huffman_Coding(root->right,rcode);
}

int main()
{
  int freq[Nsymbols] = {0};
  char *str = "this is the string need to be compressed";

  //statistic character frequency
  while (*str!='\0')
    freq[*str++]++;

  //build tree
  HTree* r = BuildTree(freq);
  Huff_code nullcode;
  nullcode.clear();
  Huffman_Coding(r,nullcode);

  for(map<char,Huff_code>::iterator it = Huff_Dic.begin(); it != Huff_Dic.end(); it++)
  {
    cout<<(*it).first<<'\t';
    Huff_code vec_code = (*it).second;
    for (vector<bool>::iterator vit = vec_code.begin(); vit!=vec_code.end();vit++)
    {
      cout<<(*vit)<<endl;
    }
  }
}



上面这段代码，我执行出来不正确。在调试的时候发现了一个问题，就是QTree优先队列中的排序出了问题，说来也是，上面的代码中，我重载小于号是对HTree object做的；而实际上我建树时用的是指针，那么优先级队列中元素为指针时该怎么办呢？





那我们将friend bool operator >(Node node1,Node node2)改动为friend bool operator >(Node* node1,Node* node2)，也就是传递的是Node的指针行不行呢?

答案是不能够，由于依据c++primer中重载操作符中讲的“程序猿仅仅能为类类型或枚举类型的操作数定义重载操作符，在把操作符声明为类的成员时，至少有一个类或枚举类型的參数依照值或者引用的方式传递”，也就是说friend bool operator >(Node* node1,Node* node2)形參中都是指针类型的是不能够的。我们仅仅能再建一个类，用当中的重载（）操作符作为优先队列的比較函数。

就得到了以下正确的代码：


/************************************************************************/
/*  File Name: Huffman.cpp
*   @Function: Lossless Compression
    @Author: Sophia Zhang
    @Create Time: 2012-9-26 10:40
    @Last Modify: 2012-9-26 12:10
*/
/************************************************************************/

#include"iostream"
#include "queue"
#include "map"
#include "string"
#include "iterator"
#include "vector"
#include "algorithm"
using namespace std;

#define NChar 8 //suppose use 8 bits to describe all symbols
#define Nsymbols 1<<NChar //can describe 256 symbols totally (include a-z, A-Z)
typedef vector<bool> Huff_code;//8 bit code of one char
map<char,Huff_code> Huff_Dic; //huffman coding dictionary

/************************************************************************/
/* Tree Class elements:
*2 child trees
*character and frequency of current node
*/
/************************************************************************/
class HTree
{
public :
  HTree* left;
  HTree* right;
  char ch;
  int weight;

  HTree(){left = right = NULL; weight=0;ch ='\0';}
  HTree(HTree* l,HTree* r,int w,char c){left = l; right = r;  weight=w; ch=c;}
  ~HTree(){delete left; delete right;}
  bool Isleaf(){return !left && !right; }
};

/************************************************************************/
/* prepare for pointer sorting*/
/*because we cannot use overloading in class HTree directly*/
/************************************************************************/
class Compare_tree
{
public:
  bool operator () (HTree* t1, HTree* t2)
  {
    return t1->weight> t2->weight;
  }
};

/************************************************************************/
/* use priority queue to build huffman tree*/
/************************************************************************/
HTree* BuildTree(int *frequency)
{
  priority_queue<HTree*,vector<HTree*>,Compare_tree> QTree;

  //1st level add characters
  for (int i=0;i<Nsymbols;i++)
  {
    if(frequency[i])
      QTree.push(new HTree(NULL,NULL,frequency[i],(char)i));      
  }

  //build
  while (QTree.size()>1)
  {
    HTree* lc  = QTree.top();
    QTree.pop();
    HTree* rc = QTree.top();
    QTree.pop();

    HTree* parent = new HTree(lc,rc,lc->weight+rc->weight,(char)256);
    QTree.push(parent);
  }
  //return tree root
  return QTree.top();
}

/************************************************************************/
/* Give Huffman Coding to the Huffman Tree*/
/************************************************************************/
void Huffman_Coding(HTree* root, Huff_code& curcode)
{
  if(root->Isleaf())
  {
    Huff_Dic[root->ch] = curcode;
    return;
  }
  Huff_code lcode = curcode;
  Huff_code rcode = curcode;
  lcode.push_back(false);
  rcode.push_back(true);

  Huffman_Coding(root->left,lcode);
  Huffman_Coding(root->right,rcode);
}

int main()
{
  int freq[Nsymbols] = {0};
  char *str = "this is the string need to be compressed";

  //statistic character frequency
  while (*str!='\0')
    freq[*str++]++;

  //build tree
  HTree* r = BuildTree(freq);
  Huff_code nullcode;
  nullcode.clear();
  Huffman_Coding(r,nullcode);

  for(map<char,Huff_code>::iterator it = Huff_Dic.begin(); it != Huff_Dic.end(); it++)
  {
    cout<<(*it).first<<'\t';
    std::copy(it->second.begin(),it->second.end(),std::ostream_iterator<bool>(cout));
    cout<<endl;
  }
}







 ​














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