香农-范诺算法（Shannon-Fano coding）原理

和Huffman-Tree一样，Shannon-Fano coding也是用一棵二叉树对字符进行编码。但在实际操作中呢，Shannon-Fano却没有大用处，这是由于它与Huffman coding相比，编码效率较低的结果（或者说香农-范诺算法的编码平均码字较大）。但是它的基本思路我们还是可以参考下的。

根据Wikipedia上面的解释，我们来看下香农范诺算法的原理：

Shannon-Fano的树是根据旨在定义一个有效的代码表的规范而建立的。实际的算法很简单：

1. 对于一个给定的符号列表，制定了概率相应的列表或频率计数，使每个符号的相对发生频率是已知。
2. 排序根据频率的符号列表，最常出现的符号在左边，最少出现的符号在右边。
3. 清单分为两部分，使左边部分的总频率和尽可能接近右边部分的总频率和。
4. 该列表的左半边分配二进制数字0，右半边是分配的数字1。这意味着，在第一半符号代都是将所有从0开始，第二半的代码都从1开始。
5. 对左、右半部分递归应用步骤3和4，细分群体，并添加位的代码，直到每个符号已成为一个相应的代码树的叶。

示例

香农-范诺编码算法

这个例子展示了一组字母的香浓编码结构（如图a所示）这五个可被编码的字母有如下出现次数:

Symbol	A	B	C	D	E
Count	15	7	6	6	5
Probabilities	0.38461538	0.17948718	0.15384615	0.15384615	0.12820513

从左到右，所有的符号以它们出现的次数划分。在字母B与C之间划定分割线，得到了左右两组，总次数分别为22,17。 这样就把两组的差别降到最小。通过这样的分割, A与B同时拥有了一个以0为开头的码字, C，D，E的码子则为1,如图b所示。 随后, 在树的左半边，于A，B间建立新的分割线，这样A就成为了码字为00的叶子节点，B的码子01。经过四次分割, 得到了一个树形编码。 如下表所示，在最终得到的树中, 拥有最大频率的符号被两位编码, 其他两个频率较低的符号被三位编码。

符号	A	B	C	D	E
编码	00	01	10	110	111

Entropy(熵，平均码字长度): 

Pseudo-code

 1:  begin 2:     count source units 3:     sort source units to non-decreasing order 4:     SF-SplitS 5:     output(count of symbols, encoded tree, symbols) 6:     write output 7:   end 8:   9:  procedure SF-Split(S)10:  begin11:     if (|S|>1) then12:      begin13:        divide S to S1 and S2 with about same count of units14:        add 1 to codes in S115:        add 0 to codes in S216:        SF-Split(S1)17:        SF-Split(S2)18:      end19:  end

想不清楚的朋友可以看下这个网站的模拟程序，很形象，perfect~

香农-范诺算法实现（Shannon-Fano coding implementation in C++）

/************************************************************************//* File Name: Shanno-Fano.cpp*  @Function: Lossless Compression@Author: Sophia Zhang@Create Time: 2012-9-26 20:20@Last Modify: 2012-9-26 20:57*//************************************************************************/#include"iostream"#include "queue"#include "map"#include "string"#include "iterator"#include "vector"#include "algorithm"#include "math.h"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)#define INF 1<<31-1typedef vector<bool> SF_Code;//8 bit code of one charmap<char,SF_Code> SF_Dic; //huffman coding dictionaryint Sumvec[Nsymbols]; //record the sum of symbol count after sortingclass 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; }};bool comp(const HTree* t1, const HTree* t2)//function for sorting{ return (*t1).weight>(*t2).weight; }typedef vector<HTree*> TreeVector;TreeVector TreeArr;//record the symbol count array after sortingvoid Optimize_Tree(int a,int b,HTree& root)//find optimal separate point and optimize tree recursively{ if(a==b)//build one leaf node {  root = *TreeArr[a-1];  return; } else if(b-a==1)//build 2 leaf node {  root.left = TreeArr[a-1];  root.right=TreeArr[b-1];  return; } //find optimizing point x int x,minn=INF,curdiff; for(int i=a;i<b;i++)//find the point that minimize the difference between left and right; this can also be implemented by dichotomy {  curdiff = Sumvec[i]*2-Sumvec[a-1]-Sumvec[b];  if(abs(curdiff)<minn){   x=i;   minn = abs(curdiff);  }  else break;//because this algorithm has monotonicity } HTree*lc = new HTree; HTree *rc = new HTree; root.left = lc;  root.right = rc; Optimize_Tree(a,x,*lc); Optimize_Tree(x+1,b,*rc);}HTree* BuildTree(int* freqency)//create the tree use Optimize_Tree{ int i; for(i=0;i<Nsymbols;i++)//statistic {  if(freqency[i])   TreeArr.push_back(new HTree (NULL,NULL,freqency[i], (char)i)); } sort(TreeArr.begin(), TreeArr.end(), comp); memset(Sumvec,0,sizeof(Sumvec)); for(i=1;i<=TreeArr.size();i++)  Sumvec[i] = Sumvec[i-1]+TreeArr[i-1]->weight; HTree* root = new HTree; Optimize_Tree(1,TreeArr.size(),*root); return root;}/************************************************************************//* Give Shanno Coding to the Shanno Tree/*PS: actually, this generative process is same as Huffman coding/************************************************************************/void Generate_Coding(HTree* root, SF_Code& curcode){ if(root->Isleaf()) {  SF_Dic[root->ch] = curcode;  return; } SF_Code lcode = curcode; SF_Code rcode = curcode; lcode.push_back(false); rcode.push_back(true); Generate_Coding(root->left,lcode); Generate_Coding(root->right,rcode);}int main(){ int freq[Nsymbols] = {0}; char *str = "bbbbbbbccccccaaaaaaaaaaaaaaaeeeeedddddd";//15a,7b,6c,6d,5e //statistic character frequency while (*str!='\0')  freq[*str++]++; //build tree HTree* r = BuildTree(freq); SF_Code nullcode; Generate_Coding(r,nullcode); for(map<char,SF_Code>::iterator it = SF_Dic.begin(); it != SF_Dic.end(); it++) {    cout<<(*it).first<<'\t';    std::copy(it->second.begin(),it->second.end(),std::ostream_iterator<bool>(cout));    cout<<endl;   }  }

Result：

以上面图中的统计数据为例，进行编码。

Reference:

1. Shannon-Fano coding. Wikipedia, the free encyclopedia
2. Claude Elwood Shannon. Wikipedia, the free encyclopedia.
3. C. E. Shannon: A Mathematical Theory of Communication. The Bell System Technical Journal, Vol. 27, July, October, 1948.
4. C. E. Shannon: Prediction and Entropy of Printed English. The Bell System Technical Journal, Vol. 30, 1951.
5. C. E. Shannon: Communication Theory of Secrecy Systems. The Bell System Technical Journal, Vol. 28, 1949.
6. http://www.stringology.org/DataCompression/sf/index_en.html