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：

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。
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。
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。
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。
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:  begin2:     count source units3:     sort source units to non-decreasing order4:     SF-SplitS5:     output(count of symbols, encoded tree, symbols)6:     write output7:   end8:  9:  procedure SF-Split(S)
10:  begin
11:     if (|S|>1) then
12:      begin
13:        divide S to S1 and S2 with about same count of units
14:        add 1 to codes in S1
15:        add 0 to codes in S2
16:        SF-Split(S1)
17:        SF-Split(S2)
18:      end
19:  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++）

我们由上面的算法可知，需要迭代地寻找一个最优点，使得树中每个节点的左右子树频率总和尽可能相近。

这里我寻找最优化点用的是顺次查找法，其实呢，我们还可以用二分法（dichotomy）达到更高的效率~

[cpp] view plaincopy

/************************************************************************/
/* 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<<31-1
typedef vector<bool> SF_Code;//8 bit code of one char
map<char,SF_Code> SF_Dic; //huffman coding dictionary
int Sumvec[Nsymbols]; //record the sum of symbol count after sorting
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; }
};
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 sorting
void 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。

符号	A	B	C	D	E
计数	15	7	6	6	5

Reference:

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