二阶Markov信源编码译码

对二阶Markov信源编码译码

题目概述

设信源可能输出的符号是a, b, c 三个字母，构成一个二阶Markov信源，且各阶条件概率如下，试编写程序可以对任意字母序列（如abbcabcb）进行基于上下文的自适应算术编码，并进行相应的译码。
零阶条件概率：
p(a)=1/3; p(b)=1/3; p©=1/3;
一阶条件概率：
p(a/a)=1/2; p(b/a )=1/4; p(c/a)=1/4;
p(a/b)=1/4; p(b/b)=1/2; p(c/b)=1/4;
p(a/c)=1/4; p(b/c)=1/4; p(c/c)=1/2;
二阶条件概率：
p(a/aa)=3/5; p(b/aa)=1/5; p(c/aa)=1/5;
p(a/ab)=1/4; p(b/ab)=1/4; p(c/ab)=1/2;
p(a/ac)=1/4; p(b/ac)=1/4; p(c/a2)=1/2;
p(a/ba)=1/2; p(b/ba)=1/4; p(c/ba)=1/4;
p(a/bb)=1/5; p(b/bb)=3/5; p(c/bb)=1/5;
p(a/bc)=1/4; p(b/bc)=1/4; p(c/bc)=1/2;
p(a/ca)=1/2; p(b/ca)=1/4; p(c/ca)=1/4;
p(a/cb)=1/2; p(b/cb)=1/4; p(c/cb)=1/4;
p(a/cc)=1/5; p(b/cc)=1/5; p(c/cc)=3/5;

题目分析

该题目是对二阶的Markov信源进行编码，译码。所以每次的概率与之前的概率都有关系，所以可以初步设想采用迭代的方法。

输入

可以创建一个一维数组（可以选择大小确定的数组，也可以使用指针创建动态数组，为了减小难度，选择大小确定的数组），通过cin输入字符。
循环数组，当为0时（注意不同于‘0’），结束循环，得到数组长度。
保证程序的健壮性，要对输入进行合法性判断，通过if语句依次判断是否在a~c， A-C之间。

编码

判断每次的字符是什么，然后每次该字符的概率依次叠加，得到该字符串最后的概率pp。
通过公式求得最低位数digit。
并通过一个for循环，每次划分新的区间，得到最后的范围。
选取上界值然后减去pp的0.01，得到近似概率。
不断*2，得到二进制编码，直到达到最低位数digit。

译码

先将二进制编码转为十进制。
依次从0倒1选取区间，判断是否在相应区间内，如果在就输出相应字符。
知道字符数与输入相同。

问题分析

1.精度问题，当输入过长，long double的精度以及不够用了，所以我选择使用二维数组存储字符，编译译码也对每一组进行编译译码，理论上可以达到无穷精度。
2.方法复杂度是n2,待改进。
3.耦合度过高，所以可以考虑将运算封装到函数中，来降低耦合度。

运行图片

代码

#include <iostream>
#include <string>
#include <cstdlib>
#include <math.h>
using namespace std;
/*
	2019.04.28.21.47
	create by Yu Zhao
	spend three hours
	the methods is not good and the logic has a little defect
	defectss:
		1.Dynamic array
		2.ergodic
		3.Determination of Bits in Decoding**
*/


char str[1000];																				//初始输入数组
char se_str[1000];																			//二进制数组
char str_end[200][5];																		//待定
char letter[3] = {'a','b','c'};																//字母表
int len,flag,flag_deno,flag_deno_sec,digit,se_len;
long double bottom,top,pp,value,start;
long double pro_one = 0.333333;																//近似1/3
long double pro_two[3][3] = {0.5,0.25,0.25,0.25,0.5,0.25,0.25,0.25,0.5};					//pro_two[分母][分子]
long double pro_two_scale[3][4]={0.0,0.5,0.75,1.0,0.0,0.25,0.75,1.0,0.0,0.25,0.5,1.0};		//同上
long double pro_three[3][3][3] ={0.6,0.2,0.2,0.25,0.25,0.5,0.25,0.25,0.5,0.5,0.25,0.25,0.2,0.6,0.2,0.25,0.25,0.5,0.5,0.25,0.25,0.5,0.25,0.25,0.2,0.2,0.6};											//pro_three【一级】【二级】【分子】										//pro_three[一级][二级][分子] 
long double pro_three_scale[3][3][4] ={0.0,0.6,0.8,1.0,0.0,0.25,0.5,1.0,0.0,0.25,0.5,1.0,0.0,0.5,0.75,1.0,0.0,0.2,0.8,1.0,0.0,0.25,0.5,1.0,0.0,0.5,0.75,1.0,0.0,0.5,0.75,1.0,0.0,0.2,0.4,1.0};		//同上																			//pro_three[一级][二级][分子] 

void ensure_flag(int a){																	//映射函数——确定标志位
	if(str[a] == 'a'||str[a] == 'A'){
		flag = 0;
	}
	if(str[a] == 'b'||str[a]=='B'){
		flag = 1;
	}
	if(str[a] == 'c'||str[a]=='C'){
		flag = 2;
	}
}
void ensure_flag_deno(int a){																//同上
	if(a != 0){
		a = a-1;
		if(str[a] == 'a'||str[a] == 'A'){
			flag_deno = 0;
		}
		if(str[a] == 'b'||str[a]=='B'){
			flag_deno = 1;
		}
		if(str[a] == 'c'||str[a]=='C'){
			flag_deno = 2;
		}
	}

}
void ensure_flag_deno_sec(int a){															//同上上
	if(a != 0&&a != 1){
		a = a-2;
		if(str[a] == 'a'||str[a] == 'A'){
			flag_deno_sec = 0;
		}
		if(str[a] == 'b'||str[a]=='B'){
			flag_deno_sec = 1;
		}
		if(str[a] == 'c'||str[a]=='C'){
			flag_deno_sec = 2;
		}
	}
}
bool input(){																				//输入
	int i = 0;
	cout <<"please input the liter(from a to c)\n";
	cin >> str;													//输入目标字符串
	while(str[i]!=0 ){											//读取长度——有一个问题
		len++;
		i++;	
	}
	cout <<"the liters' length is :"<< len<<endl;
	for(i=0;i<len;i++){											//判断输入是否合理
		if(str[i]>='a'&&str[i]<='c'||str[i]>='A'&&str[i]<='C'){
			
		}
		else {
			cout << "your liter is wrong"<<endl;
			exit(1);
			return false;
		}
	}
	cout<<"your liter is right"<<endl;
	i = 0;
	cout <<"the liters you put is :";
	for(int j = 0;j<((len/5)+1);j++){							//改进——转为二维数组——未完成
		for(int k = 0;k<5;k++){
			if(j<len/5){
				str_end[j][k] = str[i];
				i++;
				cout<<str_end[j][k];
			}else{
				str_end[j][k] = str[i];
				i++;
				cout<<str_end[j][k];
				if(i ==len%5 )
					break;
			}
			
			
		}
	}
	cout<<endl;	
	return true;
}


void bianma(){
	cout<<"************编码*************"<<endl;
	pp = 1;
	for(int i = 0;i<len;i++){											//按顺序确定区间——算法过于多余——待改进
//		cout<<"i ="<<i<<endl;
		ensure_flag(i);
		ensure_flag_deno(i);
		ensure_flag_deno_sec(i);
		if(i == 0){
			bottom = bottom + flag*pro_one;
			pp=pp*pro_one;
			top = bottom + pp;
		}else if (i == 1){
			bottom = bottom + pp*pro_two_scale[flag_deno][flag];
			pp = pp*pro_two[flag_deno][flag];
			top = bottom + pp;
		}else {
			bottom = bottom + pp*pro_three_scale[flag_deno_sec][flag_deno][flag];
			pp = pp*pro_three[flag_deno_sec][flag_deno][flag];
			top = bottom + pp;
		}
	}
	
	digit = ceil((log10(1/pp))/0.301);							//确定最低位数
	cout<<"the length of the section is :"<<"	"<<pp<<endl;
	cout<<"the digit is :"<<"	"<<digit<<endl;
	cout<<"the section is "<<"["<<bottom<<","<<top<<")"<<endl;			//待确定
	value = top-0.001*pp;												//取值
	cout<<"the value is :"<<value<<endl;
	cout<<"the end of jiema is :";
	for(i = 0;i<digit;i++){											//转为二进制
		value = value*2;
		se_str[i] = (char)((int)value+48);								//转为字符数组
		if(value>=1)
			value = value-(int)value;
	}
	cout<<se_str<<endl;
}
void yima(){
	cout<<"********************译码********************"<<endl;
	int i = 0;
	start = 0;
	int j = 0;
	value = 0;
	se_len = 0;
	pp = 1;
	bottom = 0;
	top = 0;
//	top = bottom+scale[j+1]*pp;
//	bottom = bottom+pp*scale[j];
	while(se_str[i]!=0 ){														//读取二进制的长度										
		se_len++;
		i++;	
	}
	cout <<"the se_str length is :"<<se_len<<endl;
	for(i=0;i<se_len;i++){														//转为十进制
		value = value+((int)(se_str[i])-48)*pow(2,(-(i+1)));
	}
	cout<<"the value is :"<<value<<endl;
	for(i = 0;i<len;i++){														//对应区间并输出——感觉方法不好——待改进
		if(i==0){
//			cout<<"number01"<<endl;
			if(value>=0&&value<pro_one){
				cout<<letter[0];
				bottom = 0;
				top = pro_one;
				pp = top-bottom;
				flag = 0;
			}
			else if(value>=pro_one&&value<pro_one*2){
				cout<<letter[1]<<endl;
				bottom = pro_one;
				top = pro_one*2;
				pp = top-bottom;
				flag = 1;
			}
			else{
				cout<<letter[2];
				bottom = pro_one*2;
				top = 1;
				pp = top-bottom;
				flag = 2;
			}
			start = bottom;
		}else if(i == 1){
//			cout<<"number02"<<endl;
			for(int j = 0;j<3;j++){
				top = start+pro_two_scale[flag][j+1]*pp;
				bottom = start+pp*pro_two_scale[flag][j];
				if(value>=bottom&&value<top){
					cout<<letter[j];
//					cout<<bottom<<","<<top<<endl;
					pp = top-bottom;
					start = bottom;	
					flag_deno = j;
					break;
				}
			}
		}else{
//			cout<<"number03"<<endl;
			for(int j = 0;j<3;j++){
				top = start+pro_three_scale[flag][flag_deno][j+1]*pp;
				bottom = start+pp*pro_three_scale[flag][flag_deno][j];
				if(value>=bottom&&value<top){
					cout<<letter[j];
					pp = top-bottom;
					start = bottom;	
					flag = flag_deno;
					flag_deno =j;
					break;
				}
			}
		}
	}
	cout<<endl;
}


int main(){
	input();
	bianma();
	yima();
	return 0;
}

其他

另：因为代码是自己一个人写的，所以有很多不完善的地方，请谅解。切忌照抄
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