字符串相似度算法（ Levenshtein Distance算法）

字符串相似度算法（ Levenshtein Distance算法）

题目： 一个字符串可以通过增加一个字符，删除一个字符，替换一个字符得到另外一个字符串，假设，我们把从字符串A转换成字符串B，前面3种操作所执行的最少次数称为AB相似度

如 abc adc 度为 1

ababababa babababab 度为 2

abcd acdb 度为2

字符串相似度算法可以使用 Levenshtein Distance算法(中文翻译：编辑距离算法) 这算法是由俄国科学家Levenshtein提出的。其步骤

Step	Description
1	Set n to be the length of s. Set m to be the length of t. If n = 0, return m and exit. If m = 0, return n and exit. Construct a matrix containing 0..m rows and 0..n columns.
2	Initialize the first row to 0..n. Initialize the first column to 0..m.
3	Examine each character of s (i from 1 to n).
4	Examine each character of t (j from 1 to m).
5	If s[i] equals t[j], the cost is 0. If s[i] doesn't equal t[j], the cost is 1.
6	Set cell d[i,j] of the matrix equal to the minimum of: a. The cell immediately above plus 1: d[i-1,j] + 1. b. The cell immediately to the left plus 1: d[i,j-1] + 1. c. The cell diagonally above and to the left plus the cost: d[i-1,j-1] + cost.
7	After the iteration steps (3, 4, 5, 6) are complete, the distance is found in cell d[n,m].

C++实现如下


#include <iostream>


#include <vector>


#include <string>


using namespace std;




//算法


int ldistance(const string source,const string target)


{


//step 1




int n=source.length();


int m=target.length();


if (m==0) return n;


if (n==0) return m;


//Construct a matrix


typedef vector< vector<int> > Tmatrix;


Tmatrix matrix(n+1);


for(int i=0; i<=n; i++) matrix[i].resize(m+1);




//step 2 Initialize




for(int i=1;i<=n;i++) matrix[i][0]=i;


for(int i=1;i<=m;i++) matrix[0][i]=i;




//step 3


for(int i=1;i<=n;i++)


{


const char si=source[i-1];


//step 4


for(int j=1;j<=m;j++)


{




const char dj=target[j-1];


//step 5


int cost;


if(si==dj){


cost=0;


}


else{


cost=1;


}


//step 6


const int above=matrix[i-1][j]+1;


const int left=matrix[i][j-1]+1;


const int diag=matrix[i-1][j-1]+cost;


matrix[i][j]=min(above,min(left,diag));




}


}//step7


return matrix
[m];


}


int main(){


string s;


string d;


cout<<"source=";


cin>>s;


cout<<"diag=";


cin>>d;


int dist=ldistance(s,d);


cout<<"dist="<<dist<<endl;


}


#include <iostream>


#include <vector>


#include <string>


using namespace std;




//算法


int ldistance(const string source,const string target)


{


//step 1




int n=source.length();


int m=target.length();


if (m==0) return n;


if (n==0) return m;


//Construct a matrix


typedef vector< vector<int> > Tmatrix;


Tmatrix matrix(n+1);


for(int i=0; i<=n; i++) matrix[i].resize(m+1);




//step 2 Initialize




for(int i=1;i<=n;i++) matrix[i][0]=i;


for(int i=1;i<=m;i++) matrix[0][i]=i;




//step 3


for(int i=1;i<=n;i++)


{


const char si=source[i-1];


//step 4


for(int j=1;j<=m;j++)


{




const char dj=target[j-1];


//step 5


int cost;


if(si==dj){


cost=0;


}


else{


cost=1;


}


//step 6


const int above=matrix[i-1][j]+1;


const int left=matrix[i][j-1]+1;


const int diag=matrix[i-1][j-1]+cost;


matrix[i][j]=min(above,min(left,diag));




}


}//step7


return matrix
[m];


}


int main(){


string s;


string d;


cout<<"source=";


cin>>s;


cout<<"diag=";


cin>>d;


int dist=ldistance(s,d);


cout<<"dist="<<dist<<endl;


}