Levenshtein's eidit distance

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From Wiki, the definition of Levenshtein distance is astring metric for measuring the difference between two sequences. Informally, the Levenshtein distance between two words is equal to the number of single-character edits required to change one word into the other.

Mathematically, the Levenshtein distance between two strings $a, b$ is given by $\operatorname{lev}_{a,b}(|a|,|b|)$ where

$\qquad\operatorname{lev}_{a,b}(i,j) = \begin{cases} 0 &, i=j=0 \\ i &, j = 0 \text{ and } i > 0 \\ j &, i = 0 \text{ and } j > 0 \\ \min \begin{cases} \operatorname{lev}_{a,b}(i-1,j) + 1 \\ \operatorname{lev}_{a,b}(i,j-1) + 1 \\ \operatorname{lev}_{a,b}(i-1,j-1) + [a_i \neq b_j] \end{cases} &, \text{ else}\end{cases}$

Note that the first element in the minimum corresponds to deletion(from $a$ to $b$ ), the second to insertion and the third to match or mismatch, depending on whether the respective symbols are the same.

Below is an implementation in java.

package test.LevenshteinDistance;public class LevenshteinDistance {/** * @param args */public static void main(String[] args) {// TODO Auto-generated method stubString s = "kitten";String t = "sitting";LevenshteinDistance ld = new LevenshteinDistance();//int recurlen = ld.recurLevenDistance(s, t);int dynlen = ld.dynLevenDistance(s, t);//System.out.println("The length is : " + recurlen);System.out.println("The length is : " + dynlen);System.out.println(s.length());}/*public int minimum(int a, int b, int c){return Math.min(Math.min(a, b), c);}*/public int minimum(int a, int b, int c){if(a < b && a < c) return a;if(b < a && b < c) return b;return c;}// compute the distance use recursive waypublic int recurLevenDistance(String s, String t){int slen = s.length();int tlen = t.length();int ins = 0; // for recording insert lengthint del = 0; // for recording delete lengthint sub = 0; // for substitution lengthif(slen == 0 && tlen == 0) return 0;if(slen == 0) return tlen;if(tlen == 0) return slen;/*if(s.charAt(slen - 1) == t.charAt(tlen - 1)){sub = recurLevenDistance(s.substring(0, slen - 1), t.substring(0, tlen - 1));}else{sub = recurLevenDistance(s.substring(0, slen - 1), t.substring(0, tlen - 1)) + 1;}*/sub = recurLevenDistance(s.substring(0, slen - 1), t.substring(0, tlen - 1)) + ((s.charAt(slen - 1) == t.charAt(tlen - 1)) ? 0 : 1);ins = recurLevenDistance(s.substring(0, slen - 1), t) + 1;del = recurLevenDistance(s, t.substring(0, tlen - 1)) + 1;return minimum(ins, del, sub);}// compute the distance use dynamic waypublic int dynLevenDistance(String s, String t){int[][] distance = new int[s.length() + 1][t.length() + 1];int i, j;for(i = 0; i <= s.length(); i++)distance[i][0] = i;for(j = 0; j <= t.length(); j++)distance[0][j] = j;for(i = 1; i <= s.length(); i++){for(j = 1; j <= t.length(); j++){distance[i][j] = minimum(distance[i - 1][j] + 1,distance[i][j - 1] + 1,distance[i - 1][j - 1] + ((s.charAt(i - 1) == t.charAt(j - 1)) ? 0 : 1));}}return distance[s.length()][t.length()];}}