Levenshtein distance with back tracing in PHP - php

Question

I'm trying to align strings in PHP using Levenshtein distance algorithm. The problem is that my back tracing code does not work properly for all cases. For example when the second array has inserted lines at the beginning. Then the back tracing will only go as far as when i=0.
How to properly implement back tracing for Levenshtein distance?
Levenshtein distance, $s and $t are arrays of strings (rows)
function match_rows($s, $t)
{
$m = count($s);
$n = count($t);
for($i = 0; $i <= $m; $i++) $d[$i][0] = $i;
for($j = 0; $j <= $n; $j++) $d[0][$j] = $j;
for($i = 1; $i <= $m; $i++)
{
for($j = 1; $j <= $n; $j++)
{
if($s[$i-1] == $t[$j-1])
{
$d[$i][$j] = $d[$i-1][$j-1];
}
else
{
$d[$i][$j] = min($d[$i-1][$j], $d[$i][$j-1], $d[$i-1][$j-1]) + 1;
}
}
}
// backtrace
$i = $m;
$j = $n;
while($i > 0 && $j > 0)
{
$min = min($d[$i-1][$j], $d[$i][$j-1], $d[$i-1][$j-1]);
switch($min)
{
// equal or substitution
case($d[$i-1][$j-1]):
if($d[$i][$j] != $d[$i-1][$j-1])
{
// substitution
$sub['i'][] = $i;
$sub['j'][] = $j;
}
$i = $i - 1;
$j = $j - 1;
break;
// insertion
case($d[$i][$j-1]):
$ins[] = $j;
$j = $j - 1;
break;
// deletion
case($d[$i-1][$j]):
$del[] = $i;
$i = $i - 1;
break;
}
}

Answer 1

This is not to be nit-picky, but to help you find the answers you want (and improve your implementation).
The algorithm you are using is the Wagner-Fischer algorithm, not the Levenshtein algorithm. Also, Levenshtein distance is not use to align strings. It is strictly a distance measurement.
There are two types of alignment: global and local. Global alignment is used to minimize the distance between two entire strings. Example: global align "RACE" on "REACH", you get "RxACx". The x's are gaps.
In general, this is the Needleman-Wunsch algorithm, which is very similar to the Wagner-Fischer algorithm. Local alignment finds a substring in a long string and minimizes the difference between a short string and a the substring of the long string.
Example: local align "BELL" on "UMBRELLA" and you get "BxELL" aligned on "BRELL". It is the Smith-Waterman algorithm which, again, is very similar to the Wagner-Fischer algorithm.
I hope that this is helpful in allowing you to better define the exact kind of alignment you want.

Answer 2

I think your bug is exactly what you say in your question that it is: you stop as soon as i==0, instead of going all the way to i==0 && j==0. Simply replace this condition:
while($i > 0 && $j > 0)
with
while ($i > 0 || $j > 0)
and you're halfway to your solution. The tricky bit is that if $i==0, then it's incorrect to use the array index $i-1 in the loop body. So you'll also have to change the body of the loop to something more like
while ($i || $j) {
$min = $d[$i][$j]; // or INT_MAX or something
if ($i && $j && $min > $d[$i-1][$j-1]) {
$newi = $i-1;
$newj = $j-1;
$min = $d[$newi][$newj];
}
if ($i && $min > $d[$i-1][$j]) {
$newi = $i-1;
$newj = $j;
$min = $d[$newi][$newj];
}
if ($j && $min > $d[$i][$j-1]) {
$newi = $i;
$newj = $j-1;
$min = $d[$newi][$newj];
}
// What sort of transformation is this?
if ($newi == $i && $newj == $j) {
assert(false); // should never happen
} else if ($newi == $i) {
// insertion
$ins[] = $j;
} else if ($newj == $j) {
// deletion
$del[] = $i;
} else if ($d[$i][$j] != $d[$newi][$newj]) {
// substitution
$sub['i'][] = $i;
$sub['j'][] = $j;
} else {
// identity
}
assert($newi >= 0); assert($newj >= 0);
$i = $newi;
$j = $newj;
}