PHP Program for Longest Palindromic Subsequence | DP-12

Given a sequence, find the length of the longest palindromic subsequence in it.

As another example, if the given sequence is “BBABCBCAB”, then the output should be 7 as “BABCBAB” is the longest palindromic subsequence in it. “BBBBB” and “BBCBB” are also palindromic subsequences of the given sequence, but not the longest ones. 

1) Optimal Substructure: 
Let X[0..n-1] be the input sequence of length n and L(0, n-1) be the length of the longest palindromic subsequence of X[0..n-1]. 
If last and first characters of X are same, then L(0, n-1) = L(1, n-2) + 2. 
Else L(0, n-1) = MAX (L(1, n-1), L(0, n-2)). 
Following is a general recursive solution with all cases handled. 

Dynamic Programming Solution

PHP

<?php
// A Dynamic Programming based
// PHP program for LPS problem
// Returns the length of the
// longest palindromic
// subsequence in seq

// A utility function to get
// max of two integers
// function max( $x, $y)
// { return ($x > $y)? $x : $y; }

// Returns the length of the
// longest palindromic
// subsequence in seq
function lps($str)
{
$n = strlen($str);
$i; $j; $cl;

// Create a table to store
// results of subproblems
$L[][] = array(array());


// Strings of length 1 are
// palindrome of length 1
for ($i = 0; $i < $n; $i++)
    $L[$i][$i] = 1;

    // Build the table. Note that
    // the lower diagonal values
    // of table are useless and
    // not filled in the process.
    // The values are filled in a
    // manner similar to Matrix
    // Chain Multiplication DP
    // solution (See
    // https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/).
    // cl is length of substring
    for ($cl = 2; $cl <= $n; $cl++)
    {
        for ($i = 0; $i < $n - $cl + 1; $i++)
        {
            $j = $i + $cl - 1;
            if ($str[$i] == $str[$j] &&
                            $cl == 2)
            $L[$i][$j] = 2;
            else if ($str[$i] == $str[$j])
            $L[$i][$j] = $L[$i + 1][$j - 1] + 2;
            else
            $L[$i][$j] = max($L[$i][$j - 1],
                             $L[$i + 1][$j]);
        }
    }

    return $L[0][$n - 1];
}

// Driver Code
$seq = 'GEEKS FOR GEEKS';
$n = strlen($seq);
echo "The length of the " .
      "LPS is ", lps($seq);

// This code is contributed
// by shiv_bhakt.
?>

Output

The length of the LPS is 7

Please refer complete article on Longest Palindromic Subsequence | DP-12 for more details!