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 : $y; }// Returns the length of the// longest palindromic // subsequence in seqfunction lps($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 1for ($i = 0; $i < $n; $i++) $L[$i][$i] = 1; // Build the table. Note that // the lower diagonal values // of table are useless and // not filled in the process. // The values are filled in a // manner similar to Matrix // Chain Multiplication DP // solution (See // cl is length of substring for ($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.?> |
The length of the LPS is 7
Please refer complete article on Longest Palindromic Subsequence | DP-12 for more details!
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