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.
// C# program of above approach using System;
public class GFG {
// A utility function to get max of two integers
static int max( int x, int y)
{
return (x > y) ? x : y;
}
// Returns the length of the longest palindromic subsequence in seq
static int lps( char [] seq, int i, int j)
{
// Base Case 1: If there is only 1 character
if (i == j) {
return 1;
}
// Base Case 2: If there are only 2 characters and both are same
if (seq[i] == seq[j] && i + 1 == j) {
return 2;
}
// If the first and last characters match
if (seq[i] == seq[j]) {
return lps(seq, i + 1, j - 1) + 2;
}
// If the first and last characters do not match
return max(lps(seq, i, j - 1), lps(seq, i + 1, j));
}
/* Driver program to test above function */
public static void Main()
{
String seq = "GEEKSFORGEEKS" ;
int n = seq.Length;
Console.Write( "The length of the LPS is " + lps(seq.ToCharArray(), 0, n - 1));
}
} // This code is contributed by Rajput-Ji |
The length of the LPS is 5
Dynamic Programming Solution
// A Dynamic Programming based C# Program // for the Egg Dropping Puzzle using System;
class GFG {
// A utility function to get max of
// two integers
static int max( int x, int y)
{
return (x > y) ? x : y;
}
// Returns the length of the longest
// palindromic subsequence in seq
static int lps( string seq)
{
int n = seq.Length;
int i, j, cl;
// Create a table to store results
// of subproblems
int [, ] L = new int [n, n];
// 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 (seq[i] == seq[j] && cl == 2)
L[i, j] = 2;
else if (seq[i] == seq[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 program to test above
functions */
public static void Main()
{
string seq = "GEEKS FOR GEEKS" ;
int n = seq.Length;
Console.Write( "The length of the "
+ "lps is " + lps(seq));
}
} // This code is contributed by nitin mittal. |
The length of the lps is 7
Please refer complete article on Longest Palindromic Subsequence | DP-12 for more details!