Longest Palindromic Subsequence

LeetCode 516 | Difficulty: Medium

Medium

Problem Description

Given a string s, find the longest palindromic subsequence's length in s.

A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

Example 1:

Input: s = "bbbab"
Output: 4
Explanation: One possible longest palindromic subsequence is "bbbb".

Example 2:

Input: s = "cbbd"
Output: 2
Explanation: One possible longest palindromic subsequence is "bb".

Constraints:

- `1 <= s.length <= 1000`

- `s` consists only of lowercase English letters.

Topics: String, Dynamic Programming

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

String Processing

Consider character frequency counts, two-pointer approaches, or building strings efficiently. For pattern matching, think about KMP or rolling hash. For palindromes, expand from center or use DP.

When to use

Anagram detection, palindrome checking, string transformation, pattern matching.

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Solutions

Solution 1: C# (Best: 120 ms)

Metric	Value
Runtime	120 ms
Memory	39.9 MB
Date	2021-12-20

Solution

public class Solution {
    public int LongestPalindromeSubseq(string s) {
        int n = s.Length;
    int[,] dp = new int[n+1, n+1];
    for(int i=0;i<n;i++)
    {
        for (int j = 0; j < n; j++)
        {
            dp[i+1,j+1] = (s[i] == s[n-1-j]) ? dp[i,j]+1 : Math.Max(dp[i,j+1], dp[i+1,j]);
        }
    }
    return dp[n,n];
    }
}

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Complexity Analysis

Approach	Time	Space
DP (2D)	$O(n × m)$	$O(n × m)$

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Interview Tips

Key Points

• Discuss the brute force approach first, then optimize. Explain your thought process.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.