7. Longest Palindromic Subsequence

The problem can be found at the following link: Question Link

Problem Description

Given a string s, return the length of the longest palindromic subsequence.

A subsequence is a sequence derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. A palindromic subsequence reads the same forwards and backwards.

Examples

Example 1:

Input:

s = "bbabcbcab"

Output:

7

Explanation:

The longest palindromic subsequence is "babcbab", which has a length of 7.

Example 2:

Input:

s = "abcd"

Output:

1

Explanation:

The longest palindromic subsequence is any single character, which has a length of 1.

Example 3:

Input:

s = "agbdba"

Output:

5

Explanation:

The longest palindromic subsequence is "abdba", which has a length of 5.

Constraints:

• $1 \leq s.size() \leq 1000$

• The string contains only lowercase letters.

My Approach

DP - Bottom-Up 2D Table

Algorithm Steps:

1. Create a 2D DP table, where dp[i][j] stores the length of the longest palindromic subsequence in substring s[i...j].

2. Initialize diagonal elements (dp[i][i]) to 1, since a single character is always a palindrome of length 1.

3. Iterate over substrings of increasing lengths.

4. For each substring s[i...j]:

   • If s[i] == s[j], the length extends by 2 (both characters can form a palindrome around the inner subsequence s[i+1...j-1]).

   • Otherwise, take the maximum palindromic subsequence length by either excluding s[i] or excluding s[j].

5. Final answer is stored in dp[0][n-1] — the longest palindromic subsequence in the entire string.

Time and Auxiliary Space Complexity

• Expected Time Complexity: $O(N^2)$, as we fill a 2D DP table for all substrings, and each cell is filled in constant time.

• Expected Auxiliary Space Complexity: $O(N^2)$, for storing the DP table, which tracks the longest palindromic subsequence length for each substring.

Code (C++)

class Solution {
public:
    int longestPalinSubseq(string &s) {
        int n = s.size();
        vector<vector<int>> dp(n, vector<int>(n, 0));
        for (int i = n - 1; i >= 0; i--) {
            dp[i][i] = 1;
            for (int j = i + 1; j < n; j++) {
                if (s[i] == s[j])
                    dp[i][j] = 2 + dp[i + 1][j - 1];
                else
                    dp[i][j] = max(dp[i + 1][j], dp[i][j - 1]);
            }
        }
        return dp[0][n - 1];
    }
};

⚡ Alternative Approaches

2️⃣ Space Optimized Dynamic Programming (O(N²) Time, O(N) Space)

Algorithm Steps:

• We only need the current and previous rows, so the 2D table can be reduced to two 1D arrays.

• Iterate over i (backwards) and j (forwards), and fill only the current row using the previous row.

• This reduces space from O(N²) to O(N).

class Solution {
public:
    int longestPalinSubseq(string &s) {
        int n = s.size();
        vector<int> prev(n + 1, 0), curr(n + 1, 0);
        for (int i = 1; i <= n; ++i) {
            for (int j = 1; j <= n; ++j) {
                if (s[i - 1] == s[n - j])
                    curr[j] = 1 + prev[j - 1];
                else
                    curr[j] = max(prev[j], curr[j - 1]);
            }
            swap(prev, curr);
        }
        return prev[n];
    }
};

3️⃣ Recursive + Memoization (O(N²) Time, O(N²) Space)

Algorithm Steps:

• Use recursive DFS with memoization.

• If characters match, extend the palindrome.

• Otherwise, check both possibilities (exclude either character).

• Cache results to avoid redundant work.

class Solution {
public:
    int helper(string &s, int i, int j, vector<vector<int>> &dp) {
        if (i > j) return 0;
        if (i == j) return 1;
        if (dp[i][j] != -1) return dp[i][j];
        if (s[i] == s[j])
            return dp[i][j] = 2 + helper(s, i + 1, j - 1, dp);
        return dp[i][j] = max(helper(s, i + 1, j, dp), helper(s, i, j - 1, dp));
    }

    int longestPalinSubseq(string &s) {
        int n = s.size();
        vector<vector<int>> dp(n, vector<int>(n, -1));
        return helper(s, 0, n - 1, dp);
    }
};

📊 Comparison of Approaches

Approach	⏱️ Time Complexity	🗂️ Space Complexity	✅ Pros	⚠️ Cons
2D DP Table	🟡 O(N²)	🔴 O(N²)	Simple & intuitive	High space usage
Space Optimized 1D DP	🟡 O(N²)	🟢 O(N)	Lower space	Slightly trickier to implement
Recursive + Memoization	🟡 O(N²)	🔴 O(N²)	Natural recursive logic	Recursion overhead

💡 Best Choice?

• ✅ For balanced space and time: Use Space Optimized 1D DP.

• ✅ For simplicity and clarity: Use 2D DP Table.

• ✅ For recursive enthusiasts: Use Recursive + Memoization.

Code (Java)

class Solution {
    public int longestPalinSubseq(String s) {
        int n = s.length();
        int[][] dp = new int[n][n];
        for (int i = n - 1; i >= 0; i--) {
            dp[i][i] = 1;
            for (int j = i + 1; j < n; j++) {
                if (s.charAt(i) == s.charAt(j))
                    dp[i][j] = 2 + dp[i + 1][j - 1];
                else
                    dp[i][j] = Math.max(dp[i + 1][j], dp[i][j - 1]);
            }
        }
        return dp[0][n - 1];
    }
}

Code (Python)

class Solution:
    def longestPalinSubseq(self, s):
        n = len(s)
        dp = [[0] * n for _ in range(n)]
        for i in range(n - 1, -1, -1):
            dp[i][i] = 1
            for j in range(i + 1, n):
                if s[i] == s[j]:
                    dp[i][j] = 2 + dp[i + 1][j - 1]
                else:
                    dp[i][j] = max(dp[i + 1][j], dp[i][j - 1])
        return dp[0][n - 1]

Contribution and Support:

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