Minimum Insertion Steps to Make a String Palindrome

LeetCode 1437 | Difficulty: Hard

Hard

Problem Description

Given a string s. In one step you can insert any character at any index of the string.

Return the minimum number of steps to make s palindrome.

A Palindrome String is one that reads the same backward as well as forward.

Example 1:

Input: s = "zzazz"
Output: 0
Explanation: The string "zzazz" is already palindrome we do not need any insertions.

Example 2:

Input: s = "mbadm"
Output: 2
Explanation: String can be "mbdadbm" or "mdbabdm".

Example 3:

Input: s = "leetcode"
Output: 5
Explanation: Inserting 5 characters the string becomes "leetcodocteel".

Constraints:

- `1 <= s.length <= 500`

- `s` consists 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: 92 ms)

Metric	Value
Runtime	92 ms
Memory	39.1 MB
Date	2021-12-19

Solution

public class Solution {
    public int MinInsertions(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 n-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

• Break the problem into smaller subproblems. Communicate your approach before coding.
• 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.
• LeetCode provides 2 hint(s) for this problem — try solving without them first.

💡 Hints

Hint 1: Is dynamic programming suitable for this problem ?

Hint 2: If we know the longest palindromic sub-sequence is x and the length of the string is n then, what is the answer to this problem? It is n - x as we need n - x insertions to make the remaining characters also palindrome.