Remove Palindromic Subsequences

Problem Statement

The task is to determine the minimum number of steps required to completely remove a string s, which is made up solely of the letters 'a' and 'b'. This removal is done by deleting palindromic subsequences from s step by step. A subsequence is a sequence that can be derived from the string by deleting some or no elements without changing the order of the remaining elements, and importantly, these elements do not have to be consecutive. A palindromic sequence reads the same forwards and backwards.

Examples

Example 1

Input:

s = "ababa"

Output:

1

Explanation:

s is already a palindrome, so its entirety can be removed in a single step.

Example 2

Input:

s = "abb"

Output:

2

Explanation:

"abb" -> "bb" -> "".
Remove palindromic subsequence "a" then "bb".

Example 3

Input:

s = "baabb"

Output:

2

Explanation:

"baabb" -> "b" -> "".
Remove palindromic subsequence "baab" then "b".

Constraints

• 1 <= s.length <= 1000
• s[i] is either 'a' or 'b'.

Approach and Intuition

By analyzing the constraints and examples, we can derive some insights and potential approaches:

1. If the entire string s is already a palindrome, the entire string can be removed in one step. This is shown in Example 1 where s = "ababa", which is a palindrome, thus removable in a single step.

2. If the string is not a palindrome, the strategy involves removing maximal palindromic subsequences in each step. The minimum number of steps for any non-palindromic string s can be intuitively understood as follows:

   • If s only consists of one type of character (all 'a's or all 'b's), then it is a palindrome and can be removed in one step.
   • For strings consisting of mixed characters ('a' and 'b'), consider the longest palindromic prefix or suffix that can be removed in one step. The rest of the string constitutes the following steps.

3. For a string like "abb" (from Example 2), the removal process can be visualized as:

   • First, remove the palindromic subsequence "a".
   • Second, remove the remaining "bb".
   • This results in 2 steps, implying any string that is not a palindrome initially can always first have one layer removed (if the non-palindromic parts are on the opposite ends), and then the potentially palindromic part in the subsequent steps.

4. Considering a string with interspersed characters like "baabb" (from Example 3):

   • One approach is to remove "baab", the palindromic subsequence, in the first step.
   • This leaves "b", which is itself a palindrome and thus removable in the next step.
   • Again, this leads to a conclusion of 2 steps.

These insights suggest a general rule that most strings of mixed characters can be resolved to an empty string in a maximum of two steps. Thus, optimizing the identification and removal of initial subsequences is the key to minimizing the steps. The complexity and feasibility of determining the minimum steps rest upon effectively identifying these palindromic subsequences quickly within the constraints provided.

Solutions

class Solution {
    public int deletePalindromeSubsequences(String s) {
        if (s.isEmpty()) {
            return 0;
        }
        if (checkPalindrome(s)) {
            return 1;
        }
        return 2;
    }
    
    private boolean checkPalindrome(String s) {
        int start = 0;
        int end = s.length() - 1;
        while (start < end) {
            if (s.charAt(start) != s.charAt(end)) {
                return false;
            }
            start++;
            end--;
        }
        return true;
    }
}