You are given a string s consisting only of letters 'a' and 'b'. In a single step you can remove one palindromic subsequence from s.

Return the minimum number of steps to make the given string empty.

A string is a subsequence of a given string if it is generated by deleting some characters of a given string without changing its order. Note that a subsequence does not necessarily need to be contiguous.

A string is called palindrome if is one that reads the same backward as well as forward.

For example:

• s = "ababa", output = 1 (already a palindrome)
• s = "abb", output = 2 ("a" -> "bb")
• s = "baabb", output = 2 ("baab" -> "b")

What is the most efficient way to determine the minimum steps?

Problem: Remove Palindromic Subsequences

Problem Description

You are given a string s consisting only of letters 'a' and 'b'. In a single step you can remove one palindromic subsequence from s.

Return the minimum number of steps to make the given string empty.

A string is a subsequence of a given string if it is generated by deleting some characters of a given string without changing its order. Note that a subsequence does not necessarily need to be contiguous.

A string is called palindrome if is one that reads the same backward as well as forward.

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

Example 3: Input: s = "baabb" Output: 2 Explanation: "baab" -> "b" -> "". Remove palindromic subsequence "baab" then "b".

Solution

The key insight here is that since the string s consists only of 'a' and 'b', any subsequence of the same character is a palindrome. This simplifies the problem significantly.

1. If the string is empty, the answer is 0.
2. If the string is a palindrome, the answer is 1.
3. If the string is not a palindrome, we can always remove all 'a's in one step and all 'b's in another step. Thus, the answer is 2.

def remove_palindromic_subsequences(s: str) -> int:
    if not s:
        return 0
    
    if s == s[::-1]:
        return 1
    else:
        return 2

Explanation:

• An empty string requires 0 steps.
• If the given string s is already a palindrome (reads the same forwards and backward), it can be removed in one step.
• If the given string s is not a palindrome, since s only contains the characters 'a' and 'b', we can remove all 'a's (or 'b's) in the string in one step, which will result in a palindromic subsequence. The remaining string will then consist only of 'b's (or 'a's), which is also a palindromic subsequence, and we can remove it in a second step. Therefore, any non-palindromic string consisting only of 'a' and 'b' can be removed in two steps.

Example Walkthrough:

Example 1: s = "ababa"

1. s is "ababa".
2. s is a palindrome.
3. Return 1.

Example 2: s = "abb"

1. s is "abb".
2. s is not a palindrome.
3. Return 2.

Example 3: s = "baabb"

1. s is "baabb".
2. s is not a palindrome.
3. Return 2.

Time Complexity

The time complexity is O(n), where n is the length of the string s. This is because the palindrome check s == s[::-1] takes O(n) time.

Space Complexity

The space complexity is O(1). No extra space is used that scales with the input size. The slicing operation s[::-1] can create a copy of the string, but in many Python implementations, this operation is optimized and doesn't create a new string in memory for comparison purposes, or the memory is freed immediately after the comparison.