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To solve this problem, we need to find the maximum number of non-overlapping palindromic substrings of length at least 2 in a given string. The key insight here is to use a greedy approach to select the shortest possible palindromic substrings (length 2 or 3) as early as possible, which allows us to fit more substrings into the string.

Approach

1. Greedy Selection:

   • Prefer palindromic substrings of length 2 first because they use fewer characters and allow more substrings to be formed.
   • If a length 2 palindrome is not possible starting at the current position, check for a length 3 palindrome (since it is the next shortest valid length).
   • If neither is possible, move to the next character.

2. Algorithm:

   • Iterate through the string from left to right.
   • For each position, check if the current and next character form a palindrome (length 2). If yes, increment the count and skip the next character.
   • If not, check if the current character and the one two positions ahead form a palindrome (length 3). If yes, increment the count and skip the next two characters.
   • If neither, move to the next character.

This approach ensures we maximize the number of valid substrings as it prioritizes shorter palindromes, leading to an optimal solution.

Solution Code

def max_palindromic_substrings(s):
    count = 0
    n = len(s)
    i = 0
    while i < n:
        if i + 1 < n and s[i] == s[i+1]:
            count += 1
            i += 2
        elif i + 2 < n and s[i] == s[i+2]:
            count += 1
            i += 3
        else:
            i += 1
    return count

Explanation

• Time Complexity: O(n) where n is the length of the string. We process each character at most once.
• Space Complexity: O(1) since we only use a few variables for counting and iteration, no additional data structures are needed.

This solution efficiently handles all cases, including edge cases like empty strings, strings of length 1, and strings with no valid palindromic substrings. It ensures we get the maximum number of non-overlapping palindromic substrings of length at least 2.

Example: For the string "aabbaa", the algorithm returns 3 (substrings "aa", "bb", "aa"), which is the maximum possible. For the string "ababa", it returns 1 (substring "aba"), which is correct as no more valid substrings can be formed without overlapping.

This approach is optimal and works well for large input sizes up to 100,000 characters. It is both time and space efficient, making it suitable for the problem's constraints.

Answer: The function max_palindromic_substrings returns the desired result. For the given problem, the code should be as written above. The final answer is the code provided.

def max_palindromic_substrings(s):
    count = 0
    n = len(s)
    i = 0
    while i < n:
        if i + 1 < n and s[i] == s[i+1]:
            count += 1
            i += 2
        elif i + 2 < n and s[i] == s[i+2]:
            count += 1
            i += 3
        else:
            i += 1
    return count

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