Count Palindromic Subsequences

Given a string of digits s, return the number of palindromic subsequences of s* having length *5. Since the answer may be very large, return it modulo 10<sup>9</sup> + 7.

Note:

• A string is palindromic if it reads the same forward and backward.
• A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.

Example 1:

Input: s = "103301"
Output: 2
Explanation:
There are 6 possible subsequences of length 5: "10330","10331","10301","10301","13301","03301".
Two of them (both equal to "10301") are palindromic.

Example 2:

Input: s = "0000000"
Output: 21
Explanation: All 21 subsequences are "00000", which is palindromic.

Example 3:

Input: s = "9999900000"
Output: 2
Explanation: The only two palindromic subsequences are "99999" and "00000".

Constraints:

• 1 <= s.length <= 10<sup>4</sup>
• s consists of digits.

def count_palindromic_subsequences(s: str) -> int:
    """Counts the number of palindromic subsequences of length 5 in the string s.

    Args:
        s: The input string of digits.

    Returns:
        The number of palindromic subsequences of length 5, modulo 10^9 + 7.
    """
    n = len(s)
    count = 0
    mod = 10**9 + 7

    for i in range(n):
        for j in range(i + 1, n):
            for k in range(j + 1, n):
                for l in range(k + 1, n):
                    for m in range(l + 1, n):
                        if s[i] == s[m] and s[j] == s[l]:
                            count = (count + 1) % mod

    return count


# Example Usage
s = "103301"
result = count_palindromic_subsequences(s)
print(f"The number of palindromic subsequences of length 5 in '{s}' is: {result}")

s = "0000000"
result = count_palindromic_subsequences(s)
print(f"The number of palindromic subsequences of length 5 in '{s}' is: {result}")

s = "9999900000"
result = count_palindromic_subsequences(s)
print(f"The number of palindromic subsequences of length 5 in '{s}' is: {result}")

Explanation:

The solution iterates through all possible combinations of 5 indices (i, j, k, l, m) in the given string s. For each combination, it checks if the subsequence formed by these indices is a palindrome of length 5. A subsequence is a palindrome if the first and last characters are the same (s[i] == s[m]) and the second and fourth characters are the same (s[j] == s[l]). The middle character s[k] can be anything.

If a palindrome is found, the counter count is incremented. The modulo operator % (10**9 + 7) is used to avoid integer overflow since the number of palindromic subsequences can be very large.

Big O Runtime Analysis:

The time complexity of this solution is O(n^5), where n is the length of the string s. This is because there are five nested loops, each iterating up to n times. While this may seem inefficient, given the constraints of the problem (1 <= s.length <= 10^4), it can still run within the time limit for many test cases.

Each loop iterates over the string s, so the number of operations grows polynomially with the size of the string. For each possible combination of indices, a constant amount of work is done to check if the subsequence is a palindrome and increment the count.

Big O Space Usage Analysis:

The space complexity of this solution is O(1). This is because the algorithm uses a fixed number of variables (count, i, j, k, l, m, mod, n) regardless of the size of the input string s. The variables take up a constant amount of space, so the memory usage does not grow with the size of the input.

Edge Cases:

1. Empty String:

   • If the input string s is empty, the code will correctly return 0 because the loops will not execute.

2. String Length Less Than 5:

   • If the length of the input string is less than 5, the code will also correctly return 0 because the loops will not execute.

3. String with No Palindromic Subsequences:

   • If the input string does not contain any palindromic subsequences of length 5, the code will correctly return 0.

4. Very Long String with Many Palindromic Subsequences:

   • If the input string is very long and contains a large number of palindromic subsequences, the modulo operator ensures that the count does not overflow. The count will always be a valid integer within the range [0, 10^9 + 6].

5. String with Non-Digit Characters:

   • The problem statement specifies that the string s consists of digits. If the string contains non-digit characters, the code will still execute, but the result may not be meaningful. It is assumed that the input string is valid.