Unique Length-alindromic Subsequences

1930. Unique Length-3 Palindromic Subsequences

Difficulty: Medium

Topics: Hash Table, String, Bit Manipulation, Prefix Sum

Given a string s, return the number of unique palindromes of length three that are a subsequence of s.

Note that even if there are multiple ways to obtain the same subsequence, it is still only counted once.

A palindrome is a string that reads the same forwards and backwards.

A subsequence of a string is a new string generated from the original string with some characters (can be none) deleted without changing the relative order of the remaining characters.

• For example, "ace" is a subsequence of "abcde".

Example 1:

• Input: s = "aabca"
• Output: 3
• Explanation: The 3 palindromic subsequences of length 3 are:
  • "aba" (subsequence of "aabca")
  • "aaa" (subsequence of "aabca")
  • "aca" (subsequence of "aabca")

Example 2:

• Input: s = "adc"
• Output: 0
• Explanation: There are no palindromic subsequences of length 3 in "adc".

Example 3:

• Input: s = "bbcbaba"
• Output: 4
• Explanation: The 4 palindromic subsequences of length 3 are:
  • "bbb" (subsequence of "bbcbaba")
  • "bcb" (subsequence of "bbcbaba")
  • "bab" (subsequence of "bbcbaba")
  • "aba" (subsequence of "bbcbaba")

Constraints:

• 3 5
• s consists of only lowercase English letters.

Hint:

1. What is the maximum number of length-3 palindromic strings?
2. How can we keep track of the characters that appeared to the left of a given position?

Solution:

We can use an efficient algorithm that leverages prefix and suffix character tracking to count all valid palindromic subsequences.

Approach

1. Track Prefix Characters:
   Use an array to store the set of characters encountered to the left of each position in the string. This will help in efficiently checking if a character can form the first part of a palindromic subsequence.

2. Track Suffix Characters:
   Use another array to store the set of characters encountered to the right of each position in the string. This will help in efficiently checking if a character can form the third part of a palindromic subsequence.

3. Count Palindromic Subsequences:
   For each character in the string, consider it as the middle character of a length-3 palindrome. Check for all valid combinations of prefix and suffix characters to determine unique palindromes.

4. Store Results:
   Use a hash set to store unique palindromic subsequences, ensuring no duplicates.

Let's implement this solution in PHP: 1930. Unique Length-3 Palindromic Subsequences

<?php /**
 * @param String $s
 * @return Integer
 */
function countPalindromicSubsequence($s) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo countPalindromicSubsequence("aabca") . PHP_EOL; // Output: 3
echo countPalindromicSubsequence("adc") . PHP_EOL;   // Output: 0
echo countPalindromicSubsequence("bbcbaba") . PHP_EOL; // Output: 4
?>

Explanation:

1. Prefix Array:

   • For each character at position i, prefix[i] stores all distinct characters encountered before index i.

2. Suffix Array:

   • For each character at position i, suffix[i] stores all distinct characters encountered after index i.

3. Middle Character:

   • Consider each character as the middle of the palindrome. For each combination of prefix and suffix characters that matches the middle character, form a length-3 palindrome.

4. Hash Map:

   • Use an associative array ($uniquePalindromes) to store unique palindromes, ensuring duplicates are not counted.

Complexity

• Time Complexity: O(n)

  • Traversing the string twice to compute the prefix and suffix arrays.
  • A third traversal checks valid palindromic subsequences.

• Space Complexity: O(n)

  • For prefix and suffix arrays.

Output

The code produces the correct results for the given examples:

• Input: "aabca" → Output: 3
• Input: "adc" → Output: 0
• Input: "bbcbaba" → Output: 4

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