Smallest Palindromic Rearrangement II

To solve the problem, we need to find the k-th lexicographically smallest palindromic permutation of the given string. This involves calculating possible rearrangements of half of the string and determining the exact order, using hash tables and math for efficiency.

Problem Statement

You are given a palindromic string s and an integer k. Your task is to return the k-th lexicographically smallest palindromic permutation of s. If there are fewer than k distinct palindromic permutations, return an empty string.

Note that different rearrangements yielding the same palindromic string are considered identical and counted once. Only rearranging half of the string is necessary because of the inherent symmetry of palindromes.

Examples

Example 1

Input: s = "abba", k = 2

Output: "baab"

Example 2

Input: s = "aa", k = 2

Output: ""

Example 3

Input: s = "bacab", k = 1

Output: "abcba"

Constraints

• 1 <= s.length <= 104
• s consists of lowercase English letters.
• s is guaranteed to be palindromic.
• 1 <= k <= 106

Solution Approach

Analyze Half of the String

Since a palindrome has symmetry, you only need to focus on rearranging the first half of the string. This reduces the problem size significantly, especially when the string is large.

Use Hash Table for Frequency Counting

Utilize a hash table to track the frequency of each character in the first half of the string. This allows for efficient computation of the lexicographical order of the palindromic rearrangements.

Generate Permutations and Count

Generate permutations of the characters in the first half of the string and count how many distinct palindromic permutations can be formed. Return the k-th permutation, if it exists.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the number of possible permutations of the first half of the string. Space complexity is driven by the storage of frequency counts and the need to generate permutations, which can grow based on the input size.

What Interviewers Usually Probe

• The candidate uses hash tables effectively for frequency counting.
• The candidate efficiently reduces the problem space by leveraging palindrome symmetry.
• The candidate handles permutation generation and lexicographical ordering efficiently.

Common Pitfalls or Variants

Common pitfalls

• Failing to account for the symmetry of palindromes, leading to unnecessary computation.
• Incorrectly calculating the number of distinct palindromic permutations.
• Using inefficient permutation generation techniques that do not leverage the character frequencies.

Follow-up variants

• Handling a large string where k is much larger than the number of distinct palindromic permutations.
• Optimizing the space complexity when dealing with long strings.
• Considering the case when there are fewer than k distinct palindromic permutations.

FAQ

What is the key pattern to solve the Smallest Palindromic Rearrangement II problem?

The key pattern is using hash tables and math to efficiently calculate permutations of only the first half of the palindrome, leveraging symmetry.

How does the palindrome symmetry help in solving the problem?

By focusing only on the first half of the string, you reduce the problem size, as the second half is determined by the symmetry of the palindrome.

How can GhostInterview help with generating permutations in this problem?

GhostInterview provides a structured approach to generating lexicographically ordered permutations, ensuring efficiency by leveraging character frequencies.

What happens if there are fewer than k distinct palindromic permutations?

If there are fewer than k distinct palindromic permutations, the correct output is an empty string, as there are not enough permutations to reach the k-th one.

What is the time complexity of the Smallest Palindromic Rearrangement II problem?

The time complexity depends on the number of possible permutations of the first half of the string, which is affected by the character frequencies and the length of the string.