Minimum Number of Operations to Make Word K-Periodic

Problem Description

You are given a string word of size n, and an integer k such that k divides n. In one operation, you can pick any two indices i and j, that are divisible by k, then replace the substring of length k starting at i with the substring of length k starting at j. Return the minimum number of operations required to make word k-periodic.

Key Insights

• A string is k-periodic if it can be formed by repeating a substring of length k.
• Characters in word can be grouped based on their positions modulo k.
• We need to determine how many operations are needed to make all characters in each group the same.
• The optimal replacement strategy involves replacing characters with the most frequent character in each group.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(k)

Solution

To solve this problem, we can utilize a counting approach. We first group the characters of the string based on their indices modulo k. For each group, we count the frequency of each character. The minimum number of operations required to make the characters in each group identical is the size of the group minus the frequency of the most common character in that group. We sum these minimum operations for all groups to get the final answer.

1. Divide the string into k groups based on their indices modulo k.
2. Count the frequency of each character in each group.
3. For each group, calculate the number of operations required to replace characters to make them identical.
4. Sum the operations across all groups to get the total minimum operations.

Code Solutions