Count K-Reducible Numbers Less Than N

Problem Description

You are given a binary string s representing a number n in its binary form. You are also given an integer k. An integer x is called k-reducible if performing the operation of replacing x with the count of set bits in its binary representation at most k times reduces it to 1. Return the number of positive integers less than n that are k-reducible.

Key Insights

• The operation of counting set bits continues until the number reduces to 1 or until the operation limit k is reached.
• The maximum number of iterations needed is limited by the value of k (max 5).
• Pre-computation may be useful to determine how many integers are k-reducible without redundant calculations.
• The final count must be returned modulo (10^9 + 7).

Space and Time Complexity

Time Complexity: O(n * k), where n is the number represented by the binary string and k is the maximum operations allowed.
Space Complexity: O(n), for storing intermediate results.

Solution

To solve the problem, we can use a dynamic programming approach. We will define a function that counts how many times a number can be reduced to 1 by counting its set bits. We will iterate through all numbers less than n, apply the reduction process, and check if the number can be reduced to 1 within k operations.

1. Convert the binary string s into its integer representation n.
2. For each integer x from 1 to n-1, perform the reduction process:
   • Count the number of set bits in x.
   • Keep track of how many operations are performed until x becomes 1 or until k operations are reached.
3. Increment a counter for each number that can be reduced to 1 within the allowed k operations.
4. Return the counter modulo (10^9 + 7).

Code Solutions