Number of Integers With Popcount-Depth Equal to K I

In this problem, you need to count the number of integers within a given range that have a specific popcount-depth. The popcount-depth of an integer is defined by the number of steps it takes to reduce the number to 1 by repeatedly replacing it with the popcount (number of 1-bits in its binary representation). Use dynamic programming with state transitions based on the binary representation of the number.

Problem Statement

You are given two integers, n and k. Your task is to count how many numbers between 1 and n have exactly k steps to reduce to 1 by repeatedly applying the popcount operation.

In this problem, the popcount operation replaces a number x with the number of 1-bits in its binary representation. The process is repeated until x becomes 1. The challenge is to efficiently count how many numbers in the range [1, n] take exactly k steps to reduce to 1.

Examples

Example 1

Input: n = 4, k = 1

Output: 2

The following integers in the range [1, 4] have popcount-depth exactly equal to 1: Thus, the answer is 2.

Example 2

Input: n = 7, k = 2

Output: 3

The following integers in the range [1, 7] have popcount-depth exactly equal to 2: Thus, the answer is 3.

Constraints

• 1 <= n <= 1015
• 0 <= k <= 5

Solution Approach

Dynamic Programming with State Transitions

The main approach is to use dynamic programming (DP) to track the number of valid integers with a specific popcount-depth. By utilizing digit dynamic programming (digit DP), you work with the binary representation of n. Define dp[pos][ones][tight] as the number of ways to choose bits at each position while ensuring that the popcount-depth condition holds.

Binary Representation and Transitions

Transform the problem into a binary representation problem. At each position in the binary representation of n, apply a transition to the next state based on whether you add a 1 or 0 to the current binary number. The 'tight' constraint ensures that the numbers generated do not exceed n.

Math and Combinatorics Integration

This problem combines dynamic programming with combinatorial reasoning. At each stage of the DP, you consider the number of ways bits can be chosen for a given popcount-depth, ensuring the final result is within the given range and satisfies the k-depth condition.

Complexity Analysis

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

The time and space complexity depend on the approach taken for digit DP. In the worst case, the number of states can be O(log n * k), where log n is the number of binary digits in n, and k is the depth condition. Space complexity is similarly related to the number of states tracked in the DP table.

What Interviewers Usually Probe

• Can the candidate correctly identify how to apply dynamic programming to a problem with state transitions in binary representations?
• Does the candidate understand the optimization trade-offs when handling large values of n?
• Is the candidate able to efficiently connect the combinatorial aspect of the problem with dynamic programming?

Common Pitfalls or Variants

Common pitfalls

• Misunderstanding the popcount-depth operation and treating it as a direct count of bits rather than a step-by-step reduction to 1.
• Failing to apply the tight constraint in the digit DP, which could result in generating numbers greater than n.
• Overcomplicating the DP transitions by not considering the efficient use of binary digits and bitwise operations.

Follow-up variants

• Consider a variant where the popcount-depth is required to be exactly k for numbers greater than n.
• Change the problem to count numbers with popcount-depth less than or equal to k, which introduces additional optimization challenges.
• Extend the problem to count numbers within a range [a, b] instead of just from 1 to n, requiring careful boundary handling.

FAQ

How can I optimize my solution for large values of n?

You can optimize the solution by carefully managing the DP table's state transitions and using efficient bitwise operations to minimize redundant calculations.

What is the significance of the popcount-depth in this problem?

The popcount-depth refers to the number of steps it takes to reduce a number to 1 by repeatedly applying the popcount operation (counting the 1-bits in the number's binary representation).

What does 'tight' mean in the digit DP approach?

In digit DP, 'tight' ensures that the number being formed does not exceed the original number n. It helps maintain the constraint that the generated numbers stay within the given range.

How is dynamic programming applied to binary representations in this problem?

Dynamic programming is applied by using the binary representation of n to calculate the number of valid integers. States are tracked for each binary digit position, with transitions based on whether the next bit is 0 or 1.

What happens if I don’t consider the tight constraint in my solution?

Without considering the tight constraint, you might generate numbers greater than n, leading to incorrect results. The tight constraint ensures that the solution remains within the given range.