Number of Integers With Popcount-Depth Equal to K II

The problem involves processing a sequence of operations on an array using a popcount function. Precomputing the depth for each number is key to handling multiple range queries and updates efficiently, utilizing a segment tree structure for optimal performance in this problem.

Problem Statement

You are given an integer array nums. For any positive integer x, define the following sequence: start with x, and repeatedly replace it with the number of 1-bits in its binary representation. This sequence will eventually reach the value 1. The number of steps required to reach 1 is called the 'popcount-depth' of x.

You need to answer multiple queries of two types: the first type asks for the count of integers in a subarray whose popcount-depth equals a given K, and the second type updates a value in the array. Efficiently solving these queries, especially for large arrays and multiple queries, requires utilizing a segment tree after precomputing the depth for each number.

Examples

Example 1

Input: nums = [2,4], queries = [[1,0,1,1],[2,1,1],[1,0,1,0]]

Output: [2,1]

Thus, the final answer is [2, 1] .

Example 2

Input: nums = [3,5,6], queries = [[1,0,2,2],[2,1,4],[1,1,2,1],[1,0,1,0]]

Output: [3,1,0]

Thus, the final answer is [3, 1, 0] .

Example 3

Input: nums = [1,2], queries = [[1,0,1,1],[2,0,3],[1,0,0,1],[1,0,0,2]]

Output: [1,0,1]

Thus, the final answer is [1, 0, 1] .

Constraints

• 1 <= n == nums.length <= 105
• 1 <= nums[i] <= 1015
• 1 <= queries.length <= 105
• queries[i].length == 3 or 4

queries[i] == [1, l, r, k] or, queries[i] == [2, idx, val] 0 <= l <= r <= n - 1 0 <= k <= 5 0 <= idx <= n - 1 1 <= val <= 1015

• queries[i] == [1, l, r, k] or,
• queries[i] == [2, idx, val]
• 0 <= l <= r <= n - 1
• 0 <= k <= 5
• 0 <= idx <= n - 1
• 1 <= val <= 1015

Solution Approach

Precompute Depths

First, for each number in the input array, precompute the popcount-depth using the number of 1-bits in its binary representation until reaching 1. This precomputation step allows fast lookup for each query instead of recalculating depths repeatedly.

Segment Tree for Range Queries

To efficiently answer range queries (type 1), construct a segment tree where each node stores the count of integers having specific popcount-depth values within the range. This allows range queries to be answered in logarithmic time, reducing the complexity of brute-force approaches.

Handle Updates with Segment Tree

For update queries (type 2), modify the segment tree to reflect changes in the array. The update operation involves recalculating the depth for the new value and adjusting the segment tree to maintain correct counts for the affected range.

Complexity Analysis

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

The time complexity for precomputing depths is O(n) where n is the length of the array. Each range query and update operation can be answered in O(log n) using the segment tree. Therefore, the overall time complexity depends on the number of queries and the size of the array, but the approach is efficient enough to handle the problem's constraints.

What Interviewers Usually Probe

• Candidate demonstrates an understanding of segment tree construction and updates.
• Candidate efficiently handles large input sizes by using precomputation and segment tree structures.
• Candidate solves the problem with time complexity considerations, focusing on optimal range queries and updates.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to precompute popcount-depths, leading to inefficient query processing.
• Incorrectly updating the segment tree, which can cause inaccurate results for range queries.
• Not handling the complexity of large input sizes, resulting in time limit exceeded errors.

Follow-up variants

• A variant could involve limiting the range of possible popcount-depth values, which may reduce the size of the segment tree.
• Another variant might involve using a different data structure, such as a binary indexed tree, instead of a segment tree.
• A variant could also focus on optimizing the space complexity of the solution, for example, by compressing the segment tree structure.

FAQ

How does the popcount-depth relate to the segment tree in this problem?

The popcount-depth values are precomputed and stored in the segment tree to efficiently handle range queries and updates based on the number of 1-bits in the binary representation of each number.

Why do we precompute the popcount-depth for each number?

Precomputing the popcount-depth allows us to avoid recalculating the depth for every query, making the solution more efficient, especially when dealing with multiple queries.

How do updates work with the segment tree in this problem?

When a value in the array is updated, the segment tree is modified to reflect the new popcount-depth of the updated value. This ensures that subsequent range queries give the correct results.

What is the time complexity of the solution?

The time complexity for precomputing the depths is O(n). Each query or update operation can be handled in O(log n) time, making the solution efficient for large inputs.

How does GhostInterview help with solving this problem?

GhostInterview provides a detailed approach to using segment trees for range queries, helping you optimize solutions for large input sizes, and guides you through common pitfalls in the implementation.