#3495

Hard

LeetCode Problem Workspace

Minimum Operations to Make Array Elements Zero

Minimize operations to reduce array elements to zero, focusing on array manipulation, math, and bit operations for efficient computation.

array math bit manipulation

Problem Statement

You are given a list of queries, where each query is a pair [l, r]. Each query represents an array nums consisting of integers from l to r, inclusive. In each operation, you can divide a number by 4 (if divisible), or subtract 1. Your task is to determine the minimum number of operations required to reduce all elements of the array nums to zero for each query.

Return the sum of the results for all queries. The challenge requires you to efficiently compute the minimum number of operations for large ranges of numbers. Be mindful of the problem's constraints, especially the large range of numbers involved, and aim to develop an optimal approach.

Examples

Example 1

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

Output: 3

For queries[0] : For queries[1] : The output is 1 + 2 = 3 .

Example 2

Input: queries = [[2,6]]

Output: 4

For queries[0] : The output is 4.

Constraints

1 <= queries.length <= 105
queries[i].length == 2
queries[i] == [l, r]
1 <= l < r <= 109

Solution Approach

Leverage Logarithmic Operations

The key observation is that for a number x, the number of operations to make it zero can be expressed as floor(log4(x)) + 1. This uses the property of division by 4 and helps minimize unnecessary operations when processing large ranges.

Use Efficient Range Processing

To optimize the computation across a large range of numbers, process the range [l, r] by calculating the number of operations for each number efficiently. This approach minimizes redundant calculations and ensures that the solution scales well with large inputs.

Precompute Logarithmic Values

Precompute the minimum number of operations required for numbers up to the highest value in the queries. This allows for faster lookups and ensures the solution handles large query sets efficiently.

Complexity Analysis

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

The time and space complexity depend on the final implementation. Precomputing logarithmic values may lead to improved time complexity, but the challenge lies in optimizing for the large input size while keeping space usage manageable. Efficient range handling is key for large queries.

What Interviewers Usually Probe

Ability to identify key optimization patterns like logarithmic operations for division-based problems.
Effective use of precomputation and range processing to handle large inputs efficiently.
Clear understanding of bit manipulation and its connection to operations involving powers of 4.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize the solution for large inputs, leading to time limit exceeded errors.
Misunderstanding the problem's pattern, such as incorrectly applying operations without leveraging logarithmic calculations.
Not handling the constraints effectively, especially with high numbers in the range of [l, r], which could lead to inefficiencies.

Follow-up variants

Variation where subtraction is the only allowed operation (no division by 4).
Modified problem where the operations can involve division by other factors (e.g., 2 or 3).
Scenario where the task is to calculate the number of operations to reduce an array to a constant value, rather than zero.

FAQ

How do I minimize operations for the problem 'Minimum Operations to Make Array Elements Zero'?

The solution involves using logarithmic calculations to minimize division operations, specifically using floor(log4(x)) + 1 to calculate the number of operations.

What is the primary technique for optimizing this problem?

Optimizing for this problem involves using logarithmic properties for division operations and efficiently processing large ranges of numbers within the constraints.

How does bit manipulation play a role in this problem?

Bit manipulation is important because it helps in efficiently performing operations like division by powers of 4, which is key to reducing the problem to minimal steps.

What is the time complexity of the optimal solution for this problem?

The time complexity depends on the final approach, but with logarithmic precomputation and efficient range processing, it can be optimized to O(n) or better for large inputs.

How can I handle large numbers in the range [l, r] efficiently?

Efficient handling involves precomputing the minimum number of operations for numbers up to the largest value in the queries, which reduces the need for redundant calculations.

terminal

Solution

Solution 1: Prefix Sum

According to the problem description, suppose the minimum number of operations required to make an element $x$ become $0$ is $p$, where $p$ is the smallest integer such that $4^p > x$.

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class Solution:
    def minOperations(self, queries: List[List[int]]) -> int:
        def f(x: int) -> int:
            res = 0
            p = i = 1
            while p <= x:
                cnt = min(p * 4 - 1, x) - p + 1
                res += cnt * i
                i += 1
                p *= 4
            return res

        ans = 0
        for l, r in queries:
            s = f(r) - f(l - 1)
            mx = f(r) - f(r - 1)
            ans += max((s + 1) // 2, mx)
        return ans

Continue Practicing

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