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Minimum Operations to Make Array Elements Zero

Difficulty: Hard

Problem Description

You are given a 2D array queries, where queries[i] is of the form [l, r]. Each queries[i] defines an array of integers nums consisting of elements ranging from l to r, both inclusive. In one operation, you can select two integers a and b from the array and replace them with floor(a / 4) and floor(b / 4). Your task is to determine the minimum number of operations required to reduce all elements of the array to zero for each query. Return the sum of the results for all queries.

Key Insights

  • The operation significantly reduces the values of the integers, especially for larger numbers.
  • The time complexity can be optimized by calculating the number of operations required for each integer directly instead of simulating each operation.
  • The maximum integer in the range [l, r] determines the number of operations needed, as larger integers take more iterations to reach zero.
  • Utilizing a greedy approach to always reduce the largest available numbers first will minimize the total number of operations.

Space and Time Complexity

Time Complexity: O(n * log(m)), where n is the number of queries and m is the maximum value of r in the queries.
Space Complexity: O(1), as we only need a constant amount of additional space for calculations.

Solution

To solve the problem, we will calculate the number of operations required for each number in the range [l, r] using the properties of logarithms. Specifically, for any integer x, the number of operations needed to reduce it to zero can be determined by the number of times you can divide it by 4 until it reaches zero. This is equivalent to the logarithm base 4 of x.

We will iterate over each query, calculate the sum of operations for the range [l, r], and return the total.

Code Solutions

def min_operations_to_zero(queries):
    def operations_count(x):
        count = 0
        while x > 0:
            x //= 4
            count += 1
        return count

    total_operations = 0
    for l, r in queries:
        for num in range(l, r + 1):
            total_operations += operations_count(num)
    
    return total_operations

# Example usage:
queries = [[1,2],[2,4]]
print(min_operations_to_zero(queries))  # Output: 3
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