3495. Minimum Operations to Make Array Elements Zero

3495. Minimum Operations to Make Array Elements Zero

Difficulty: Hard

Topics: Array, Math, Bit Manipulation, Weekly Contest 442

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.
• Replace them with floor(a / 4) and floor(b / 4).

Your task is

3495. Minimum Operations to Make Array Elements Zero

Difficulty: Hard

Topics: Array, Math, Bit Manipulation, Weekly Contest 442

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.
• 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.

Example 1:

• Input: queries = [[1,2],[2,4]]
• Output: 3
• Explanation:
  • For queries[0]:
    • The initial array is nums = [1, 2].
    • In the first operation, select nums[0] and nums[1]. The array becomes [0, 0].
    • The minimum number of operations required is 1.
  • For queries[1]:
    • The initial array is nums = [2, 3, 4].
    • In the first operation, select nums[0] and nums[2]. The array becomes [0, 3, 1].
    • In the second operation, select nums[1] and nums[2]. The array becomes [0, 0, 0].
    • The minimum number of operations required is 2.
  • The output is 1 + 2 = 3.

Example 2:

• Input: queries = [[2,6]]
• Output: 4
• Explanation:
  • For queries[0]:
    • The initial array is nums = [2, 3, 4, 5, 6].
    • In the first operation, select nums[0] and nums[3]. The array becomes [0, 3, 4, 1, 6].
    • In the second operation, select nums[2] and nums[4]. The array becomes [0, 3, 1, 1, 1].
    • In the third operation, select nums[1] and nums[2]. The array becomes [0, 0, 0, 1, 1].
    • In the fourth operation, select nums[3] and nums[4]. The array becomes [0, 0, 0, 0, 0].
    • The minimum number of operations required is 4.
  • The output is 4.

Constraints:

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

Hint:

1. For a number x, the number of "/4" operations to change it to 0 is floor(log4(x)) + 1.
2. Always pair the 2 numbers with the maximum "/4" operations needed.

Solution:

We need to compute the minimum number of operations to reduce all numbers to zero. Here’s the plan:

Approach:

1. Precompute Ranges: For each possible value of k (the number of operations needed to reduce a number to zero), determine the range of numbers that require exactly k operations. This is done by noting that numbers in the range [4(k-1), 4k - 1] require k operations.
2. Process Queries: For each query [l, r], calculate the total number of operations T needed by summing the contributions of all numbers in the range [l, r]. Each number in a precomputed range [low, high] contributes k to T. Also, track the maximum k value M encountered in the range.
3. Compute Operations: The minimum number of operations required for the query is the maximum of ceil(T / 2) and M. This accounts for the fact that each operation can process two numbers, but a number requiring M operations must be processed in at least M operations.
4. Sum Results: Sum the results for all queries and return the total.

Let’s implement this solution in PHP: 3495. Minimum Operations to Make Array Elements Zero

<?php
/**
 * @param Integer[][] $queries
 * @return Integer
 */
function minOperations($queries) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$queries1 = [[1,2],[2,4]];
echo minOperations($queries1) . PHP_EOL; // 3

$queries2 = [[2,6]];
echo minOperations($queries2) . PHP_EOL; // 4
?>

Explanation:

1. Precomputation: The code first precomputes the ranges of numbers that require exactly k operations to become zero. For each k from 1 to 16, it calculates the range [4(k-1), 4k - 1].
2. Query Processing: For each query, it checks which precomputed ranges overlap with the query range [l, r]. For each overlapping range, it calculates the count of numbers in the overlap and adds k * count to the total operations S. It also updates the maximum k value M found in the query range.
3. Operations Calculation: The number of operations needed for the query is the maximum of ceil(S / 2) and M. This ensures that both the total steps and the maximum required steps for any number are considered.
4. Result Summation: The results for all queries are summed up and returned as the final answer.

This approach efficiently processes each query by leveraging precomputed ranges and ensures optimal performance even for large inputs. The complexity is linear with respect to the number of queries, making it suitable for the given constraints.

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