Problem Description

In this problem, you are given an array of integers nums and another integer threshold. Your task is to find the smallest positive integer divisor such that when each element of the array nums is divided by this divisor, the results of the division are rounded up to the nearest integer, and these integers are summed up, the sum is less than or equal to the given threshold. The division rounding should round up to the nearest integer greater than or equal to the result of the division. For instance, 7 divided by 3 should be rounded up to 3 (since 7/3 = 2.33, and rounding up gives us 3), and similarly, 10 divided by 2 should be rounded to 5. The problem ensures that there will always be an answer.

Intuition

The key observation here is that as the divisor increases, the resultant sum after the division decreases. This relationship between the divisor and the sum is monotonous; hence we can employ a binary search strategy to efficiently find the smallest divisor which results in a sum that is less than or equal to the threshold.

We can start our binary search with the left boundary (l) set to 1, since we are looking for positive divisors only, and the right boundary (r) set to the maximum value in nums, because dividing by a number larger than the largest number in nums would result in every division result being 1 (or close to 1), which would be the smallest possible sum. We then repeatedly divide the range by finding the midpoint (mid) and testing if the sum of the rounded division results is less than or equal to the threshold. If it is, we know that we can possibly reduce our divisor even further; thus we adjust our right boundary to mid. If the sum exceeds threshold, we need a larger divisor, so we adjust our left boundary to mid + 1.

By iteratively applying this binary search, we converge upon the smallest possible divisor that satisfies the condition. When our search interval is reduced to a single point (l == r), we can return l as our answer, as it represents the smallest divisor for which the sum of the division results does not exceed the threshold.

Learn more about Binary Search patterns.

Solution Approach

The solution uses a binary search algorithm, which is a classic method to find a target value within a sorted array by repeatedly dividing the search interval in half. This pattern relies on the property that the array is monotonous, meaning the function we are trying to optimize (in this case, the sum of the division results) moves in a single direction as we traverse the array.

Binary Search Implementation

The search starts with setting up two pointers, l and r, which represent the boundaries of the potential divisor values. The left boundary l is initialized to 1 (since we are looking for a positive divisor), while the right boundary r is set to the maximum value in nums, as the sum will certainly be less than or equal to the threshold when divided by the maximum value or any larger number since the division results will be close to 1.

The binary search proceeds by computing the midpoint mid = (l + r) >> 1 (this is equivalent to (l + r) / 2, but uses bit shifting for potentially faster computation in some languages). For each element x in nums, we divide x by the current mid and then round up the result to the nearest integer by performing (x + mid - 1) // mid. We take the sum of all these numbers and compare them to threshold.

• If the sum is less than or equal to threshold, then mid can be a valid candidate for the smallest divisor, but there might be a smaller one. Hence, we move the right boundary r to mid to narrow down the search towards smaller divisors.

• On the contrary, if the sum is greater than threshold, mid is too small of a divisor, resulting in a larger sum. Thus, we move the left boundary l to mid + 1 to search among larger divisors.

This search process loops until l becomes equal to r, meaning that the space for search has been narrowed down to a single element which is the smallest possible divisor to meet the condition. As soon as this condition is met, the implementation returns l as the solution.

The patterns and data structures used in this solution are fairly simple—a single array to iterate through the integers of nums, and integer variables to manage the binary search boundaries and the running sum. This results in an efficient solution that requires only O(log(max(nums)) * len(nums)) operations, as each binary search iteration involves summing len(nums) elements, and there can be at most log(max(nums)) such iterations.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach.

Suppose we have nums = [4, 9, 8] and threshold = 6.

Initial Setup

We initialize our binary search bounds: l starts at 1 because we want a positive divisor, and r starts at 9, which is the maximum value in nums, since it is the largest the divisor can be to still produce a sum <= threshold.

First iteration:

• l = 1, r = 9
• mid = (l + r) // 2. Here, (1 + 9) // 2 = 5.
• We now divide each element of nums by mid and round up:
  • For 4: (4 + 5 - 1) // 5 yields 1.
  • For 9: (9 + 5 - 1) // 5 yields 3.
  • For 8: (8 + 5 - 1) // 5 yields 2.
• The sum of rounded divisions is 1 + 3 + 2 = 6 which is equal to threshold.
• Hence, we can still try to find a smaller divisor. So we move the right boundary r to mid, so r = 5.

Second iteration:

• l = 1, r = 5
• mid = (l + r) // 2. Here, (1 + 5) // 2 = 3.
• Divide and round up:
  • For 4: (4 + 3 - 1) // 3 yields 2.
  • For 9: (9 + 3 - 1) // 3 yields 4.
  • For 8: (8 + 3 - 1) // 3 yields 3.
• The sum is 2 + 4 + 3 = 9, which is greater than threshold.
• Because the sum exceeded the threshold, we adjust our left boundary to mid + 1, which makes l = 4.

Third iteration:

• l = 4, r = 5
• mid = (l + r) // 2. Here, (4 + 5) // 2 = 4.
• Divide and round up:
  • For 4: (4 + 4 - 1) // 4 yields 1.
  • For 9: (9 + 4 - 1) // 4 yields 3.
  • For 8: (8 + 4 - 1) // 4 yields 3.
• The sum is 1 + 3 + 3 = 7, which is greater than threshold.
• Therefore, l = mid + 1, making l = 5.

Fourth iteration:

• At this point, l = 5 and r = 5, so we have reached our single point where l and r are equal.
• The loop terminates as the space for search has been narrowed down to a single element.

We conclude that the smallest divisor given nums = [4, 9, 8] and threshold = 6 is 5 since any smaller divisor would result in a sum greater than the threshold, and it is the smallest divisor for which the division results, when summed up, are equal to threshold.

Solution Implementation

Time and Space Complexity

Time Complexity

The given algorithm utilizes a binary search, and within each iteration of this search, it computes the sum of the quotients obtained by dividing each number in the nums list by the current divisor (mid). The binary search varies the divisor mid from 1 to max(nums), shrinking this range by half with each iteration until it converges to the smallest possible divisor that satisfies the condition. Since we're halving the range in which we're searching, the binary search has a logarithmic complexity relative to the range of possible divisors. The complexity of the binary search part can be expressed as O(log M), where M is the maximum value in nums.

For each iteration of binary search, we perform a sum operation which involves iterating over all n elements in nums, calculating the quotient for each element, and summing those quotients up. This process has a linear time complexity concerning the number of elements, which is O(n).

Combining the two parts, the total computational complexity is the product of the complexity of the binary search and the complexity of the sum operation performed at each step. Therefore, the time complexity of the entire algorithm is O(n log M).

Space Complexity

The space complexity of the algorithm refers to the amount of additional memory space that the algorithm requires aside from the input itself. This algorithm uses a constant amount of extra space for variables such as l, r, mid, and the space used for the running total in the sum operation. Since this does not grow with the size of the input list nums, the space complexity is O(1).

Learn more about how to find time and space complexity quickly using problem constraints.