Problem Description

In this problem, we have an array of non-negative integers called nums and two positive integers x and y. We are allowed to perform operations on the array with the aim of reducing all the elements to zero or less. In each operation, we can choose an index i and decrease the element at that index nums[i] by x and simultaneously decrease all other elements by y. The task is to find and return the minimum number of such operations required to achieve the target condition where every element of the array is less than or equal to zero.

Intuition

To arrive at the solution, we need to recognize that decreasing all elements in the array besides the chosen index i by y in each operation will set a pace for how quickly the array can reach <= 0. This pace is constrained by the smallest element since it will hit zero first due to the decrement by y in each operation. The chosen element decremented by x will go down faster, so we want to figure out how often we need to 'hit' an element with the x decrement to bring it down to zero along with the rest of the elements.

A sound approach here is to perform a binary search on the number of operations t we would need since the answer will range between 0 and the maximum element in nums. For any guess t, we calculate if it's possible to bring all elements to less than or equal to zero within t operations. To do this, for each v in nums, we check if v > t * y, meaning the decay by y alone wouldn't be enough. If so, we need additional decrements of x - y to bring it down, which we calculate using ceil((v - t * y) / (x - y)). If the sum of additional decrements for all elements is less than or equal to t, we've found a viable candidate for the number of operations, and we can try to lower our guess. If not, we need to increase it. The first t that satisfies the condition for all elements in nums will be our answer.

This binary search loop continues until we narrow down the number of operations needed to a single value, which is then returned as the minimum number of operations required.

Learn more about Binary Search patterns.

Solution Approach

The solution uses a binary search algorithm to efficiently narrow down the minimum number of operations. The reasoning behind using binary search is that we're dealing with a monotonic condition: if we can bring down all elements to zero or less in t operations, we can also do it in t+1, t+2, etc. So, we try to find the smallest t for which the condition holds true.

Here are the steps of the implemented solution:

1. Define an inner function check(t: int) -> bool that will verify if we can achieve the target with t operations. It performs the following steps:

   • Initialize a counter variable cnt to 0.
   • Loop through each element v in nums.
   • Within the loop, check if v is greater than t * y. If it is, calculate the extra decrements needed by subtracting t * y from v, and then dividing the rest by x - y using ceil. This ensures that we take into account the integer division rounding.
   • Increment cnt by the number computed. This represents the number of times x decrement should be applied to each element.
   • If the sum of decrements cnt is less than or equal to t, return True; otherwise return False.

2. Conduct the binary search between l = 0 and r = max(nums), as the answer is in this range:

   • While l < r, calculate mid which is the midpoint of l and r.
   • Call check(mid) to verify if t = mid is a viable number of operations.
   • If check(mid) returns True, we can potentially reduce the number of operations, so set r to mid.
   • Else, if check(mid) returns False, we need more operations to reach the target, so set l to mid + 1.

3. The loop ends when l is equal to r, meaning we've found the smallest possible t where check(t) is True. Return l as the minimum number of operations required.

The reason we use (l + r) >> 1 to calculate the midpoint is that it is equivalent to (l + r) / 2, but faster in terms of execution as it is a bit shift operation.

The time complexity of this solution is O(nlog(maxElement)), where n is the number of elements in nums. The log factor comes from the binary search, and for each step of the search, we iterate through the array once to check if the current t is viable. The space complexity is O(1) since we only use a few variables for the binary search and the checks.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have an array nums = [3, 19, 10] and the integers x = 7 and y = 5.

The target condition is to reduce all elements to zero or less. According to the solution approach, we can perform a binary search to find the minimum number of operations, t.

Step by step, here's how the solution works with this example:

1. We estimate the bounds [l, r] for the binary search. The upper bound r can be the maximum element in nums, which is 19. We start with l = 0.

2. We define our check function, which will determine if a given number of operations t is sufficient to reduce all the elements in nums to zero or less.

3. We begin the binary search. With l = 0 and r = 19, we calculate the midpoint mid to check. mid is (0 + 19) >> 1 = 9.

4. We call check(9) to verify if t = 9 is a viable number of operations:

   • For nums[0] = 3, since 3 <= 9 * 5, we don't need any extra decrements because y decrements alone will bring it down to zero.
   • For nums[1] = 19, since 19 > 9 * 5, we do the math ceil((19 - 9 * 5) / (7 - 5)) = ceil((19 - 45) / 2) = ceil(-26 / 2) = 0. So no extra decrements are needed.
   • For nums[2] = 10, since 10 > 9 * 5, we do ceil((10 - 9 * 5) / (7 - 5)) = ceil((10 - 45) / 2) = ceil(-35 / 2) = 0.
   • Sum of extra decrements needed is 0 which is <= 9, so check(9) returns True.

5. Since check(9) returns True, we can potentially reduce the number of operations. Now our new upper bound r is set to our current mid, which is 9. We repeat the binary search with new bounds l = 0 and r = 9 and find the midpoint again.

6. After some iterations, we might find that lower values of t also pass the check function (in this example, t = 1 could also work since none of the elements need extra decrements). Binary search will continue until l == r.

7. Once we find the smallest t such that check(t) is True, that will be our answer for the minimum number of operations required.

This detailed example walkthrough shows us how the binary search can efficiently find the minimum number of operations to reduce all array elements to zero or less. In this instance, given x and y are much larger than the array elements, even one operation is sufficient—hence the answer is t = 1.

Solution Implementation

Time and Space Complexity

The given code snippet is implementing a binary search algorithm to find the minimum number of operations needed to reduce certain elements in the array nums so that no element is greater than t * y. To determine if a given target t is feasible, the check function is employed, which computes the number of operations needed for each element in nums. If the element is greater than t * y, the number of operations to reduce it is added to cnt.

The check function is called with a time complexity of O(n) where n is the number of elements in nums. This is because it iteratively checks each element once.

The binary search runs from 0 to the maximum value in nums. The time complexity for binary search is O(log m), where m is the range within which the binary search operates. Here m corresponds to the maximum value in nums.

The overall time complexity of the code is therefore O(n log m), since the binary search is the outer loop and the check function is called within each iteration of this binary search.

As for the space complexity, the space used is constant and does not depend on the input size, besides the input array itself. No additional data structures that grow with input size are created. Thus, the space complexity is O(1).

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