Problem Description

The goal is to search within an array of integers (nums) for a contiguous subsequence (a subarray) of length k, such that every element within this subarray is greater than a certain threshold divided by k. You must determine k such that this condition holds true for the subarray. If no such subarray exists, the function should return -1.

Intuition

The solution utilizes a classic problem-solving technique known as 'monotonic stack' to find the maximum k for which the threshold condition is satisfied. This approach is useful for problems involving the next greater or smaller elements in an array.

Here's the approach broken down:

1. We use two monotonic stacks to keep track of the indices of elements in the nums array, which represent the boundaries of potential subarrays. Specifically, we identify the next smaller element to the right (right) and to the left (left) for every element in nums. This helps us determine possible values of k for each element as the potential maximum value for a subarray.

2. We initially fill the left array with -1 and the right array with n (length of nums), since if there's no smaller element found for a particular index, the boundaries are considered to be the start and end of the nums array.

3. We traverse the nums array twice, once from the start to the end and then from the end to the start, to populate the left and right arrays using the monotonic stacks. Whenever we find a smaller element, we update the left or right indices. This way, each element will have its left and right boundaries where a smaller element exists.

4. After determining the left and right boundaries, we iterate through the nums array and calculate k for each element as right[i] - left[i] - 1, which defines the length of the maximum subarray where nums[i] is the smallest number.

5. For each element, we check if nums[i] is greater than threshold / k. If it is, we have found our subarray of size k, which fits the condition, and we return that k.

6. If we go through the entire nums array and find no such element that fits the condition for any subarray, we return -1 to indicate no subarray meets the criteria.

This approach is efficient since it traverses the array a constant number of times, giving us a linear time complexity, which is reasonable for such problems.

Learn more about Stack, Union Find and Monotonic Stack patterns.

Solution Approach

The implementation of the solution follows these main steps:

1. Initialize Boundary Arrays: Two arrays left and right are initialized to keep track of the smaller elements to the left and to the right of each element in nums. They are initially filled with -1 (indicating no smaller element to the left) and n (length of nums, indicating no smaller element to the right), respectively.

2. Fill left Array with Monotonic Stack:

   • A stack stk is used to iterate over the array from left to right.
   • For each element i, the stack is used to find the most recent smaller element. As long as the top of the stack has an element greater than or equal to nums[i], it's popped out.
   • If the stack is not empty after popping, it means the current top of the stack is the index of the nearest smaller element to the left of i. This index is stored in left[i].
   • The index i is then pushed onto the stack.

3. Fill right Array with Monotonic Stack:

   • The stack stk is reused for the reverse iteration, from right to left.
   • A similar process is followed as for the left array. Now, the stack helps in finding the nearest smaller element to the right.
   • This time, when a smaller element is found, the right array is updated with the index of this nearest smaller element to the right of i.
   • Indices are again stored in the stack during the traversal.

4. Iterate to Find the Result:

   • We iterate through nums using the index i.
   • For each element, we compute k using right[i] - left[i] - 1. This k represents the maximum length of the subarray where the current nums[i] would be the smallest element.
   • We then check if this nums[i] is greater than threshold / k. If it is, we've found a valid k and return it.

5. Return -1 if No Subarray Matches:

   • If no such element is found that satisfies the condition for any subarray, -1 is returned after the iteration.

The critical part of this solution is the use of the monotonic stack, which allows us to keep track of the elements in a way that we can access the nearest smaller elements in O(n) time. This stack maintains a sorted order such that for any new element, it's easy to find and record predecessors strictly decreasing in value.

In summary, a combination of monotonic stacks for efficient lookup of boundaries and a linear scan through the array allows us to calculate the size of the required subarray efficiently.

Example Walkthrough

Let us take a smaller example to illustrate the solution approach:

Given:

• An array nums = [3, 1, 5, 4, 2]
• A threshold T = 8

We're looking for the largest k such that there exists a subarray of length k where all elements are greater than T/k.

First, we initialize the left and right arrays:

left =  [-1, -1, -1, -1, -1]
right = [ 5,  5,  5,  5,  5] (5 is the length of `nums`)

Now let’s fill the left array using the monotonic stack:

• Start with an empty stack stk and iterate through nums from index 0 to 4.
• For i = 0 (nums[i] = 3), the stk is empty, so left[0] remains -1. Push i to stk.
• For i = 1 (nums[i] = 1), pop 0 from stk since nums[0] >= nums[1], and push i to stk. So left[1] remains -1.
• Continue the iteration, and the stk will help in finding the nearest smaller element for each index.

left = [-1, -1,  1,  2,  3]
stk = []

Next, let's fill the right array:

• Reuse the stk and iterate through nums reversely, from index 4 to 0.
• For i = 4 (nums[i] = 2), the stk is empty, so right[4] remains 5. Push i to stk.
• For i = 3 (nums[i] = 4), pop 4 from stk since nums[4] < nums[3], and set right[3] = 4. Push 3 to stk.
• Complete the iteration similarly for the rest of the elements.

right = [5,  5,  3,  4,  5]
stk = []

Now find the result:

• Iterate through nums and for each i, calculate k = right[i] - left[i] - 1 and check if nums[i] is greater than T/k.
• For i = 0, k = right[0] - left[0] - 1 = 5 - (-1) - 1 = 5, we check if 3 > 8/5 which is not true.
• For i = 1, k = right[1] - left[1] - 1 = 5 - (-1) - 1 = 5, again 1 > 8/5 is not true.
• For i = 2, k = right[2] - left[2] - 1 = 3 - 1 - 1 = 1, check if 5 > 8 which is not true.
• For i = 3, k = right[3] - left[3] - 1 = 4 - 2 - 1 = 1, check if 4 > 8 which is not true.
• For i = 4, since nums[i] is the last element and it doesn't form a subarray, we do not need to check it.

Since there was no i for which nums[i] was greater than T/k, we return -1.

Therefore, according to this approach and the example provided, the function should return -1 as no subarray meeting the requirements can be found.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code consists of two major for-loops that iterate over the list nums. Both for-loops iterate through each element of the list once from different directions (left-to-right and right-to-left).

The first loop processes each element to find the "left" limit for each number in nums, while the second loop finds the "right" limit. Even though the loops contain while-loops that might suggest a nested loop complexity, the while-loops are only popping elements from the stack that were previously added in the same iteration. Each element is pushed and popped from the stack at most once, which means that the internal while-loops do not increase the asymptotic complexity; instead, they operate in an amortized O(1) time per loop iteration.

After setting the left and right limits, a third loop iterates over each element to calculate k and check if the value meets the condition to determine the valid subarray size.

The complexity of these operations, considering the amortized analysis for the while-loops, is O(n) for each of the three loops, leading to an overall time complexity of O(n).

Space Complexity

The space complexity is determined by the additional memory used by the algorithm aside from the input. Here, we have several data structures to consider:

1. left list, which holds the left boundaries for each element: O(n)
2. right list, which holds the right boundaries for each element: O(n)
3. stk stack, which is at most the size of the list nums at any time: O(n)

Combining these, the total space complexity is O(n). There is no additional memory usage that grows with the size of the input aside from these data structures. Therefore, the overall space complexity is O(n).

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