Subarray With Elements Greater Than Varying Threshold

Subarray With Elements Greater Than Varying Threshold

What is this problem about?

This "Hard" rated problem asks you to find any subarray of length L such that every element in that subarray is strictly greater than a threshold defined as threshold / L. If such a subarray exists, you return its length; otherwise, you return -1. This is a complex optimization problem because the threshold itself changes based on the length of the subarray you are considering.

Why is this asked in interviews?

Companies like Amazon, TikTok, and Google ask this because it tests high-level data structure knowledge. It can be solved using several advanced techniques: Monotonic Stacks, Union-Find, or Segment Trees. It evaluates how a candidate handles "dynamic" constraints where the target condition depends on the size of the window being examined. It's a test of both mathematical insight and technical implementation.

Algorithmic pattern used

There are two popular patterns for this problem. The Monotonic Stack approach involves finding the largest window in which each element is the minimum. For each element arr[i], if it is the minimum in a window of size L, you check if arr[i] > threshold / L. The Union-Find approach involves sorting the elements in descending order and adding them to the array one by one, merging adjacent elements into components. When a component of size L is formed, you check the condition using the smallest element added so far.

Example explanation (use your own example)

Array: [1, 3, 4, 3, 1], threshold = 6.

1. Consider subarray [3, 4, 3]. Length L = 3.
2. Threshold for L=3 is 6 / 3 = 2.
3. Are all elements [3, 4, 3] > 2? Yes.
4. We found a valid subarray. Return length 3. If we chose [4], length L=1, threshold = 6/1=6. Since 4 is not > 6, [4] is not valid alone.

Common mistakes candidates make

One common mistake is trying to use a simple sliding window, which doesn't work because the condition is not monotonic with respect to the window size. Another mistake is failing to correctly calculate the boundaries (left and right) for each element using a monotonic stack. Some candidates also forget that the problem asks for any valid subarray, so they might overcomplicate the logic trying to find the "best" or "longest" one.

Interview preparation tip

To excel in the Subarray With Elements Greater Than Varying Threshold interview question, study the "Largest Rectangle in Histogram" problem. The monotonic stack logic used there is almost identical to what is needed here. Being able to explain why you are finding the "nearest smaller element" to define a range for each number will show the interviewer that you have a deep grasp of stack-based optimizations.