There is indeed an algorithm that can do this in O(N) time with no dependence on the window size w. The idea is to use a clever data structure that supports the following operations:
- Enqueue, which adds a new element to the structure,
- Dequeue, which removes the oldest element from the structure, and
- Find-max, which returns (but does not remove) the minimum element from the structure.
This is essentially a queue data structure that supports access (but not removal) of the maximum element. Amazingly, as seen in this earlier question, it is possible to implement this data structure such that each of these operations runs in amortized O(1) time. As a result, if you use this structure to enqueue w elements, then continuously dequeue and enqueue another element into the structure while calling find-max as needed, it will take only O(n + Q) time, where Q is the number of queries you make. If you only care about the minimum of each window once, this ends up being O(n), with no dependence on the window size.
Hope this helps!
