Why Monotonic Deque Works

For sliding window maximum:

$1.$ Store indices (not values) in the deque, so you can check if elements are in the current window.

$2.$ Maintain decreasing order: nums[deque.front()] >= nums[deque.back()].

$3.$ When adding index $i$, remove indices from the back while nums[i] >= nums[deque.back()]. Smaller elements can't be the maximum while $i$ is in the window.

$4.$ Remove indices from the front that are outside the window (index $< i - k + 1$).

$5.$ The front is always the maximum of the current window.