##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
Monotonic increasing queue
Use a monotonic increasing queue of indices by prefix sum value. Two key observations:
If and , you never need . Why? Index is closer to future values AND has equal or smaller prefix sum, so it gives a shorter valid subarray.
Once index forms a valid subarray with (meaning ), you can remove from the deque. Future positions would give longer subarrays with .
