Shortest Subarray - The Idea

Use a monotonic increasing queue of indices by prefix sum value. Two observations:

$1.$ If $P[i_1] \geq P[i_2]$ and $i_1 < i_2$, you never need $i_1$. Why? Index $i_2$ is closer to future $j$ values AND has equal or smaller prefix sum, so it gives a shorter valid subarray.

$2.$ Once index $i$ forms a valid subarray with $j$ (meaning $P[j] - P[i] \geq k$), you can remove $i$ from the deque. Future positions $j' > j$ would give longer subarrays with $i$.