Classifying Pattern and Feature Properties to Get a $Θ(n)$ Checker and Reformulation for Sliding Time-Series Constraints

Given, a sequence $\mathcal{X}$ of $n$ variables, a time-series constraint ctr using the Sum aggregator, and a sliding time-series constraint enforcing the constraint ctr on each sliding window of $\mathcal{X}$ of $m$ consecutive variables, we describe a $Θ(n)$ time complexity checker, as well as a $Θ(n)$ space complexity reformulation for such sliding constraint.