Approximate Hotspots of Orthogonal Trajectories

In this paper we study the problem of finding hotspots, i.e. regions in which a moving entity has spent a significant amount of time, for polygonal trajectories. The fastest optimal algorithm, due to Gudmundsson, van Kreveld, and Staals (2013) finds an axis-parallel square hotspot of fixed side length in $O(n^2)$. Limiting ourselves to the case in which the entity moves in a direction parallel either to the $x$ or the $y$-axis, We present an approximation algorithm with the time complexity $O(n \log^3 n)$ and approximation factor $1/2$.