Approximate Hotspots of Orthogonal Trajectories

TL;DR

This paper introduces an efficient approximation algorithm for identifying hotspots in axis-aligned polygonal trajectories, significantly improving computational time while maintaining a reasonable approximation quality.

Contribution

It presents the first approximation algorithm with sub-quadratic time complexity for axis-aligned trajectory hotspots, reducing computation from quadratic to near-linearithmic time.

Findings

Achieves a 1/2 approximation factor.

Runs in O(n log^3 n) time.

Applicable to axis-parallel trajectories.

Abstract

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 $1/2$.