# Approximate Discontinuous Trajectory Hotspots

## TL;DR

This paper introduces an efficient algorithm to find approximate hotspots in polygonal trajectories, significantly reducing computation time while maintaining near-optimal accuracy in identifying regions of maximum entity presence.

## Contribution

The paper presents a novel $(1 + ext{epsilon})$-approximate hotspot algorithm with improved time complexity for polygonal trajectories.

## Key findings

- Algorithm achieves $O({n o ext{ratio of average edge length to } s} imes rac{1}{ ext{epsilon}} imes ext{log}({n o ext{ratio of average edge length to } s} imes rac{1}{ ext{epsilon}}))$ time complexity.
- Provides a practical method for approximate hotspot detection with theoretical guarantees.
- Reduces computational complexity compared to exact hotspot algorithms.

## Abstract

A hotspot is an axis-aligned square of fixed side length $s$, the duration of the presence of an entity moving in the plane in which is maximised. An exact hotspot of a polygonal trajectory with $n$ edges can be found in $O(n^2)$. Defining a $c$-approximate hotspot as an axis-aligned square of side length $cs$, in which the duration of the entity's presence is no less than that of an exact hotspot, in this paper we present an algorithm to find a $(1 + \epsilon)$-approximate hotspot of a polygonal trajectory with the time complexity $O({n\phi \over \epsilon} \log {n\phi \over \epsilon})$, where $\phi$ is the ratio of average trajectory edge length to $s$.