# Looking for Bird Nests: Identifying Stay Points with Bounded Gaps

## TL;DR

This paper introduces algorithms to identify stay points in trajectories allowing bounded gaps, improving the naturalness of stay point detection in trajectory analysis.

## Contribution

It presents an $O(n \, log \, n)$ exact algorithm for 1D trajectories and a $(1 + \\epsilon)$-approximation for 2D trajectories, addressing the bounded gap stay point problem.

## Key findings

- Efficient exact algorithm for 1D trajectories.
- Approximate algorithm for 2D trajectories with bounded gaps.
- Algorithm complexity depends on trajectory length and gap ratio.

## Abstract

A stay point of a moving entity is a region in which it spends a significant amount of time. In this paper, we identify all stay points of an entity in a certain time interval, where the entity is allowed to leave the region but it should return within a given time limit. This definition of stay points seems more natural in many applications of trajectory analysis than those that do not limit the time of entity's absence from the region. We present an $O(n \log n)$ algorithm for trajectories in $R^1$ with $n$ vertices and a $(1 + \epsilon)$-approximation algorithm for trajectories in $R^2$ to identify all such stay points. Our algorithm runs in $O(kn^2)$, where $k$ depends on $\epsilon$ and the ratio of the duration of the trajectory to the allowed gap time.