Abstract
Abstract A hotspot is an axis-aligned square of fixed side length s, where the amount of time a moving entity spends within it is maximised. An exact hotspot of a polygonal trajectory with n edges can be found with time complexity O(n 2). We define a c-approximate hotspot as an axis-aligned square of side length cs, in which the presence duration of the entity is no less than that of an exact hotspot. In this paper we present an algorithm to find a (1 + ϵ)-approximate hotspot of a polygonal trajectory with time complexity O ( n ϕ ϵ log n ϕ ϵ ) O\left( {{{n\phi } \over \varepsilon }\log {{n\phi } \over \varepsilon }} \right) , where ϕ is the ratio of average trajectory edge length to s.
Highlights
Many objects on earth move and huge collections of trajectory data have been collected by tracking these objects with technologies such as GPS devices
A c-duration-approximate hotspot is a square of side length s, whose weight is at least c times the weight of an exact hotspot
We show that r′ is a (1+ ε)-approximate hotspot
Summary
Many objects on earth move and huge collections of trajectory data have been collected by tracking these objects with technologies such as GPS devices. We focus on one of their definitions, which is given as follows: a hotspot is an axis-aligned square of some pre-specified side length, in which the entity (or entities) spends the maximum possible duration, and the presence of the entity in the region can be discontinuous. For this problem and for a trajectory with n edges, they presented an exact O(n2) algorithm. A c-size-approximate (or c-approximate for brevity) hotspot is a square whose weight is at least the weight of an exact hotspot and its side length is c times s
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