A trajectory is a function that specifies the location of a moving entity through a certain time interval. In this paper we study the problem of finding the hotspots of polygonal two-dimensional trajectories,  regions in which the entity has spent a significant amount of time. The best optimal algorithm, due to Gudmundsson, van Kreveld, and Staals (2013), finds an axis-parallel square hotspot with fixed side length in O(n^2). We present an approximation algorithm with  time complexity O(n \log n) and approximation factor 1/4 for orthogonal trajectories, in which the entity moves in a direction parallel either to the x or to the y-axis  .
 In this paper we study the problem of finding hotspots of polygonal two-dimensional trajectories, i.e. regions in which a moving entity has spent a significant amount of time. 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). We present an approximation algorithm with the time complexity O(n \log n) and approximation factor 1/4 for orthogonal trajectories, in which the entity moves in a direction parallel either to the x or to the y-axis . We also present a 1/4-approximation algorithm for finding axis-parallel cube hotspots of fixed side length for orthogonal three-dimensional trajectories .
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