Before revision:  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.
After revision:  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 present an approximation algorithm with the time complexity O(n \log^3 n) and approximation factor 1/ 4-approximation algorithm for finding axis-parallel cube hotspots of fixed side length for orthogonal three-dimensional trajectories.