Before revision:  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  .
After revision:  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 . We also present a 1/4-approximation algorithm for finding axis-parallel cube hotspots of fixed side length for orthogonal three-dimensional trajectories .