Query Shortest Paths Amidst Growing Discs

The determination of collision-free shortest paths among growing discs has previously been studied for discs with fixed growing rates [1, 2, 3]. Here, we study a more general case of this problem, where: (1) the speeds at which the discs are growing are polynomial functions of degree β, and (2) the source and destination points are given as query points. We show how to preprocess the n growing discs so that, for two given query points s and d, a shortest path from s to d can be found in O(n2 log(βn)) time. The preprocessing time of our algorithm is O(n2 log n+ k log k) where k is the number of intersections between the growing discs and the tangent paths (straight line paths which touch the boundaries of two growing discs). We also prove that k ∈ O(n3β).