The safest path via safe zones

We define and study Euclidean and spatial network variants of a new path finding problem: given a set of safe zones, find paths that minimize the distance traveled outside the safe zones. In this problem, the entire space with the exception of the safe zones is unsafe, but passable, and it differs from problems that involve unsafe regions to be strictly avoided. As a result, existing algorithms are not effective solutions to the new problem. To solve the Euclidean variant, we devise a transformation of the continuous data space with safe zones into a discrete graph upon which shortest path algorithms apply. A naive transformation yields a very large graph that is expensive to search. In contrast, our transformation exploits properties of hyperbolas in the Euclidean space to safely eliminate graph edges, thus improving performance without affecting the shortest path results. To solve the spatial network variant, we propose a different graph-to-graph transformation that identifies critical points that serve the same purpose as do the hyperbolas, thus avoiding the creation of extraneous edges. This transformation can be extended to support a weighted version of the problem, where travel in safe zones has non-zero cost. We conduct extensive experiments using both real and synthetic data. The results show that our approaches outperform baseline approaches by more than an order of magnitude in graph construction time, storage space and query response time.