Grid Pathfinding on the 2k Neighborhoods.

Grid pathfinding, an old AI problem, is central for the development of navigation systems for autonomous agents. A surprising fact about the vast literature on this problem is that very limited neighborhoods have been studied. Indeed, only the 4- and 8-neighborhoods are usually considered, and rarely the 16-neighborhood. This paper describes three contributions that enable the construction of effective grid path planners for extended 2(k)-neighborhoods. First, we provide a simple recursive definition of the 2(k)-neighborhood in terms of the 2(k-1)-neighborhood. Second, we derive distance functions, for any k > 1, which allow us to propose admissible heurisitics which are perfect for obstacle-free grids. Third, we describe a canonical ordering which allows us to implement a version of A* whose performance scales well when increasing k. Our empirical evaluation shows that the heuristics we propose are superior to the Euclidean distance (ED) when regular A* is used. For grids beyond 64 the overhead of computing the heuristic yields decreased time performance compared to the ED. We found also that a configuration of our A*-based implementation, without canonical orders, is competitive with the "any-angle" path planner Theta* both in terms of solution quality and runtime.