Video-game designers often tessellate continuous 2-dimensional terrain into a grid of blocked and unblocked square cells. The three main ways to calculate short paths on such a grid are to determine truly shortest paths, shortest vertex paths and shortest grid paths, listed here in decreasing order of computation time and increasing order of resulting path length. We show that, for both vertex and grid paths on both 4-neighbor and 8-neighbor grids, placing vertices at cell corners rather than at cell centers tends to result in shorter paths. We quantify the advantage of cell corners over cell centers theoretically with tight worst-case bounds on the ratios of path lengths, and empirically on a large set of benchmark test cases. We also quantify the advantage of 8-neighbor grids over 4-neighbor grids.

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.