Journal of Artificial Intelligence Research 67 (2020) 81-113 Submitted 03/2019; published 01/2020 The 2 k Neighborhoods for Grid Path Planning Nicol´ as Rivera nicolas.rivera@cl.cam.ac.uk Computer Laboratory, University of Cambridge, Cambridge, UK Carlos Hern´ andez carlos.hernandez.u@unab.cl Nicol´ as Hormaz´ abal nic.hormazabal@uandresbello.edu Departamento de Ciencias de la Ingenier´ ıa, Universidad Andr´ es Bello, Santiago, Chile Jorge A. Baier jabaier@ing.puc.cl Departamento de Ciencia de la Computaci´ on Pontificia Universidad Cat´ olica de Chile Santiago, Chile Abstract Grid path planning is an important problem in AI. Its understanding has been key for the development of autonomous navigation systems. An interesting and rather surprising fact about the vast literature on this problem is that only a few neighborhoods have been used when evaluating these algorithms. Indeed, only the 4- and 8-neighborhoods are usu- ally 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; that is, neighborhoods that admit 2 k neighbors per state, where k is a parameter. 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 ≥ 2, which allow us to propose admissible heuristics that are perfect for obstacle-free grids, which generalize the well-known Manhattan and Octile distances. Third, we define the notion of canonical path for the 2 k -neighborhood; this allows us to incorporate our neighborhoods into two versions of A*, namely Canonical A* and Jump Point Search (JPS), whose performance, we show, scales well when increasing k. Our empirical evaluation shows that, when increasing k, the cost of the solution found improves substantially. Used with the 2 k -neighborhood, Canon- ical A* and JPS, in many configurations, are also superior to the any-angle path planner Theta ∗ both in terms of solution quality and runtime. Our planner is competitive with one implementation of the any-angle path planner, ANYA in some configurations. Our main practical conclusion is that standard, well-understood grid path planning technology may provide an effective approach to any-angle grid path planning. 1. Introduction Grid path planning is one of the most well-known problems in AI. It arises naturally when modeling the problem of goal-directed navigation over a two-dimensional space as a graph search problem over a grid in which each cell of the grid can either be an obstacle or be free. c 2020 AI Access Foundation. All rights reserved.