A Suboptimality Bound for 2k Grid Path Planning

Abstract

The 2^k neighborhood has been recently proposed as an alternative to optimal any-angle path planning over grids. Even though it has been observed empirically that the quality of solutions approaches the cost of an optimal any-angle path as k is increased, no theoretical bounds were known. In this paper we study the ratio between the solutions obtained by an any-angle path and the optimal path in the 2^kk, that generalizes previously known bounds for the 4- and 8-connected grids. We analyze two cases: when vertices of the search graph are placed (1) at the corners of grid cells, and (2) when they are located at their centers. For case (1) we obtain a suboptimality bound of 1 + 1/8k² + O(1/k³), which is tight; for (2), however, worst-case suboptimality is a fixed value, for every k ≤ 3. Our results strongly suggests that vertices need to be placed in corners in order to obtain near-optimal solutions. In an empirical analysis, we compare theoretical and experimental suboptimality.