AAAI Publications, Eleventh Annual Symposium on Combinatorial Search

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A Suboptimality Bound for 2k Grid Path Planning
Benjamín Kramm, Nicolas Rivera, Carlos Hernandez, Jorge A. Baier

Last modified: 2018-07-02


The 2k 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 2kk, 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.

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