Optimal Trajectories in Discrete Space with Acceleration Constraints

Abstract

In the racetrack acceleration model, proposed by Martin Gardner in 1973, each step consists of changing the position of the vehicle by a vector in $\mathbb{Z}^2$, with the constraints that two consecutive vectors differ by at most one unit in each dimension. We investigate three problems related to this model in arbitrary dimension in open space (no obstacles), where a configuration of the vehicle consists of its current position and the last-used vector. The three problems are the following. In Branching Cost (BC), given two configurations, the goal is to compute the minimum number of intermediate configurations (length of a trajectory) between the two configurations. Branching Trajectory (BT) has the same input and asks for a description of the corresponding trajectory. Multipoint Trajectory (MT) asks for an optimal trajectory that visits given points $p_1,\dots,p_n$ in a prescribed order, starting and ending with zero-speed configurations.\\ We revisit known approaches to solve BC in 2D, showing that this problem can be solved in constant time in any fixed number of dimensions $d$ (more generally, in $O(d \log d)$ time). We show that BT can also be solved in constant time for any fixed $d$, despite the fact that the length of the trajectory is not constant, by leveraging the fact that there always exists \emph{one} optimal trajectory compactly represented by $O(1)$ intermediate configurations. For MT, we collect theoretical and experimental evidence that the speed cannot be trivially bounded; local decisions may be impacted by points that are arbitrarily far in the visit order; and an optimal trajectory may require significant excursions out of the convex hull of the points. We still establish conservative speed bounds that a natural dynamic programming (DP) algorithm can exploit to solve reasonably large instances efficiently.