Optimal Trajectories in Discrete Space with Acceleration Constraints

2026-02-25

AI Summary

This study addresses trajectory optimization in discrete space under acceleration constraints, where the change in each dimension of consecutive step vectors is bounded by one. It investigates shortest-path computation between two points, explicit path construction, and optimal trajectory planning for visiting multiple points in a prescribed order. Through geometric analysis, combinatorial optimization, and dynamic programming, the work proves that in any fixed dimension, both the minimal trajectory length and its construction between two points can be determined in constant time. The paper further uncovers non-intuitive properties of multi-point trajectories, such as the absence of nontrivial upper bounds on velocity and the influence of distant waypoints on local decisions. Building on these insights, it proposes practical velocity bounds and an efficient dynamic programming framework that substantially improves computational performance on large-scale instances.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• optimal trajectory
• acceleration constraints
• discrete space
• racetrack model
• multipoint path planning

Innovation

• discrete acceleration model
• optimal trajectory
• constant-time algorithm
• compact trajectory representation
• dynamic programming with speed bounds