# Asymptotic Optimality of a Time Optimal Path Parametrization Algorithm

## TL;DR

This paper proves that a linear-time algorithm for Time Optimal Path Parametrization is asymptotically optimal for all problems solvable by convex optimization, extending its known optimality beyond a specific subclass.

## Contribution

It establishes the asymptotic optimality of a linear-time algorithm for all convex-optimized problems in Time Optimal Path Parametrization and characterizes the problem's optimum.

## Key findings

- The linear-time algorithm is asymptotically optimal for all convex-optimized problems.
- Characterization of the problem's optimal solution.
- Extension of optimality proof beyond a specific subclass.

## Abstract

Time Optimal Path Parametrization is the problem of minimizing the time interval during which an actuation constrained agent can traverse a given path. Recently, an efficient linear-time algorithm for solving this problem was proposed. However, its optimality was proved for only a strict subclass of problems solved optimally by more computationally intensive approaches based on convex programming. In this paper, we prove that the same linear-time algorithm is asymptotically optimal for all problems solved optimally by convex optimization approaches. We also characterize the optimum of the Time Optimal Path Parametrization Problem, which may be of independent interest.