The Provable Virtue of Laziness in Motion Planning

TL;DR

This paper analyzes LazySP motion-planning algorithms, demonstrating they are asymptotically optimal in minimizing edge evaluations through probabilistic bounds and matching lower bounds.

Contribution

It provides the first theoretical analysis establishing the asymptotic optimality of LazySP algorithms in edge evaluation minimization.

Findings

LazySP algorithms are asymptotically optimal in the worst case.

An analytical upper bound on edge evaluations is derived.

A matching lower bound confirms optimality.

Abstract

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.