On the Complexity of the Plantinga-Vegter Algorithm

TL;DR

This paper applies advanced numerical and probabilistic tools to analyze the complexity of the Plantinga-Vegter subdivision algorithm, achieving polynomial time estimates in average and smoothed cases.

Contribution

It introduces a novel combination of numerical analysis, probability, and continuous amortization to analyze the algorithm's complexity beyond worst-case bounds.

Findings

Polynomial time complexity in average-case analysis

Polynomial time complexity in smoothed analysis

Extension of complexity bounds to finite-precision implementations

Abstract

We introduce tools from numerical analysis and high dimensional probability for precision control and complexity analysis of subdivision-based algorithms in computational geometry. We combine these tools with the continuous amortization framework from exact computation. We use these tools on a well-known example from the subdivision family: the adaptive subdivision algorithm due to Plantinga and Vegter. The only existing complexity estimate on this rather fast algorithm was an exponential worst-case upper bound for its interval arithmetic version. We go beyond the worst-case by considering both average and smoothed analysis, and prove polynomial time complexity estimates for both interval arithmetic and finite-precision versions of the Plantinga-Vegter algorithm.