The Complexity of Subdivision for Diameter-Distance Tests

TL;DR

This paper introduces a framework for analyzing the complexity of subdivision algorithms based on diameter-distance tests, providing new bounds and applying them to implicit curve and surface approximation algorithms.

Contribution

It formalizes diameter-distance tests, offers the first complexity analysis for a specific implicit curve and surface approximation algorithm, and proves the bounds are tight.

Findings

Provides non-adaptive and adaptive complexity bounds.

Applies framework to analyze Plantinga and Vegeter's algorithm.

Shows bounds are tight with constructed hypersurface families.

Abstract

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight.