The complexity of subdivision for diameter-distance tests. (November 2020)

• Record Type: Journal Article
• Authors: Burr, Michael; Gao, Shuhong; Tsigaridas, Elias
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 1
• Page End: 27
• Publication Date: 2020-11
• Subjects: Algorithmic complexity -- Worst case bit-complexity -- Subdivision algorithms -- Continuous amortization -- Adaptive complexity -- Separation bounds; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.06.004
• Languages: English
• ISSNs: 0747-7171

Abstract

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.