Plantinga-Vegter Algorithm takes Average Polynomial Time

Abstract

We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. The first complexity analysis of the Plantinga-Vegter Algorithm is due to Burr, Gao and Tsigaridas who proved a $\mathcal{O}\big(2^{\tau d^{4}\log d}\big)$ worst-case cost bound for degree $d$ plane curves with maximum coefficient bit-size $\tau$. This exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the Plantinga-Vegter Algorithm is bounded by $O(d^7)$ for real, degree $d$, plane curves. We also exhibit a smoothed analysis of the Plantinga-Vegter Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation.