# Plantinga-Vegter algorithm takes average polynomial time

## TL;DR

This paper provides a condition-based analysis showing that the Plantinga-Vegter algorithm for plane curves has an expected polynomial time complexity of O(d^7), contrasting with its worst-case exponential bound, supported by probabilistic and smoothed analysis.

## Contribution

It introduces the first expected polynomial time complexity analysis for the PV Algorithm using probabilistic and smoothed analysis techniques.

## Key findings

- Expected time complexity is bounded by O(d^7) for degree d curves.
- Smoothed analysis yields similar polynomial bounds.
- The approach combines probabilistic methods with condition numbers and continuous amortization.

## Abstract

We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. The first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a $O\big(2^{\tau d^{4}\log d}\big)$ 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 PV Algorithm is bounded by $O(d^7)$ for real, degree $d$, plane curves. We also exhibit a smoothed analysis of the PV 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.