  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.
