720 research outputs found
Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves
Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive
epsilon, we want to compute an epsilon-isotopic polygonal approximation to the
restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on
subdivision algorithms because of their adaptive complexity and ease of
implementation. Plantinga and Vegter gave a numerical subdivision algorithm
that is exact when the curve S is bounded and non-singular. They used a
computational model that relied only on function evaluation and interval
arithmetic. We generalize their algorithm to any bounded (but possibly
non-simply connected) region that does not contain singularities of S. With
this generalization as a subroutine, we provide a method to detect isolated
algebraic singularities and their branching degree. This appears to be the
first complete purely numerical method to compute isotopic approximations of
algebraic curves with isolated singularities
Deformation of string topology into homotopy skein modules
Relations between the string topology of Chas and Sullivan and the homotopy
skein modules of Hoste and Przytycki are studied. This provides new insight
into the structure of homotopy skein modules and their meaning in the framework
of quantum topology. Our results can be considered as weak extensions to all
orientable 3-manifolds of classical results by Turaev and Goldman concerning
intersection and skein theory on oriented surfaces.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.htm
Plantinga-Vegter algorithm takes average polynomial time
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 worst-case cost bound for degree plane curves with maximum
coefficient bit-size . 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
for real, degree , 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.Comment: 8 pages, correction of typo
The Complexity of Subdivision for Diameter-Distance Tests
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
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