Complexity of a Single Face in an Arrangement of s-Intersecting Curves
Computational Geometry
2011-08-23 v1 Combinatorics
Abstract
Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(\lambda_s(n)), O(\lambda_{s+1}(n)), and O(\lambda_{s+2}(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; \lambda_k(n) is the maximum length of a Davenport-Schinzel sequence of order k on an alphabet of n symbols. Our bounds asymptotically match the known worst-case lower bounds. Our proof settles the still apparently open case of semi-infinite curves. Moreover, it treats the three cases in a fairly uniform fashion.
Keywords
Cite
@article{arxiv.1108.4336,
title = {Complexity of a Single Face in an Arrangement of s-Intersecting Curves},
author = {Boris Aronov and Dmitriy Drusvyatskiy},
journal= {arXiv preprint arXiv:1108.4336},
year = {2011}
}
Comments
9 pages, 5 figures