Optimal randomized incremental construction for guaranteed logarithmic planar point location
Abstract
Given a planar map of $n$ segments in which we wish to efficiently locate points, we present the first randomized incremental construction of the wellknown trapezoidalmap searchstructure that only requires expected $O(n \log n)$ preprocessing time while deterministically guaranteeing worstcase linear storage space and worstcase logarithmic query time. This settles a long standing open problem; the best previously known construction time of such a structure, which is based on a directed acyclic graph, socalled the history DAG, and with the above worstcase space and querytime guarantees, was expected $O(n \log^2 n)$. The result is based on a deeper understanding of the structure of the history DAG, its depth in relation to the length of its longest search path, as well as its correspondence to the trapezoidal search tree. Our results immediately extend to planar maps induced by finite collections of pairwise interior disjoint wellbehaved curves.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.5602
 Bibcode:
 2014arXiv1410.5602H
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 The article significantly extends the theoretical aspects of the work presented in http://arxiv.org/abs/1205.5434