Optimal randomized incremental construction for guaranteed logarithmic planar point location

Given a planar map of n segments in which we wish to efficiently locate points, we present the first randomized incremental construction of the well-known trapezoidal-map search-structure that only requires expected O(n logn) preprocessing time while deterministically guaranteeing worst-case linear storage space and worst-case 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, so-called the history DAG, and with the above worst-case space and query-time guarantees, was expected O(n log 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 well-behaved curves. The article significantly extends the theoretical aspects of the work presented in http://arxiv.org/abs/1205.5434. ∗This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), by the GermanIsraeli Foundation (grant no. 1150-82.6/2011), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. †Institute of Operating Systems and Computer Networks, University of Technology Braunschweig, Braunschweig, Germany; [email protected] ‡School of Computer Science, Tel Aviv University, Tel Aviv, Israel; [email protected], [email protected] 1 ar X iv :1 41 0. 56 02 v1 [ cs .C G ] 2 1 O ct 2 01 4