Optimal randomized incremental construction for guaranteed logarithmic planar point location

TL;DR

This paper introduces a randomized incremental algorithm for constructing a trapezoidal map for planar point location that guarantees optimal expected preprocessing time, worst-case space, and query time, solving a longstanding open problem.

Contribution

It presents the first randomized incremental construction with expected O(n log n) preprocessing and worst-case linear space and logarithmic query time for planar point location.

Findings

Expected O(n log n) preprocessing time achieved

Deterministic worst-case linear space and logarithmic query time guaranteed

Extends to planar maps from disjoint well-behaved curves

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 well-known trapezoidal-map search-structure that only requires expected $O(n \log n)$ 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 $O(n \log^2 n)$. 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 well-behaved curves.