An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

Abstract

Let [Formula: see text] be a set of [Formula: see text] labeled points in the plane. The radial system of [Formula: see text] describes, for each [Formula: see text], the order in which a ray that rotates around [Formula: see text] encounters the points in [Formula: see text]. This notion is related to the order type of [Formula: see text], which describes the orientation (clockwise or counterclockwise) of every ordered triple in [Formula: see text]. Given only the order type, the radial system is uniquely determined and can easily be obtained. The converse, however, is not true. Indeed, let [Formula: see text] be the radial system of [Formula: see text], and let [Formula: see text] be the set of all order types with radial system [Formula: see text] (we define [Formula: see text] for the case that [Formula: see text] is not a valid radial system). Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) show that [Formula: see text] may contain up to [Formula: see text] order types. They also provide polynomial-time algorithms to compute [Formula: see text] when only [Formula: see text] is given. We describe a new algorithm for finding [Formula: see text]. The algorithm constructs the convex hulls of all possible point sets with the radial system [Formula: see text]. After that, orientation queries on point triples can be answered in constant time. A representation of this set of convex hulls can be found in [Formula: see text] queries to the radial system, using [Formula: see text] additional processing time. This is optimal. Our results also generalize to abstract order types.