A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal.

We present a new priority queue data structure, the queap, that executes insertion in O(1) amortized time and extract-min in O(log(k + 2)) amortized time if there are k items that have been in the heap longer than the item to be extracted. Thus if the operations on the queap are rst-in rst-out, as on a queue, each operation will execute in constant time. This idea of trying to make operations on the least recently accessed items fast, which we call the queueish property, is a natural complement to the working set property of certain data structures, such as splay trees and pairing heaps, where operations on the most recently accessed data execute quickly. However, we show that the queueish property is in some sense more dicult than the working set property by demonstrating that it is impossible to create a queueish binary search tree, but that many search data structures can be made almost queueish with a O(log log n) amortized extra cost per operation.

We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time1. We also design three variants of this index. The first supports point location using lg n +2 √ lg n + O(lg 1/4 n) point-line comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H +1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg 2 n) time. 1

Abstract. A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is ˜ H + O ( ˜ H2/3 + 1) where ˜ H = ˜ H(G, D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy. 1

Suppose there are n post offices p1,...pn in a city. Someone who is located at a position q within the city would like to know which post office is closest to him. Modeling the city as a planar region, we think of p1,...pn and q as points in the plane. Denote the set of post offices by P = {p1,...pn}. Figure 10.1: Closest post offices for various query points. While the locations of post offices are known and do not change so frequently, we do not know in advance for which—possibly many—query locations the closest post office is to be found. Therefore, our long term goal is to come up with a data structure on top of P that allows to answer any possible query efficiently. The basic idea is to apply a so-called locus approach: we partition the query space into regions on which is the answer is the same. In our case, this amounts to partition the plane into regions such that for all points within a region the same point from P is closest (among all points from P). As a warmup, consider the problem for two post offices pi, pj ∈ P. For which query locations is the answer pi rather than pj? This region is bounded by the bisector of pi and pj, that is, the set of points which have the same distance to both points. 106 CG 2012 10.2. Voronoi Diagram Proposition 10.1 For any two distinct points inR d the bisector is a hyperplane, that is, inR 2 it is a line. Proof. Let p = (p1,...,pd) and q = (q1,...,qd) be two points inR d. The bisector of p and q consists of those points x = (x1,...,xd) for which ||p−x| | = ||q−x| | ⇐ ⇒ ||p−x| | 2 = ||q−x| | 2 ⇐ ⇒ ||p| | 2 −||q| | 2 = 2(p−q) ⊤ x. As p and q are distinct, this is the equation of a hyperplane. pj pi H(pi, pj) Figure 10.2: The bisector of two points. Denote by H(pi, pj) the closed halfspace bounded by the bisector of pi and pj that contains pi. InR 2, H(pi, pj) is a halfplane; see Figure 10.2. Exercise 10.2 a) What is the bisector of a line ℓ and a point p ∈R 2 \ ℓ, that is, the set of all points x ∈R 2 with ||p−x| | = ||p−ℓ| | ( = minq∈ℓ||p−q||)? b) For two points p ̸ = q ∈R 2, what is the region that contains all points whose distance to p is exactly twice their distance to q? 10.2 Voronoi Diagram In the following we work with a set P = {p1,..., pn} of points inR 2. Definition 10.3 (Voronoi cell) For pi ∈ P denote the Voronoi cell VP(i) of pi by VP(i): = { q ∈R 2 ∣ ||q−pi| | � ||q−p| | for all p ∈ P}.