Randomized incremental constructions of three-dimensional convex hulls and planar voronoi diagrams, and approximate range counting

Abstract

We present new algorithms for approximate range counting, where, for a specified ε > 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instances of range counting: halfspaces in R 3 and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a random permutation of the input.A major technical step in our analysis, which we believe to be of independent interest, is a bound of O(n log n) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of n points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lower envelope of n planes in R 3 , or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of n points in R 3 , that are generated during a randomized incremental construction of the lower envelope or of the hull, respectively. All these bounds are tight in the worst case.The first bound leads to an algorithm that, for a query point x ∈ R 2 , efficiently retrieves the sequence of nearest neighbors of x in P , over the random insertion process. A query takes O(log n) expected time, and the expected storage size is O(n log n). Similarly, the other bounds lead to an algorithm that, for a query direction ω ∈ S 2 , efficiently retrieves the sequence of the convex hull vertices that are touched by the planes with outward direction ω that support the convex hull during the random insertion process. Again, a query takes O(log n) expected time, and the expected storage size is O(n log n).These algorithms are used as the main component in the approximate range counting technique that we present, for ranges that are halfspaces in R 3 or disks in the plane. Our algorithms have the best performance known to date; they slightly improve upon the previous technique of Aronov and Har-Peled [5], and appear to be conceptually simpler. *

Relative (p,ε)-Approximations in Geometry

“…Observe that Theorems 5.3 and 5.4 improve over the previous results in [6,25], which have query time Ω( 1 ε 2 log 2 n). It would also be interesting to compare these results to the recent technique of Aronov and Sharir [7]; as presented, this technique caters only to range searching in four and higher dimensions, but it can be adapted to two or three dimensions too.…”

“…As in the planar case, Theorem 5.5 improves over the previous results [6,25], which require Ω( 1 ε 2 log 2 n) time to answer a query. However, in a subsequent work, Afshani and Chan [2] managed to obtain an improved solution.…”

“…Here, for a set P of n points in R d , for d ≥ 2, the best known algorithm for exact halfspace range counting with nearlinear storage takes O(n 1−1/d ) time [33]. As shown in several recent papers, if we only want to approximate the count, as in (1), there exist faster solutions, in which the query time is close to O(n 1−1/ d/2 ) (and is polylogarithmic in two and three dimensions) [6,7,[25][26][27].…”

The Overlay of Minimization Diagrams in a Randomized Incremental Construction

“…In an earlier version of this paper [16], we have obtained this bound (namely, the bound O(n log n)) only for d = 3, using a more involved proof. The present version extends the result to all dimensions, and simplifies the proof.…”

“…In an earlier version of the paper [16], which has catered only to the 3-dimensional case, we have presented an algorithm that implements Cohen's technique by constructing the overlay along the lines described above. For the halfspace minimum range searching problem, the algorithm requires O(n log n) expected storage and preprocessing, and answers a query in O(log n) expected time; the storage and preprocessing bounds derive from our bound on the overlay complexity, as stated above.…”