The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R^d, the maximum complexity of its Voronoi diagram under the L1 metric, and also under a simplicial distance function, are both shown to be \Theta(n dd=2e ). The upper bound for the case of the L1 metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R^d. This complexity is \Theta(n dd=2e ), for d 1, and it improves to \Theta(n bd=2c ), for d 2, if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R³ is shown to be \Theta(n 2 ). We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in d...

. We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3-space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k 3 + kn log k log n) expected time. Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539793250755 1. Combinatorial bounds. Let P = {P 1 , . . . , P k } be a family of k convex polyhedra in 3-space, let n i be the number of faces of P i , and let n = # k i=1 n i . Put U = # P. By the combinatorial complexity of a polyhedral set we mean the total number of its vertices, edges, and faces. Our main result is the followin...

. Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A 1 , . . . , A k with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums P i = A i # (-B), for i = 1, . . . , k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be ## nk#(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P 1 , . . . , P k . We also derive an e#cient randomized algorithm that constructs this configuration space in expected time O(nk log k log n). Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms, algorithmic motion planning AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539794266602 1. Introduction. Let A 1 , . . . , A k be k close...

Abstract We present new algorithms for approximate range counting,where, for a specified "> 0, we want to count the number of data points in a query range, up to relative error of". We first describe a general framework, adapted from Cohen [10], for this task, and then specialize it to two important instances of range counting: halfspaces in R3 anddisks in the plane. The technique reduces the approximate

We study the motion-planning problem for a convex m-gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3-dimensional space of all free placements of P in Q) in time that is near-quadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.-Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...

In approximate halfspace range counting, one is given a set P of n points in R d, and an ε> 0, and the goal is to preprocess P into a data structure which can answer efficiently queries of the form: Given a halfspace h, compute an estimate N such that (1−ε)|P ∩h | ≤ N ≤ (1+ε)|P ∩h|. Several recent papers have addressed this problem, including a study by the authors [18], which is based, as is the present paper, on Cohen’s technique for approximate range counting [9]. In this approach, one chooses a small number of random permutations of P, and then constructs, for each permutation π, a data structure that answers efficiently minimum range queries: Given a query halfspace h, find the minimum-rank element (according to π) in P ∩ h. By repeating this process for all chosen permutations, the approximate count can be obtained, with high probability, using a certain averaging process over the minimum-rank outputs. In the previous study, the authors have constructed such a data structure in R 3, using a combinatorial result about the overlay of minimization diagrams in a randomized incremental construction of lower envelopes. In the present work, we propose an alternative approach to the range-minimum problem,

In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in R d, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n ⌊d/2 ⌋ log n), for d odd, and O(n ⌊d/2 ⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d = 3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.