On lazy randomized incremental construction

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Computing the Maximum Overlap of Two Convex Polygons Under Translations.

International audienceLet P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O(n^2+m^2+min(nm^2+n^2m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n+m)log(n+m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q. We also prove that the position which translates the centroid of Q on the centroid of P always realizes an overlap of 9/25 of the maximum overlap and that this overlap may be as small as 4/9 of the maximum

The Overlay ofLower Envelopes and its Applications

Let F and G be two collections of a total of n bivariate algebraic functions of constant maximum degree. The minimization diagrams of F, G are the planar maps obtained by the xy-projections of the lower envelopes of F, G, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of F and of G is O(n 2+ "), for any ">0. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an e cient and simple divide-and-conquer algorithm for constructing lower envelopes in three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simplyshaped convex sets in three dimensions

A Deterministic Algorithm for the Three-Dimensional Diameter Problem

We give a deterministic algorithm for computing the diameter of an n point set in three dimensions with O(n log^c n) running time, c a constant

Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications

Let \Sigma be a collection of n algebraic surface patches o

Linear optimization queries*

Let F be a set of n halfspaces in Ed (where the di-mension d ~ 3 is fixed) and c a d-component vector. We denote by LP(I’, c) the linear programming prob-lem of minimizing the function c. x over the intersec-tion of all ha~spaces of I’. We show that r can be preprocessed in time and space O(ml+j) (for any fixed &>0, m is an adjustable parameter, n < m < nldizj) so that given c c Ed, LP(I’, c) can be solved in ttme O((m,/~./,J + lrq 1) log2d+1 n). The data structure can be dynamically maintained under insertions and dele-tions of hyperplanes from 17, in 0(m1+3/n) amortized i!ime per update operation. We use a multidimensional version of Megiddo ’s parametric search technique. In connection with an output-sensitive algorithm of Seidel, we get that a convex hull of an n-point set in Ed (d ~ 4) can be computed in time 0(n2”*+h + h log n), where h is the number of faces of the con-vex hull. We also show that given an n-point set P in Ed, one can determine the extreme points of P m time 0(n2-*f6) (for any fixed 6> o). *This extended abstract combines a paper [Mat91d] of the first author with an improvement and simplification achieved by th

Computing Many Faces in Arrangements of Lines and Segments

We present randomized algorithms for computing many faces in an arrangement of lines or of segments in the plane, which are considerably simpler and slightly faster than the previously known ones. The main new idea is a simple randomized O(n log n) expected time algorithm for computing p n cells in an arrangement of n lines. 1 Introduction Given a finite set of lines, L, in the plane, the arrangement of L, denoted as A(L), is the cell complex induced by L. The 0-faces (or vertices) of A(L) are the intersection points of L, the 1-face (or edges) are maximal portions of lines of L that do not contain any vertex, and the 2-faces (called cells) are the connected components of R 2 \Gamma S L. For a finite set S of segments we define the arrangement, A(S), in an analogous manner. Notice that while the cells are convex in a line arrangement, they need not even be simply connected in an arrangement of segments. Line and segment arrangements have been extensively studied in computatio..