ABSTRACT VORONOI DIAGRAMS WITH DISCONNECTED REGIONS

Voronoi diagrams, AVDs for short, are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and distance. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD axioms, structural results and efficient algorithms become available without further effort; for example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way. One of these axioms stated that all Voronoi regions must be pathwise connected, a property quite useful in divide&conquer and randomized incremental construction algorithms. Yet, there are concrete Voronoi diagrams where this axiom fails to hold. In this paper we consider, for the first time, abstract Voronoi diagrams with disconnected regions. By combining a randomized incremental construction technique with trapezoidal decomposition we obtain an algorithm that runs in expected time [Formula: see text], where s is the maximum number of faces a Voronoi region in a subdiagram of three sites can have, and where mj denotes the average number of faces per region in any subdiagram of j sites. In the connected case, where s = 1 = mj , this results in the known optimal bound [Formula: see text].

RANDOMIZED EXTERNAL-MEMORY ALGORITHMS FOR LINE SEGMENT INTERSECTION AND OTHER GEOMETRIC PROBLEMS

We show that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires [Formula: see text] expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to derive algorithms for other geometric problems: 3-d half-space intersections, 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched planar point location; these algorithms require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an external-memory model with multiple disks.

AN IMPROVED HYPERCUBE BOUND FOR MULTISEARCHING AND ITS APPLICATIONS

We give a result that implies an improvement by a factor of log log n in the hypercube bounds for the geometric problems of batched planar point location, trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. Whereas the previous best solution to this problem took O( log n( log log n)3) time on an n-processor hypercube, the solution given here takes O( log n( log log n)2) time on an n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O( log n) bits, so that a processor knows its ID.

A Variable Projection Method for Solving Separable Nonlinear Least Squares Problems

<p>Consider the separable nonlinear least squares problem of finding ~a in R^n and ~alpha in R^k which, for given data (y_i, t_i) i=1,...,m and functions varphi_j(~alpha,t) j=1,2,...n (m&gt;n), minimize the functional</p><p>r(~a,~alpha) = ||~y - Phi(~alpha)~a||_(2)^(2)</p><p>where Phi(~alpha)_(i,j) = varphi_(j)(~alpha,t_j). This problem can be reduced to a nonlinear least squares problem involving $\mathovd{\mathop{\alpha}\limits_{\textstyle\tilde{}}}$ only and a linear least squares problem involving ~a only. the reduction is based on the results of Colub and Pereyra, <em>SIAM J. Numerical Analysis</em>, April 1973, and on the trapezoidal decomposition of Phi, in which an orthogonal matrix Q and a permutation matrix P are found such that</p><p>\begin{displaymath} Q Phi R = R &amp; S 0 &amp; 0 \end{array}\right) \begin{array}{l} \rbrace\, r \\ \mbox{} \end{array} \end{displaymath}</p><p>where R is nonsingular and upper trianular. To develop an algorithm to solve the nonlinear least squares probelm a formula is proposed for the Frechet derivation D(Phi_(2) (~alpha)) where Q i partioned into</p>