this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as

Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p). We consider the problem of positioning a point x inside a compact region R # R 2 such that min{ w(p) -1 d(x, p) ; p # S } is maximized. Based on the parametric search paradigm, we give the first subquadratic algorithms for this problem, with running time O(n log 4 n). Furthermore, we shall introduce the concept of `exact approximation' as the bit model counterpart to parametric search. Exploiting ideas from exact computation, we show that the considered problem can be solved in time O(L(L)n log n), where L denotes the maximal bit-size of input numbers, and (L) the complexity of multiplying two L-bit integers.

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width such that no point p ∈ S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n 3 log n) time and O(n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k ∈ O(n) our algorithm runs in O(n 3 log n) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n log n) time and O(n) space. 1

This paper deals with efficient methods for mapping arbitrary parallel algorithms onto faulty general purpose VLSI/WSI data-driven array. First, a brief overview of several architectural designs of the array is given. Next, three directions for the algorithmic improvement of a certain mapping scheme are presented. None of these directions takes into account the possibility of the defects in the array. Therefore, we present two methods which can be used to adapt any of the above algorithmic improvements for the case where defects are present in the array. In the first Map-onto-faulty-array method, faulty cells are taken into consideration during all the phases of the mapping/improvement process. In contrast, the second Map-and-correct method initially ignores faulty cells and takes care of them in the correction phase following the mapping/improvement process. Keywords: Mapping, fault-tolerance, processor array, heuristic algorithms, optimization. ? This work has been supported by Min...

We develop ecient algorithms for locating an obnoxious facility in a simple polygonal region and for locating a desirable facility in a simple polygonal region. Many realistic facility location problems require the facilities to be constrained to lie in a simple polygonal region.

An α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets. We present an efficient O(n log³ n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known. We also propose an O(n³ log² n)-time algorithm for the widest arbitrarily-oriented version and an Θ(n log n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point.

Two strategies for efficient dynamic program allocation on a computer with mesh-connected architecture are described. A brief description of a toolbox implementing both strategies, and some experimental results are also given.

The Delaunay triangulation is generated from a points set and a structuring element of type disc. In the Delaunay triangulation definition, replacing the disc by a planar-connected region (we call a molecule), which is a union of a fixed number of discs, allows construction of what we name the molecular graphs. In a finite points set, the molecular graphs record the empty regions which are identical to the molecule, independently of translation, rotation and scaling transforms. The molecular graphs are applied to pattern recognition problem. Knowing a template (an input pattern represented by a molecule), the addressed problem is to identify the existing patterns, whose shapes are similar to the template, in a given input points set. The proposed solution is based on a generalization of the α-shapes; the disc of radius α in the ordinary α-shape is replaced, in the generalized version, by a template of size depending on α.

A 1-corner corridor through a set S of points is an open subset of CH(S) containing no points from S and bounded by a pair of parallel polygonal lines each of which contains two segments. Given a set of n points in the plane, we consider the problem of computing a widest empty 1-corner corridor. We describe an algorithm that solves the problem in O(n4 log n) time and O(n) space. We also present an approximation algorithm that computes in O(n log n/ε1/2 + n2 /ε) time a solution with width at least a fraction (1 − ε) of the optimal, for any small enough ε>0.