Steven Fortune

Parallelism in random access machines

A model of computation based on random access machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of traditional models. In particular, deterministic parallel RAMs can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAMs can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines. Similar results hold for other classes. The effect of limiting the size of the common memory is also considered.

Voronoi diagrams and Delaunay triangulations

The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations

A sweepline algorithm for Voronoi diagrams

We present a transformation that can be used to compute Voronoi diagrams with a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms have <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) worst case running time and use <italic>&Ogr;</italic>(<italic>n</italic>) space.

WISE design of indoor wireless systems: practical computation and optimization

Designing a low-power system for wireless communication within a building might seem simple. Not so-walls can affect signal strength in ways that are hard to calculate. The paper considers how AT&Ts WISE software uses CAD, computational geometry, and optimization to quickly plan where to place base-station transceivers. >

Efficient exact arithmetic for computational geometry

We experiment with exact integer arithmetic to implement primitives for geometric algorithms. Naive use of exact arithmetic—either modular or multiprecision integer—increases execution time dramatically over the use of floating-point arithmetic. By combining tuned multiprecision integer arithmetic and a floating-point filter based on interval analysis, we can obtain the effect of exact integer arithmetic at a cost close to that of floating-point arithmetic. We describe an experimental expression compiler that conveniently packages our techniques.

Static analysis yields efficient exact integer arithmetic for computational geometry

Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extended-precision arithmetic is required.

Image segmentation by shape-directed covers

A technique for image segmentation using shape-directed covers is described and applied to the fully automatic analysis of complex printed-page layouts. The structure of the background (white space) is analyzed, assisted by an enumeration of all maximal white rectangles. For this enumeration, the most computationally expensive step, an algorithm has been developed that, aside from a sort, achieves an expected runtime linear in the number of black connected components. The crucial engineering decision is the specification of a partial order on white rectangles to express domain-specific knowledge of preferred shapes and sizes. This order determines a sequence of partial covers of the background, and thus, a sequence of nested page segmentations. In experimental trials on Manhattan layouts, good segmentations often occur early in this sequence, using a simple and uniform shape-direction rule. This is a global-to-local strategy, which for some tasks is superior to strategies currently emphasized in the literature, including bottom-up and top-down.<<ETX>>

Unbounded fan-in circuits and associative functions

We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function <italic>f</italic>, circuits of size <italic>O</italic>(<italic>nf</italic><supscrpt>−1</supscrpt>(<italic>n</italic>)) and constant depth exist for the semigroup product of <italic>n</italic> elements. The depth depends upon the choice of the primitive recursive function <italic>f</italic>. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two <italic>n</italic>-bit numbers.

Stable maintenance of point set triangulations in two dimensions

Geometric algorithms are explored, assuming that arithmetic is done approximately. Stable algorithms are described for two geometric problems. The first algorithm computes two-dimensional convex hulls. The main result is that a triangulation of a set of points in the plane can be maintained stably. The second algorithm deals with line arrangements in the plane.<<ETX>>

An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials

We discuss an iterative algorithm that approximates all roots of a univariate polynomial. The iteration is based on floating-point computation of the eigenvalues of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about log?/??(P) iterations, where ? is the relative error of floating-point arithmetic, ? is the relative separation of the roots, and ?(P) is the condition number of the polynomial. Each iteration requires an n×n floating-point eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floating-point accuracy at up to n points.We describe a careful implementation of the algorithm, including many techniques that contribute to the practical efficiency of the algorithm. On some hard examples of ill-conditioned polynomials, e.g. high-degree Wilkinson polynomials, the implementation is an order of magnitude faster than the Bini?Fiorentino implementation mpsolve.