Showing papers by "Michael T. Goodrich published in 1995"

Almost optimal set covers in finite VC-dimension

Abstract: We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.

External-memory graph algorithms

Abstract: We present a collection of new techniques for designing and analyzing e cient external-memory algorithms for graph problems and illustrate how these techniques can be applied to a wide variety of speci c problems. Our results include: Proximate-neighboring. We present a simple method for deriving external-memory lower bounds via reductions from a problem we call the \proximate neighbors" problem. We use this technique to derive non-trivial lower bounds for such problems as list ranking, expression tree evaluation, and connected components. PRAM simulation. We give methods for e ciently simulating PRAM computations in external memory, even for some cases in which the PRAM algorithm is not work-optimal. We apply this to derive a number of optimal (and simple) external-memory graph algorithms. Time-forward processing. We present a general technique for evaluating circuits (or \circuit-like" computations) in external memory. We also use this in a deterministic list ranking algorithm. Department of Computer Science, Box 1910, Brown University, Providence, RI 02912{1910. y Supported in part by the National Science Foundation, by the U.S. Army Research O ce, and by the Advanced Research

Planar separators and parallel polygon triangulation

Abstract: We show how to construct an O(√n)-separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree, where each node corresponds to a subgraph of G and stores an O(√n)-separator of that subgraph. We also show how to construct an O(nϵ)-way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n12+ϵ)-separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n/log n) processors on a CRCW PRAM.

Efficient piecewise-linear function approximation using the uniform metric

Abstract: We given anO(n logn)-time method for finding a bestk-link piecewise-linear function approximating ann-point planar point set using the well-known uniform metric to measure the error, ??0, of the approximation. Our methods is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "? space" followed by several applications of the parametric-searching technique. The previous best running time for this problems wasO(n2).

Computing faces in segment and simplex arrangements

Abstract: For a set S of n line segments in the plane, we give the first work-optimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log2 n) time using O(n logn + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divide-and-conquer alternative to the optimal sequential “plane-sweep” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2(n) logn) time, which improves on a previous O(n log n) time algorithm. For collections of simplices in IRd, we give methods for constructing a set ofm = O(nd 1 logc n+k) cells of constant descriptive complexity that covers their arrangement, where c > 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(logn) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(logn) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log2 n) in IR and O(log3 n) in IRd, d > 3), and also the size and construction cost of the covering (from O(nd 1+ + k)).

Topology B-Trees and Their Applications

Abstract: The well-known B-tree data structure provides a mechanism for dynamically maintaining balanced binary trees in external memory. We present an external-memory dynamic data structure for maintaining arbitrary binary trees. Our data structure, which we call the topology B-tree, is an external-memory analogue to the internal-memory topology tree data structure of Frederickson. It allows for dynamic expression evaluation and updates as well as various tree searching and evaluation queries. We show how to apply this data structure to a number of external-memory dynamic problems, including approximate nearest-neighbor searching and closest-pair maintenance.

On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra (Preliminary Version)

Abstract: We show that several well-known computational geometry problems involving 3-dimensional convex polyhedra are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron.