Showing papers by "Michael T. Goodrich published in 1994"

Parallel algorithms for higher-dimensional convex hulls

Abstract: We give fast randomized and deterministic parallel methods for constructing convex hulls in R/sup d/, for any fixed d. Our methods are for the weakest shared-memory model, the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In particular, we show that the convex hull of n points in R/sup d/ can be constructed in O(log n) time using O(n log n+n/sup [d/2]/) work, with high probability. We also show that it can be constructed deterministically in O(log/sup 2/ n) time using O(n log n) work for d=3 and in O(log n) time using O(n/sup [d/2]/ log/sup c([d/2]-[d/2]/) n) work for d/spl ges/4, where c>0 is a constant which is optimal for even d/spl ges/4. We also show how to make our 3-dimensional methods output-sensitive with only a small increase in running time. These methods can be applied to other problems as well. >

Practical methods for approximate geometric pattern matching under rigid motions: (preliminary version)

Abstract: We present practical methods for approximate geometric pattern matching in d-dimensions along with experimental data regarding the quality of matches and running times of these methods versus those of a branch-and-bound search. Our methods are faster than previous methods but still produce good matches.

Almost optimal set covers in finite VC-dimension: (preliminary version)

Abstract: We give a deterministic polynomial time method for finding a set cover in a set system (X,ℜ) of VC-dimension d such that the size of our cover is at most a factor of O(dlog(dc)) from the optimal size, c. For constant VC-dimension set systems, which are common in computational geometry, our method gives an O(logc) approximation factor. This improves the previous Θ(log |X|) bound of the greedy method and beats recent complexity-theoretic lower bounds for set covers (which don't make any assumptions about VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those that arise in 3-d polytope approximation and 2-d disc covering, we can quickly find O(c)-sized covers.

Efficient piecewise-linear function approximation using the uniform metric: (preliminary version)

Abstract: We give an O(nlogn)-time method for finding a best k-link piecewise-linear function approximating an n-point planar data set using the well-known uniform metric to measure the error, e≥0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in “e space” followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n2).

Optimal parallel approximation for prefix sums and integer sorting

Abstract: Parallel prefix computation is perhaps the most frequently used subroutine in parallel algorithms today. Its time complexity on the CRCW PRAM is O(lg n/ lg lg n) using a polynomial number of processors, even in a randomized setting. Nevertheless, there are a number of non-trivial applications that have been shown to be solvable using only an approximate version of the prefix sums problem. In this paper we resolve the issue of approximating parallel prefix by introducing an algorithm that runs in O(lg* n) time with very high probability, using n/ lg’ n processors, which is optimal in terms of both work and running time. Our approximate prefix sums are guaranteed to come within a factor of (1 + E) of the values of the true sums in a “consistent fashion”, where E is o(1). We achieve this result through the use of a number of interesting new techniques, such as overcertification and estimate-focusing, as well as through new adaptations of known techniques, such as failure-sweeping and bit-thinning. We give a number of non-trivial applications of our approximate parallel prefix routine. Perhaps the most interesting application is for padded integer sorting, an approximation version of another fundamental problem in parallel algorithm design-integer sorting-where one wishes to sort R integers into an array of size 0(n), allowing for gaps between consecutive elements. We show that this problem can also be solved in O(lg’ n) time, with very high probability, using a linear amount of work, which is also optimal in both time and work. Finally, we show several applications ‘Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218. Email: goodrich& . jhu. edu. This research supported by the NSF and DARPA under Grant CCR8908092, and by the NSF under Grants IRI-9116843 and CCR-

Parallel algorithms for evaluating sequences of set-manipulation operations

Abstract: Given an off-line sequence S of n set-manipulation operations, we investigate the parallel complexity of evaluating S (i.e., finding the response to every operation in S and returning the resulting set). We show that the problem of evaluating S is in NC for various combinations of common set-manipulation operations. Once we establish membership in NC (or, if membership in NC is obvious), we develop techniques for improving the time and/or processor complexity.

Biased finger trees and three-dimensional layers of maxima: (preliminary version)

Abstract: We present a method for maintaining biased search trees so as to support fast finger updates (i.e., updates in which one is given a pointer to the part of the tree being changed). We illustrate the power of such biased finger trees by showing how they can be used to derive an optimal O(nlogn) algorithm for the 3-dimensional layers-of-maxima problem and also obtain an improved method for dynamic point location.

Characterization and Recognition of Point-Halfspace and Related Orders

Abstract: We characterize four classes of geometric membership and containment orders-structurally and in terms of forbidden subposets-and present linear- or near linear-time recognition algorithms for each class. We also show that recognizing point-halfspace orders in ℝ2 is NP-hard for d≥2.