12263445
Approximation algorithms for convex hulls
The problem of constructing the convex hull of a finite point set in a Euclidean space arises in many applications. In this paper we study a set of algorithms for constructing approximate convex hulls. We show that an E-approximate hull of N points in the plane can be constructed in O(n + I/e) time. The planar algorithm has been implemented and is very fast on point sets that arise in practice. The method can be generalized to compute hulls of point sets in higher dimensional spaces. CR
13915150
Deterministic P-RAM simulation with constant redundancy
In this paper, we show that distributing the memory of a parallel computer and, thereby, decreasing its granularity allows a reduction in the redundancy required to achieve polylog simulation time for each PRAM step. Previously, realistic models of parallel computation assigned one memory module to each processor and, as a result, insisted on relatively coarse-grain memory. We propose, on the other hand, a more flexible, but equally valid model of computation, the distributed-memory, bounded-degree network (DMBDN) model. This model allows the use of fine-grain memory while maintaining the realism of a bounded-degree interconnection network. We describe a P-RAM simulation scheme, which is admitted under the DMBDN model, that exploits the increased memory bandwidth provided by a two-dimensional mesh of trees (2DMOT) network to achieve an overhead in memory redundancy lower than that required by other fast, deterministic P-RAM simulations. Specifically, for a deterministic simulation of an n-processor P-RAM on a bounded-degree network, we are able to reduce the number of copies of each variable from O(logn/loglogn) to ®(1) and still simulate each P-RAM step in polylog time.
1890410
On the power of universal bases in sequencing by hybridization
Sequencing by hybridization is a novel DNA sequencing technique in which an array (SBH chip) of short sequences of nucleotides (probes) is brought in contact with a solution of (replicas of) the target DNA sequence. A biochemical method determines the subset of probes that bind to the target sequence (the spectrum of the sequence), and a combinatorial method is used to reconstruct the DNA sequence from the spectrum. Since technology limits the number of probes on the SBH chip, a challenging combinatorial question is the design of a smallest set of probes that can sequence an arbitrary DNA string of a given length. We show in this work that the use of universal bases (bases that bind to any nucleotide [LB94]) can drastically improve the performance of the SBH process. We present a novel probe design with performance that asymptotically approaches the information-theoretical bound up to a constant factor, and, for any number of probes, is significantly better than previously analyzed probe patterns. Furthermore, the sequencing algorithm we use is substantially simpler than the Eulerian path method used in previous work.
40454105
Area-Time Optimal VLSI Integer Multiplier with Minimum Computation Time
According to VLSI theory, [logn, √n] is the range of computation times for which there may exist an AT2-optimal multiplier of n-bit integers. Such networks were previously known for the time range [Ω(log2n), 0(√n)]; in this paper we settle this theoretical question, by exhibiting a-class of AT2-optimal multipliers with computation times [Ω(logn), 0(n1/2)]. Our designs are based on the DFT on a Fermat ring, whose elements are represented in a redundant radix-4 form to ensure 0(1) addition time.
205093415
Dynamic Planar Point Location with Optimal Query Time
We present a new dynamic technique for locating a point in a convex planar subdivision whose n vertices lie on a fixed set of N horizontal lines. The supported update operations are insertion/deletion of vertices and edges, and (horizontal) translation of vertices. Our method achieves query time O(log n + log N), space O(N + n log N), and insertion/deletion time O(log n log N). Hence, for N=O (n), the query time is O(log n), which is optimal. The proposed technique, based on the trapezoid method, provides an efficient solution to many significant applications where the most frequent operation is the point location query, while updates are more rarely executed.
215848405
The cube-connected cycles: a versatile network for parallel computation
An interconnection pattern of processing elements, the cube-connected cycles (CCC), is introduced which can be used as a general purpose parallel processor. Because its design complies with present...
45504051
Horizons of Parallel Computation
This paper considers the ultimate impact of fundamental physical limitations—notably, speed of light and device size—on parallel computing machines. Although we fully expect an innovative and very gradual evolution to the limiting situation, we take here the provocative view of exploring the consequences of the accomplished attainment of the physical bounds. The main result is that scalability holds only for neighborly interconnections, such as the square mesh, of bounded-size synchronous modules, presumably of the area-universal type. We also discuss the ultimate infeasibility of latency-hiding, the violation of intuitive maximal speedups, and the emerging novel processor-time tradeoffs.
1952504
Data structures and algorithms for the string statistics problem
Given a textstringx of lengthn, theMinimal Augmented Suffix Tree T (x) ofx is a digital-search index that returns, for anyquery stringw and in a number of comparisons bounded by the length ofw, the maximum number of nonoverlapping occurrences ofw inx. It is shown that, denoting the length ofx byn, T(x) can be built in timeO(n log2n) and spaceO(n logn), off-line on a RAM.
41033149
Parallel Restructuring and Evaluation of Expressions
Abstract In this paper we describe a boolean network of size O ( N 2 log N ) which accepts a fully parenthesized N -variable expression over a given semiring and produces its value in O (log N ) time. The network consists of two components: a preprocessor and a universal evaluator. The preprocessor computes the destinations of the expression terms and routes them to the correct input terminals of the universal evaluator.
18686062
Sequencing-by-hybridization revisited: the analog-spectrum proposal
All published approaches to DNA sequencing by hybridization (SBH) consist of the biochemical acquisition of the spectrum of a target sequence (the set of its subsequences conforming to a given probing pattern) followed by the algorithmic reconstruction of the sequence from its spectrum. In the "standard" or "uniform" approach, the probing pattern is a string of length L and the length of reliably reconstructible sequences is known to be m/sub len/ = O(2/sup L/). For a fixed microarray area, higher sequencing performance can be achieved by inserting nonprobing gaps ("wild-cards") in the probing pattern. The reconstruction, however, must cope with the emergence of fooling probes due to the gaps and algorithmic failure occurs when the spectrum becomes too densely populated, although we can achieve m/sub comp/ = O(4/sup L/). Despite the combinatorial success of gapped probing, all current approaches are based on a biochemically unrealistic spectrum-acquisition model (digital-spectrum). The reality of hybridization is much more complex. Departing from the conventional model, in this paper, we propose an alternative, called the analog-spectrum model, which more closely reflects the biochemical process. This novel modeling reestablishes probe length as the performance-governing factor, adopting "semidegenerate bases" as suitable emulators of currently inadequate universal bases. One important conclusion is that accurate biochemical measurements are pivotal to the success of SBH. The theoretical proposal presented in this paper should be a convincing stimulus for the needed biotechnological work.
1765683
The Role of Arithmetic in Fast Parallel Matrix Inversion
Abstract. In the last two decades several NC algorithms for solving basic linear algebraic problems have appeared in the literature. This interest was clearly motivated by the emergence of a parallel computing technology and by the wide applicability of matrix computations. The traditionally adopted computation model, however, ignores the arithmetic aspects of the applications, and no analysis is currently available demonstrating the concrete feasibility of many of the known fast methods. In this paper we give strong evidence to the contrary, on the sole basis of the issue of robustness, indicating that some theoretically brilliant solutions fail the severe test of the ``Engineering of Algorithms.'' We perform a comparative analysis of several well-known numerical matrix inversion algorithms under both fixed- and variable-precision models of arithmetic. We show that, for most methods investigated, a typical input leads to poor numerical performance, and that in the exact-arithmetic setting no benefit derives from conditions usually deemed favorable in standard scientific computing. Under these circumstances, the only algorithm admitting sufficiently accurate NC implementations is Newton's iterative method, and the word size required to guarantee worst-case correctness appears to be the critical complexity measure. Our analysis also accounts for the observed instability of the considered superfast methods when implemented with the same floating-point arithmetic that is perfectly adequate for the fixed-precision approach.