62245037
Processor—Time Tradeoffs under Bounded-Speed Message Propagation: Part I, Upper Bounds
Upper bounds are derived for the processor—time tradeoffs of machines such as linear arrays and two-dimensional meshes, which are compatible with the physical limitation expressed by bounded-speed propagation of messages (due to the finiteness of the speed of light). It is shown that parallelism and locality combined may yield speedups superlinear in the number of processors. The speedups are inherent, due to the optimality of the obtained tradeoffs as established in a companion paper.Simulations of multiprocessor machines are developed by analogous machines with fewer processors. A crucial role is played by the hierarchical nature of the memory system. A divide-and-conquer technique for hierarchical memories is developed, based on the graph-theoretic notion of a topological separator. For multiprocessors, this technique also requires a careful balance of memory access and interprocessor communication costs, which leads to nonintuitive orchestrations of the simulation process.
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.
930169
Sequencing-by-hybridization at the information-theory bound: an optimal algorithm
In a recent paper [PFU99) we have introduced a novel probing scheme for DNA sequencing by hybridization (SBH). The new gapped-probe scheme combines natural and universal bases in a well defined periodic pattern. It was shown in [PFU99] that the performance of the gapped-probe scheme (in terms of the length of a sequence that can be uniquely reconstructed using a given library size of probes) is significantly better than the standard scheme based on oligomer probes.
In this paper we present and analyze a new, more powerful, sequencing algorithm for the gapped-probe scheme. We prove that the new algorithm exploits the full potential of the SBH technology with high-confidence performance, that comes within a small constant factor (about 2) of the information-theory bound. Moreover, this performance is achieved while maintaining running time linear in the target sequence length.
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.
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.
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
17687657
Work-Preserving Speed-Up of Parallel Matrix Computations
Brent's scheduling principle provides a general simulation scheme when fewer processors are available than specified by the fastest parallel algorithm. Such a scheme preserves, under slow-down, the actual number of executed operations, also called work. In this paper we take the complementary viewpoint, and rather than consider the work-preserving slow-down of some fast parallel algorithm, we investigate the problem of the achievable speed-ups of computation while preserving the work of the best-known sequential algorithm for the same problem. The proposed technique, eminently applicable to problems of matrix-computational flavor, achieves its result through the interplay of two algorithms with significantly different features. Analogous but structurally different "interplays" have been used previously to improve the algorithmic efficiency of graph computations, selection, and list ranking. We demonstrate the efficacy of our technique for the computation of path algebras in graphs and digraphs and various fundamental computations in linear algebra. Some of the fundamental new algorithms may have practical value; for instance, we substantially improve the algorithmic performance of the parallel solution of triangular and Toeplitz linear systems of equations and the computation of the transitive closure of digraphs.
40271942
Deterministic Simulation of Idealized Parallel Computers on More Realistic Ones
We describe a deterministic simulation of PRAMs on module parallel computers (MPCs) and on processor networks of bounded degree. The simulating machines have the same number n of processors as the simulated PRAM, and if the size of the PRAM's shared memory is polynomial in n, each PRAM step is simulated by O(log n) MPC steps or by O((log n)2) steps of the bounded degree network. This improves upon a previous result by Upfal and Wigderson. We also prove an Ω((log n)2/log log n) lower bound on the number of steps needed to simulate one PRAM step on a bounded degree network under the assumption that the communication in the network is point-to-point.
20488631
Improved time and space bounds for Boolean matrix multiplication
SummaryUsing modular arithmetic we obtain the following improved bounds on the time and space complexities for n × n Boolean matrix multiplication: O(nlog27 lognlogloglognloglogloglogn) bit operations and O(n2loglog n) bits of storage on a logarithmic cost RAM having no multiply or divide instruction; O(nlog27(logn)2−1/2log27(loglog n)1/2log27−1) bit operations and O(n2log n) bits of storage on a RAM which can use indirect addressing for table lookups. The first algorithm can be realized as a Boolean circuit with O(nlog27lognlogloglognloglogloglogn) gates. Whenever n×n arithmetic matrix multiplication can be performed in less than O(nlog27) arithmetic operations, our results have corresponding improvements.
2327802
Robust proximity queries: an illustration of degree-driven algorithm design
In the context of methodologies intended to confer robustness to geometric algo- rithms, we elaborate on the exact-computation paradigm and formalize the notion of degree of a geometric algorithm as a worst-case quantication of the precision (number of bits) to which arith- metic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach where algorithm design is driven also by the degree, we consider the important classical problem of proximity queries in two and three dimensions and develop a new technique for the ecient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique oers both low degree and fast query time and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations, and arithmetic degree.