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Las Vegas algorithms for matrix groups

Las Vegas algorithms for matrix groups

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annual ieee symposium on Foundations of computer science

作者： R. Beals L. Babai Department of CIS University of Oregon Eugene OR USA Department of Computer Science University of Chicago Chicago IL USA Eötvös University Budapest Hungary

We consider algorithms in finite groups, given by a list of generators. We give polynomial time Las Vegas algorithms (randomized, with guaranteed correct output) for basic problems for finite matrix groups over the rationals (and over algebraic number fields): testing membership, determining the order, finding a presentation (generators and relations), and finding basic building blocks: center, composition factors, and Sylow subgroups. these results extend previous work on permutation groups into the potentially more significant domain of matrix groups. Such an extension has until recently been considered intractable. In case of matrix groups G of characteristic p, there are two basic types of obstacles to polynomial-time computation: number theoretic (factoring, discrete log) and large Lie-type simple groups of the same characteristic p involved in the group. the number theoretic obstacles are inherent and appear already in handling abelian groups. they can be handled by moderately efficient (subexponential) algorithms. We are able to locate all the nonabelian obstacles in a normal subgroup N and solve all problems listed above for G/N.< >

关键词： Polynomials Computational Intelligence Society Testing Marine vehicles Statistical analysis computer science Mathematics Complexity theory Galois fields Packaging

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Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment

Near-quadratic bounds for the motion planning problem for a ...

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annual ieee symposium on Foundations of computer science

作者： D. Halperin M. Sharir Robotics Laboratory Department of Computer Science University of Stanford USA School of Mathematical Sciences Tel-Aviv University Israel Courant Institute of Mathematical Sciences New York University USA

We consider the problem of planning the motion of an arbitrary k-sided polygonal robot B, free to translate and rotate in a polygonal environment V bounded by n edges. We show that the combinatorial complexity of a single connected component of the free configuration space of B is k/sup 3/n/sup 2/2/sup O(log(2/3)/ n). this is a significant improvement of the naive bound O((kn)/sup 3/); when k is constant, which is often the case in practice, this yields a near-quadratic bound on the complexity of such a component, which almost settles (in this special case) a long-standing conjecture regarding the complexity of a single cell in a three-dimensional arrangement of surfaces. We also present an algorithm that constructs a single component of the free configuration space of B in time O(n/sup 2+/spl epsi//), for any /spl epsi/>0, assuming B has a constant number of sides. this algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion planning problem, within the same asymptotic running time.< >

关键词： Motion planning Robot kinematics Orbital robotics Pulp manufacturing Research and development computer science Upper bound

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the complexity and distribution of hard problems

The complexity and distribution of hard problems

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annual ieee symposium on Foundations of computer science

作者： D.W. Juedes J.H. Lutz Department of Computer Science Iowa State University Ames IA USA

Measure-theoretic aspects of the /spl les//sub m//sup P/-reducibility structure of exponential time complexity classes E=DTIME(2/sup linear/) and E/sub 2/=DTIME(2/sup polynomial/) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are /spl les//sub m//sup P/-hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. the upper bounds say that the /spl les//sub m//sup P/-hard languages for E are unusually simple in, the sense that they have smaller complexity cores than most languages in E. It follows that the /spl les//sub m//sup P/-complete languages for E form a measure 0 subset of E (and similarly in E/sub 2/). this latter fact is seen to be a special case of a more general theorem, namely, that every /spl les//sub m//sup P/-degree (e.g. the degree of all /spl les//sub m//sup P/-complete languages for NP) has measure 0 in E and in E/sub 2/.< >

关键词： Time measurement Size measurement Particle measurements Polynomials Lifting equipment computer science Upper bound

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Scale-sensitive dimensions, uniform convergence, and learnability

Scale-sensitive dimensions, uniform convergence, and learnab...

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annual ieee symposium on Foundations of computer science

作者： N. Alon S. Ben-David N. Cesa-Bianchi D. Haussler Department of Mathematics R and B. Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv Israel Department of Computer Science Technion-Israel Institute of Technology Haifa Israel Department of Computer Science Universitaà di Milano Department of Computer Science University of California Santa Cruz Santa Cruz CA USA

Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. these laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform Gliveako-Cantelli classes. In this paper we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes. Our characterization yields Dudley, Gine, and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a simple combinatorial quantity generalizing the Vapnik-Chervonenkis dimension. We apply this result to characterize PAC learnability in the statistical regression framework of probabilistic concepts, solving an open problem posed by Kearns and Schapire. Our characterization shows that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class.< >

关键词： Convergence computer science Mathematical model Predictive models Size measurement Meteorology Mathematics Power measurement Minimization methods Weather forecasting

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A framework for cost-scaling algorithms for submodular flow problems

A framework for cost-scaling algorithms for submodular flow ...

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annual ieee symposium on Foundations of computer science

作者： H.N. Gabow Department of Computer Science University of Colorado Boulder Boulder CO USA

the submodular flow problem includes such problems as minimum-cost network flow, dijoin, edge-connectivity orientation and others. We present a cost-scaling algorithm for submodular flow problems. the algorithm applies to these problems in general; we also examine its efficiency for the dijoin and edge-connectivity orientation problems. A minimum-cost dijoin is found in time O(min{m/sup 1/2/, n/sup 2/3/}nmlog(nN)), where n, m and N denote the number of vertices, number of edges and largest magnitude of an integral edge cost. the previous best-known bound is O(n/sup 2/m) if fast matrix multiplication is not used. A k-edge-connected orientation is found in time O(kn/sup 2/(/spl radic/(kn)+k/sup 2/log(n/k))). A minimum-cost k-edge-connected orientation is found on the above time bound for dijoins when k=O(1) (and a more complicated bound for general k). the scaling algorithm uses a transformation that eliminates vertex weights in edge-capacitated graphs. It also incorporates a scheme to limit the growth in the size of intermediate solutions, using a dual minimum-cost network flow problem.< >

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A simple local-control approximation algorithm for multicommodity flow

A simple local-control approximation algorithm for multicomm...

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annual ieee symposium on Foundations of computer science

作者： B. Awerbuch T. Leighton Laboratory for Computer Science Massachusetts Institute of Technology Cambridge MA USA Department of Mathematics and Laboratory for Computer Science Massachusetts Institute of Technology Cambridge MA USA

In this paper, we describe a very simple (1+/spl epsi/)-approximation algorithm for the multicommodity flow problem. the algorithm runs in time that is polynomial in N (the number of nodes in the network) and /spl epsiv//sup -1/ (the closeness of the approximation to optimal). the algorithm is remarkable in that it is much simpler than all known polynomial time flow algorithms (including algorithms for the special case of one-commodity flow). In particular, the algorithm does not rely on augmenting paths, shortest paths, min-cost paths, or similar techniques to push flow through a network. In fact, no explicit attempt is ever made to push flow towards a sink during the algorithm. Because the algorithm is so simple, it can be applied to a variety of problems for which centralized decision making and flow planning is not possible. For example, the algorithm can be easily implemented with local control in a distributed network and it can be made tolerant to link failures. In addition, the algorithm appears to perform well in practice. Initial experiments using the DIMACS generator of test problems indicate that the algorithm performs as well as or better than previously known algorithms, at least for certain test problems.< >

关键词： Approximation algorithms Contracts Polynomials Testing Laboratories computer science Distributed control Performance evaluation Mathematics Marine vehicles

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throughput-competitive on-line routing

Throughput-competitive on-line routing

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annual ieee symposium on Foundations of computer science

作者： B. Awerbuch Y. Azar S. Plotkin Laboratory for Computer Science MIT USA Tel-Aviv University DEC SRC Israel DEC Systems Research Center Palo Alto CA USA Department of Computer Science University of Stanford USA

We develop a framework that allows us to address the issues of admission control and routing in high-speed networks under the restriction that once a call is admitted and routed, it has to proceed to completion and no reroutings are allowed. the "no rerouting" restriction appears in all the proposals for future high-speed networks and stems from current hardware limitations, in particular the fact that the bandwidth-delay product of the newly developed optical communication links far exceeds the buffer capacity of the network. In case the goal is to maximize the throughput, our framework yields an on-line O(log nT)-competitive strategy, where n is the number of nodes in the network and T is the maximum call duration. In other words, our strategy results in throughput that is within O(log nT) factor of the highest possible throughput achievable by an omniscient algorithm that knows all of the requests in advance. Moreover, we show that no on-line strategy can achieve a better competitive ratio. Our framework leads to competitive strategies applicable in several more general settings. Extensions include assigning each connection an associated "profit" that represents the importance of this connection, and addressing the issue of call-establishment costs.< >

关键词： Routing computer science Bandwidth Contracts throughput High-speed networks Delay Admission control Proposals Hardware

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Universal emulations with sublogarithmic slowdown

Universal emulations with sublogarithmic slowdown

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annual ieee symposium on Foundations of computer science

作者： C. Kaklamanis D. Krizanc S. Rao DIMACS Center Rutgers University Piscataway NJ USA School of Computer Science Carleton University Ottawa ONT Canada NEC Research Institute Inc. Princeton NJ USA

the existence of bounded degree networks which can emulate the computation of any bounded degree network of the same size with logarithmic slowdown is well-known. the butterfly is an example of such a universal network. Leiserson was the first to introduce the concept of an area-universal network: a network with VLSI layout area A which can emulate any network of the same size and layout area with logarithmic slowdown. His results imply the existence of an N-node network with layout area O(N log/sup 2/ N) which can emulate any N-node planar network with O(log N) slowdown. the main results of this paper are: there exists an N-node network with layout area O(N log/sup 2/ N) which can emulate any N-node planar network with O(loglogN) slowdown. the N-node butterfly (and hypercube) can emulate any network with VLSI layout area N/sup 2-/spl epsiv// (/spl epsiv/>0) with O(loglogN) slowdown. We also discuss sublogarithmic bounds for the slowdown of emulations of arbitrary bounded degree networks.< >

关键词： Emulation computer networks Grid computing National electric code Very large scale integration Multiprocessor interconnection networks Computational modeling computer science Hypercubes Design engineering

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Refining a triangulation of a planar straight-line graph to eliminate large angles

Refining a triangulation of a planar straight-line graph to ...

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annual ieee symposium on Foundations of computer science

作者： S.A. Mitchell Applied and Numerical Mathematics Department Sandia National Laboratories Albuquerque NM USA

We show that any planar straight line graph (PSLG) with v vertices can be triangulated with no angle larger than 7/spl pi//8 by adding O(v/sup 2/log v) Steiner points in O(v/sup 2/log/sup 2/ v) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. We follow a lazy strategy of starting from an obtuse angle and exploring the triangulation in search of a sequence of Steiner points that will satisfy a local angle condition. Explorations may either terminate successfully (for example at a triangle vertex), or merge. Some PSLGs require /spl Omega/(v/sup 2/) Steiner points in any triangulation achieving any largest angle bound less than /spl pi/. Hence the number of Steiner points added by our algorithm is within a log v factor of worst case optimal. For most inputs the number of Steiner points and running time would be considerably smaller than in the worst case.< >

关键词： Finite element methods Laboratories Shape Steiner trees Mathematics Postal services Numerical analysis Convergence Interpolation computer graphics

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An O(nlog/sup 3/ n) algorithm for the real root problem

An O(nlog/sup 3/ n) algorithm for the real root problem

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annual ieee symposium on Foundations of computer science

作者： J.H. Reif Department of Computer Science Duke University USA

Given a univariate complex polynomial f(x) of degree n with rational coefficients expressed as a ratio of two integers

Given a univariate complex polynomial f(x) of degree n with rational coefficients expressed as a ratio of two integers <2/sup m/, the root problem is to find all the roots of f(x) up to specified precision 2/sup -/spl mu//. In this paper we assume the arithmetic model for computation. We give an algorithm for the real root problem: where all the roots of the polynomial are real. Our real root algorithm has time cost of O(nlog/sup 2/ n(log n+log b)), where b=m+/spl mu/, thus has time bound O(nlog/sup 3/ n) even in the case of high precision m+/spl mu//spl les/n/sup O(1/). this is within a small polylog factor of optimality, thus (perhaps surprisingly) upper bounding the arithmetic complexity of our real root problem to nearly the same as basic arithmetic operations on polynomials. We require only /spl pi/=O(n(/spl mu/+m+n)) bits of precision to carry out our computations. the Boolean complexity of our algorithm is a multiplicative factor of M(/spl pi/)=O(/spl pi/(log /spl pi/)loglog /spl pi/) more.< >

关键词： Eigenvalues and eigenfunctions Polynomials Arithmetic Symmetric matrices Read-write memory Contracts Algebra Computational modeling Postal services computer science

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