1,655 research outputs found

    A Time-Space Tradeoff for Triangulations of Points in the Plane

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    In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm

    New Codes and Inner Bounds for Exact Repair in Distributed Storage Systems

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    We study the exact-repair tradeoff between storage and repair bandwidth in distributed storage systems (DSS). We give new inner bounds for the tradeoff region and provide code constructions that achieve these bounds.Comment: Submitted to the IEEE International Symposium on Information Theory (ISIT) 2014. This draft contains 8 pages and 4 figure

    An Exponential Lower Bound on the Complexity of Regularization Paths

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    For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal parameter much easier. It has been assumed that these piecewise linear solution paths have only linear complexity, i.e. linearly many bends. We prove that for the support vector machine this complexity can be exponential in the number of training points in the worst case. More strongly, we construct a single instance of n input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) = \Theta(2^d) many distinct subsets of support vectors occur as the regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure

    Memory-Adjustable Navigation Piles with Applications to Sorting and Convex Hulls

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    We consider space-bounded computations on a random-access machine (RAM) where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the input is NN elements, the length of the output is limited by the computation, and the capacity of the workspace is O(S)O(S) bits for some predetermined parameter SS. We present a state-of-the-art priority queue---called an adjustable navigation pile---for this restricted RAM model. Under some reasonable assumptions, our priority queue supports minimum\mathit{minimum} and insert\mathit{insert} in O(1)O(1) worst-case time and extract\mathit{extract} in O(N/S+lgS)O(N/S + \lg{} S) worst-case time for any SlgNS \geq \lg{} N. We show how to use this data structure to sort NN elements and to compute the convex hull of NN points in the two-dimensional Euclidean space in O(N2/S+NlgS)O(N^2/S + N \lg{} S) worst-case time for any SlgNS \geq \lg{} N. Following a known lower bound for the space-time product of any branching program for finding unique elements, both our sorting and convex-hull algorithms are optimal. The adjustable navigation pile has turned out to be useful when designing other space-efficient algorithms, and we expect that it will find its way to yet other applications.Comment: 21 page

    Self-Improving Algorithms

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    We investigate ways in which an algorithm can improve its expected performance by fine-tuning itself automatically with respect to an unknown input distribution D. We assume here that D is of product type. More precisely, suppose that we need to process a sequence I_1, I_2, ... of inputs I = (x_1, x_2, ..., x_n) of some fixed length n, where each x_i is drawn independently from some arbitrary, unknown distribution D_i. The goal is to design an algorithm for these inputs so that eventually the expected running time will be optimal for the input distribution D = D_1 * D_2 * ... * D_n. We give such self-improving algorithms for two problems: (i) sorting a sequence of numbers and (ii) computing the Delaunay triangulation of a planar point set. Both algorithms achieve optimal expected limiting complexity. The algorithms begin with a training phase during which they collect information about the input distribution, followed by a stationary regime in which the algorithms settle to their optimized incarnations.Comment: 26 pages, 8 figures, preliminary versions appeared at SODA 2006 and SoCG 2008. Thorough revision to improve the presentation of the pape
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