328 research outputs found

    Deterministic Sampling and Range Counting in Geometric Data Streams

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    We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

    Blocking Coloured Point Sets

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    This paper studies problems related to visibility among points in the plane. A point xx \emph{blocks} two points vv and ww if xx is in the interior of the line segment vwˉ\bar{vw}. A set of points PP is \emph{kk-blocked} if each point in PP is assigned one of kk colours, such that distinct points v,wPv,w\in P are assigned the same colour if and only if some other point in PP blocks vv and ww. The focus of this paper is the conjecture that each kk-blocked set has bounded size (as a function of kk). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets {n1,n2,n3,n4}\{n_1,n_2,n_3,n_4\} such that some 4-blocked set has exactly nin_i points in the ii-th colour class. Amongst other results, for infinitely many values of kk, we construct kk-blocked sets with k1.79...k^{1.79...} points

    Generalizations of Tucker-Fan-Shashkin lemmas

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    Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.Comment: 10 pages, 2 figure

    Multidimensional Quasi-Monte Carlo Malliavin Greeks

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    We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fourni\'e et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.Comment: 22 pages, 6 figure

    Cutting the same fraction of several measures

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    We study some measure partition problems: Cut the same positive fraction of d+1d+1 measures in Rd\mathbb R^d with a hyperplane or find a convex subset of Rd\mathbb R^d on which d+1d+1 given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

    On Compact Routing for the Internet

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    While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

    Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

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    In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).Comment: 16 page

    Statistical mechanics of budget-constrained auctions

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    Finding the optimal assignment in budget-constrained auctions is a combinatorial optimization problem with many important applications, a notable example being the sale of advertisement space by search engines (in this context the problem is often referred to as the off-line AdWords problem). Based on the cavity method of statistical mechanics, we introduce a message passing algorithm that is capable of solving efficiently random instances of the problem extracted from a natural distribution, and we derive from its properties the phase diagram of the problem. As the control parameter (average value of the budgets) is varied, we find two phase transitions delimiting a region in which long-range correlations arise.Comment: Minor revisio

    Unsplittable coverings in the plane

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    A system of sets forms an {\em mm-fold covering} of a set XX if every point of XX belongs to at least mm of its members. A 11-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body CC, there exists a constant m=m(C)m=m(C) such that every mm-fold covering of the plane with translates of CC splits into 22 coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every mm, an unsplittable mm-fold covering of the plane with translates of any open convex body CC which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets CC do not misbehave, they satisfy the conjecture: every 33-fold covering of any region of the plane by translates of such a set CC splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant c>0c>0 such that, for any positive integer mm, every mm-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} c2m/2c2^{m/2} sets
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