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32
Consistent digital rays
 Discrete Comput. Geom
"... Given a fixed origin o in the ddimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axio ..."
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Given a fixed origin o in the ddimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n × n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital starshaped regions centered at o which we use to design efficient algorithms for image processing problems. 1
A Counterexample to Beck’s Conjecture on the Discrepancy of Three Permutations
, 2011
"... Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. József Beck conjectured (c. 1982) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3 ..."
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Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. József Beck conjectured (c. 1982) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3 k, we exhibit three permutations whose corresponding set system has discrepancy Ω(log n). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck’s conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to ℓ permutations has discrepancy O(√ℓ) [SST01].
Project Description 1. Results from Prior NSF Support
"... approach to geometric functional analysis and applications”. The amount of this award is $94,790 ..."
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approach to geometric functional analysis and applications”. The amount of this award is $94,790
Planification De Mouvement Par échantillonnage Aléatoire
"... Introduction Un syst`eme robotique agit par le mouvement dans un monde physique. La capacit'e de planification de mouvement est donc une composante essentielle de l'autonomie du syst`eme. La planification de mouvement constitue un domaine de recherche tr`es actif ces vingt derni`eres ann'ees dans l ..."
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Introduction Un syst`eme robotique agit par le mouvement dans un monde physique. La capacit'e de planification de mouvement est donc une composante essentielle de l'autonomie du syst`eme. La planification de mouvement constitue un domaine de recherche tr`es actif ces vingt derni`eres ann'ees dans la communaut'e robotique depuis les travaux pionniers de LozanoPerez au MIT. Dans sa version la plus simple, le probl`eme est intrins `equement difficile ; sa complexit'e croit exponentiellement avec le nombre de degr'es de libert'e du (ou des) mobile(s). Traditionnellement l'algorithmique du mouvement conduisait soit `a des m'ethodes exactes et compl`etes, soit `a des m'ethodes heuristiques [23]. La complexit'e des premi`eres rendait leur utilisation r'edhibitoire pour des syst`emes de dimension 'elev'ee (sup'erieure `a quatre). Les secondes souffraient des inconv'enients li'es `a la non compl'etude. Depuis une dizaine d'ann'ees sont apparues de nouvelles approches non d'eterminis
Lowdiscrepancy Lattice Sets and QMC Integration
, 2002
"... Many lowdiscrepancy sets are suitable for quasiMonte Carlo integration. ..."
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Many lowdiscrepancy sets are suitable for quasiMonte Carlo integration.
Mergeable Coresets
, 2011
"... We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a summary on the two data sets combined together, while preserving the error and size guarantees. This property means tha ..."
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We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a summary on the two data sets combined together, while preserving the error and size guarantees. This property means that the summary can be treated like other algebraic objects such as sum and max, which is especially useful for computing summaries on massive distributed data. Many data summaries are trivially mergeable by construction, most notably those based on linear transformations. But some other fundamental ones like those for heavy hitters and quantiles, are not (known to be) mergeable. In this paper, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for εapproximate heavy hitters, there is a deterministic mergeable summary log(εn)) that has a restricted form of mergeability, and a randomized one of size O ( 1 ε ε) with full mergeability. We also extend our results to geometric summaries such as εapproximations and εkernels. of size O(1/ε); for εapproximate quantiles, there is a deterministic summary of size O ( 1 ε 1
Discrepancy after adding . . .
"... We show that the hereditary discrepancy of a hypergraph F on n pointsincreases by a factor of at most O(log n) when one adds a new edge to F. ..."
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We show that the hereditary discrepancy of a hypergraph F on n pointsincreases by a factor of at most O(log n) when one adds a new edge to F.
Quantum Lower Bounds by Entropy Numbers
, 2006
"... We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of informationbased complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp ..."
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We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of informationbased complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp spaces and of Sobolev embeddings. 1
MultiPole Field Persistent Routing with . . .
, 2011
"... This work addresses the problem of balancing the spatial distribution of the routingload among the nodes in a given sensor network and the tradeoff that can be achieved for providing certain level of quality of service (QoS) guarantees. For highdensity density networks, several studies have propos ..."
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This work addresses the problem of balancing the spatial distribution of the routingload among the nodes in a given sensor network and the tradeoff that can be achieved for providing certain level of quality of service (QoS) guarantees. For highdensity density networks, several studies have proposed field fieldbased routing paradigms to uniformly distribute the traffic load throughout the network. However, as network density decreases, we observe major shortcomings of the current st stateoftheart: (i) pathmerging merging leads to a reduction of path diversity, and (ii) the paths directed towards the border of the network merge into a single path along the border. These path merging effects decrease significantly the energy balance, and as consequence, onsequence, the lifetime of the network. In this article, we propose a novel mechanism to enable better load balancing for singlesource single and multiplesource source scenarios, while minimizing the cost of the tradeoff for bounding the endtoend end packet delivery la latencies. tencies. Our evaluations demonstrate that by using the proposed methodology, the network lifetime can be significantly prolonged, when longterm pointtopoint queries are considered.
QUASIMONTE CARLO METHODS IN FINANCE
"... We review the basic principles of QuasiMonte Carlo (QMC) methods, the randomizations that turn them into variancereduction techniques, and the main classes of constructions underlying their implementations: lattice rules, digital nets, and permutations in different bases. QMC methods are designed t ..."
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We review the basic principles of QuasiMonte Carlo (QMC) methods, the randomizations that turn them into variancereduction techniques, and the main classes of constructions underlying their implementations: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate integrals over the sdimensional unit hypercube, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of lowdimensional smooth functions). Such QMCfriendly integrals are encountered frequently in computational finance and risk analysis. We give examples and provide computational results that illustrate the efficiency improvement achieved. 1