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77
Quasi-Randomized Path Planning
- In Proc. IEEE Int’l Conf. on Robotics and Automation
, 2001
"... We propose the use of quasi-random sampling techniques for path planning in high-dimensional conguration spaces. Following similar trends from related numerical computation elds, we show several advantages oered by these techniques in comparison to random sampling. Our ideas are evaluated in the con ..."
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Cited by 74 (8 self)
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We propose the use of quasi-random sampling techniques for path planning in high-dimensional conguration spaces. Following similar trends from related numerical computation elds, we show several advantages oered by these techniques in comparison to random sampling. Our ideas are evaluated in the context of the probabilistic roadmap (PRM) framework. Two quasi-random variants of PRM-based planners are proposed: 1) a classical PRM with quasi-random sampling, and 2) a quasi-random Lazy-PRM. Both have been implemented, and are shown through experiments to oer some performance advantages in comparison to their randomized counterparts. 1 Introduction Over two decades of path planning research have led to two primary trends. In the 1980s, deterministic approaches provided both elegant, complete algorithms for solving the problem, and also useful approximate or incomplete algorithms. The curse of dimensionality due to high-dimensional conguration spaces motivated researchers from the 199...
Range counting over multidimensional data streams
- Discrete & Computational Geometry
, 2004
"... \Lambda \Lambda Abstract We consider the problem of approximate range counting over streams of d-dimensional points. In the data stream model, the algorithm makes a single scan of the data, which is presented in an arbitrary order, and computes a compact summary (called a sketch). The sketch, whose ..."
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Cited by 30 (0 self)
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\Lambda \Lambda Abstract We consider the problem of approximate range counting over streams of d-dimensional points. In the data stream model, the algorithm makes a single scan of the data, which is presented in an arbitrary order, and computes a compact summary (called a sketch). The sketch, whose size depends on the approximation parameter ", can be used to count the number of points inside a query range within additive error "n, where n is the size of the stream. We present several results, deterministic and randomized, for both rectangle and halfplane ranges. 1 Introduction Data streams have emerged as an important paradigm for processing data that arrives and needs to be processed continuously. For instance, telecom service providers routinely monitor packet flows through their networks to infer usage patterns and signs of attack, or to optimize their routing tables. Financial markets, banks, web servers, and news organizations also generate rapid and continuous data streams.
A randomized quasi-Monte Carlo simulation method for Markov chains
- Operations Research
, 2007
"... Abstract. We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional highly-unifor ..."
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Cited by 27 (9 self)
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Abstract. We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional highly-uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. This technique is effective in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.
Mergeable Summaries
"... 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 single summary on the union of the two data sets, while preserving the error and size guarantees. This property means t ..."
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Cited by 22 (7 self)
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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 single summary on the union of the two data sets, while preserving the error and size guarantees. This property means that the summaries can be merged in a way like other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the data sets. 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 ε-approximate heavy hitters, there is a deterministic mergeable summary of size O(1/ε); for ε-approximate quantiles, there is a deterministic summary of size O ( 1 log(εn)) that has a restricted form of mergeability, ε and a randomized one of size O ( 1 1 log3/2) with full merge-ε ε ability. We also extend our results to geometric summaries such as ε-approximations and ε-kernels. We also achieve two results of independent interest: (1) we provide the best known randomized streaming bound for ε-approximate quantiles that depends only on ε, of size O ( 1 1 log3/2), and (2) we demonstrate that the MG and the ε ε SpaceSaving summaries for heavy hitters are isomorphic. Supported by NSF under grants CNS-05-40347, IIS-07-
From Discrepancy to Declustering: Near-optimal multidimensional declustering strategies for range queries (Extended Abstract)
, 2001
"... Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Given a declustering scheme D, its response time with respect to a query Q, rt(Q), is defined to be the maximum number of disk blocks of the query stored by the scheme in any one of the disks. If |Q| is the ..."
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Cited by 22 (2 self)
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Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Given a declustering scheme D, its response time with respect to a query Q, rt(Q), is defined to be the maximum number of disk blocks of the query stored by the scheme in any one of the disks. If |Q| is the number of data blocks in Q and M is the number of disks then rt(Q) is at least |Q|/M. One way to evaluate the performance of D with respect to a set of queries Q is to measure its additive error- the maximum difference between rt(Q) from |Q|/M over all range queries Q ∈ Q. In this paper, we consider the problem of designing declustering schemes for uniform multidimensional data arranged in a d-dimensional grid so that their additive errors with respect to range queries are as small as possible. It has been shown that such declustering schemes will have an additive error of Ω(log M) when d = 2 and Ω(log d−1 2 M) when d> 2 with respect to range queries. Asymptotically optimal declustering schemes exist for 2dimensional data. For data in larger dimensions, however, the best bound for additive errors is O(M d−1), which is extremely large. In this paper, we propose the two declustering schemes based on low discrepancy points in d-dimensions. When d is fixed, both schemes have an additive error of O(log d−1 M) with respect to range queries provided certain conditions are satisfied: the first scheme requires d ≥ 3 and M to be a power of a prime where the prime is at least d while the second scheme requires the size of the data to grow within some polynomial of M, with no restriction on
The Geometry of Differential Privacy: The Sparse and Approximate Cases
, 2012
"... In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work [BLR08,RR10,DRV10,HT10,HR10,LHR+10,BDKT12]. For a given set ..."
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Cited by 16 (5 self)
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In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work [BLR08,RR10,DRV10,HT10,HR10,LHR+10,BDKT12]. For a given set of d linear queries over a database x ∈ RN, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, [HT10, BDKT12] give an O(log2 d) approximation to the optimal mechanism. Our first contribution is to give an O(log2 d) approximation guarantee for the case of (ε, δ)-differential privacy. Our mechanism is simple, efficient and adds carefully chosen correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of [MN12], using tools from convex geometry. We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when d> n, ‖x‖1. The lower bounds used in the previous approximation algorithm no longer apply, and in fact better mechanisms are known in this setting [BLR08,RR10,HR10,GHRU11,GRU12]. Our second main contribution is to give an (ε, δ)-differentially private mechanism that for a given query set A and an upper bound n on ‖x‖1, has mean squared error within polylog(d,N) of the optimal for A and n. This approximation is achieved by coupling the Gaussian noise addition approach with linear regression over the `1 ball. Additionally, we show a similar polylogarithmic approximation guarantee for the best ε-differentially private mechanism in this sparse setting. Our work also shows that for arbitrary counting queries, i.e. A with entries in {0, 1}, there is an ε-differentially private mechanism with expected error Õ(√n) per query, improving on the Õ(n 2 3) bound of [BLR08], and matching the lower bound implied by [DN03] up to logarithmic factors. The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix A.
Shape Fitting on Point Sets with Probability Distributions
"... Abstract. We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prov ..."
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Cited by 14 (6 self)
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Abstract. We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prove approximation guarantees for the algorithms with respect to the full probability distributions. We then empirically demonstrate that our algorithms are simple and practical, solving for a constant hidden by asymptotic analysis so that a user can reliably trade speed and size for accuracy. 1
Asymptotically optimal declustering schemes for range queries
- in 8th International Conference on Database Theory, Lecture Notes In Computer Science
, 2001
"... d\Gamma 1 2) for d-dim schemes and to \Omega (log M) for 2-dim schemes, thus proving that the ..."
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Cited by 12 (2 self)
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d\Gamma 1 2) for d-dim schemes and to \Omega (log M) for 2-dim schemes, thus proving that the
A weighted error metric and optimization method for antialiasing patterns
- COMPUTER GRAPHICS FORUM
, 2006
"... Displaying a synthetic image on a computer display requires determining the colors of individual pixels. To avoid aliasing, multiple samples of the image can be taken per pixel, after which the color of a pixel may be computed as a weighted sum of the samples. The positions and weights of the sample ..."
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Cited by 11 (1 self)
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Displaying a synthetic image on a computer display requires determining the colors of individual pixels. To avoid aliasing, multiple samples of the image can be taken per pixel, after which the color of a pixel may be computed as a weighted sum of the samples. The positions and weights of the samples play a major role in the resulting image quality, especially in real-time applications where usually only a handful of samples can be afforded per pixel. This paper presents a new error metric and an optimization method for antialiasing patterns used in image reconstruction. The metric is based on comparing the pattern against a given reference reconstruction filter in spatial domain and it takes into account psychovisually measured angle-specific acuities for sharp features.
