Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

In the emerging area of sensor-based systems, a significant challenge is to develop scalable, fault-tolerant methods to extract useful information from the data the sensors collect. An approach to this data management problem is the use of sensor database systems, exemplified by TinyDB and Cougar, which allow users to perform aggregation queries such as MIN, COUNT and AVG on a sensor network. Due to power and range constraints, centralized approaches are generally impractical, so most systems use in-network aggregation to reduce network traffic. Also, aggregation strategies must provide fault-tolerance to address the issues of packet loss and node failures inherent in such a system. An unfortunate consequence of standard methods is that they typically introduce duplicate values, which must be accounted for to compute aggregates correctly. Another consequence of loss in the network is that exact aggregation is not possible in general. With this in mind, we investigate the use of approximate in-network aggregation using small sketches. Our contributions are as follows: 1) we generalize well known duplicateinsensitive sketches for approximating COUNT to handle SUM (and by extension, AVG and other aggregates), 2) we present and analyze methods for using sketches to produce accurate results with low communication and computation overhead (even on low-powered CPUs with little storage and no floating point operations), and 3) we present an extensive experimental validation of our methods. 1

Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

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Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases ɛ when the distance between the distributions is small (less than max ( 2 32 3 √ n, ɛ 4 √)) or large (more n than ɛ) in L1-distance. We also give an Ω(n 2/3 ɛ −2/3) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well.

We present a new and fast method, called the cross-entropy method, for finding the optimal solution of combinatorial and continuous nonconvex optimization problems with convex bounded domains. To find the optimal solution we solve a sequence of simple auxiliary smooth optimization problems based on Kullback-Leibler cross-entropy, importance sampling, Markov chain and Boltzmann distribution. We use importance sampling as an important ingredient for adaptive adjustment of the temperature in the Boltzmann distribution and use Kullback-Leibler cross-entropy to find the optimal solution. In fact, we use the mode of a unimodal importance sampling distribution, like the mode of beta distribution, as an estimate of the optimal solution for continuous optimization and Markov chains approach for combinatorial optimization. In the later case we show almost surely convergence of our algorithm to the optimal solution. Supporting numerical results for both continuous and combinatorial optimization problems are given as well. Our empirical studies suggest that the crossentropy method has polynomial in the size of the problem running time complexity.

This paper presents a hardware architecture for non-uniform random number generation, which allows the generator’s distribution to be modified at run-time without reconfiguration. The architecture is based on a piecewise linear approximation, using just one table lookup, one comparison and one subtract operation to map from a uniform source to an arbitrary non-uniform distribution, resulting in very low area utilisation and high speeds. Customisation of the distribution is fully automatic, requiring less than a second of CPU time to approximate a new distribution, and around 1000 cycles to switch distributions at run-time. Comparison with Gaussian specific generators show that the new architecture uses less than half the resources, provides a higher sample rate, and retains statistical quality for up to 50 billion samples, but can also generate other distributions. 1.

We initiate the cryptographic study of order-preserving symmetric encryption (OPE), a primitive suggested in the database community by Agrawal et al. (SIGMOD ’04) for allowing efficient range queries on encrypted data. Interestingly, we first show that a straightforward relaxation of standard security notions for encryption such as indistinguishability against chosen-plaintext attack (IND-CPA) is unachievable by a practical OPE scheme. Instead, we propose a security notion in the spirit of pseudorandom functions (PRFs) and related primitives asking that an OPE scheme look “as-random-as-possible ” subject to the order-preserving constraint. We then design an efficient OPE scheme and prove its security under our notion based on pseudorandomness of an underlying blockcipher. Our construction is based on a natural relation we uncover between a random order-preserving function and the hypergeometric probability distribution. In particular, it makes black-box use of an efficient sampling algorithm for the latter. 1

In the emerging area of sensor-based systems, a significant challenge is to develop scalable, fault-tolerant methods to extract useful information from the data the sensors collect. An approach to this data management problem is the use of sensor “database” systems, which allow users to perform aggregation queries on the readings of a sensor network. Due to power and range constraints, centralized approaches are generally impractical, so most systems use in-network aggregation to reduce network traffic. However, these aggregation strategies become bandwidthintensive when combined with the fault-tolerant, multi-path routing methods often used in these environments. In order to avoid this expense, we investigate the use of approximate in-network aggregation using small sketches and we survey robust and scalable methods for computing duplicate-sensitive aggregates.

C--RAND is a system of Turbo--C routines and functions intended for use on microcomputers. It contains up--to--date random number generators for more than thirty univariate distributions. For some important distributions the user has the choice between extremely fast but rather complicated methods and somewhat slower but also much simpler procedures. Menu driven demo programs allow to test and analyze the generators with regard to speed and quality of the output. 1.