The Complexity Of Nonuniform Random Number Generation Pdf Download

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Non-Uniform Random Variate Generation

'.. This is 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 algorith ..'

Methods in simulation (e.g. For comparing similar systems with common random numbers) 1, 34, 21. True randomness may nevertheless be used for selecting the seed s 0. Then, the RNG can be viewed as an extensor of randomness, stretching a short random seed into a long sequence of random-looking numbers. Jan 01, 1976  The complexity of nonuniform random number generation. Citations (188) Save 📖 Papers frequently viewed together. Interval algorithm for random number generation. 1997 3.21 IEEE Transactions on Information Theory. 2 Authors (Te Sun Hao, M. Hoshi) 110 Citations. Non-uniform random variate generation. 1 Author (Luc Devroye.

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Abstract - Cited by 1019 (26 self) - Add to MetaCart
This is 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.

A new approach to the minimum cut problem

'.. Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..'
Abstract - Cited by 127 (9 self) - Add to MetaCart
Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the
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Boltzmann Samplers For The Random Generation Of Combinatorial Structures

'.. This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..'
Abstract - Cited by 107 (4 self) - Add to MetaCart
This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
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RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS

'.. We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for pro ..'
Abstract - Cited by 101 (12 self) - Add to MetaCart
We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k).
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Approximations of General Independent Distributions

'.. We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform boolean-valued random var ..'
Abstract - Cited by 34 (4 self) - Add to MetaCart
We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform boolean-valued random variables. Our results yield efficient constructions of small sets with low discrepancy in high dimensional space and have applications to derandomizing randomized algorithms. 1 Introduction The problem of constructing small sample spaces that "approximate" the independent distribution on n random variables has received considerable attention recently (cf. [6, Chor Goldreich] [8, Karp Wigderson], [11, Luby], [1, Alon Babai Itai], [13, Naor Naor], [2, Alon Goldreich Hastad Peralta], [3, Azar Motwani Naor]). The primary motivation for this line of research is that random variables that are "approximately" independent suffices for the analysis of many interesting randomized algorithm and hence c..

Symbolic and parametric model checking of discrete-time markov Chains

'.. daws at cs.ru.nl Abstract. We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rat ..'
Abstract - Cited by 31 (0 self) - Add to MetaCart
daws at cs.ru.nl Abstract. We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational, and rational functions when some probabilities are left unspecified as parameters of the system. This allows for parametric model checking by evaluating the regular expression for different parameter values, for instance, to study the influence of a lossy channel in the overall reliability of a randomized protocol. 1
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On the Information Rate of Secret Sharing Schemes

'.. We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ff ..'
Abstract - Cited by 30 (5 self) - Add to MetaCart
We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ffl, where ffl is an arbitrary positive constant. We also consider the problem of testing if one of these access structures is a sub-structure of an arbitrary access structure and we show that this problem is NP-complete. We provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rateOmegaGammate/3 n)=n). 1 Introduction A secret sharing scheme is a method to distribute a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P ; non-qualified to know s; cannot ..

On universal types

'.. We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978 ..'
Abstract - Cited by 25 (6 self) - Add to MetaCart
We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978). We show that the empirical probability distributions of any finite order of two sequences of the same universal type converge, in the variational sense, as the sequence length increases. Consequently, the normalized logarithms of the probabilities assigned by any kth order probability assignment to two sequences of the same universal type, as well as the kth order empirical entropies of the sequences, converge for all k. We study the size of a universal type class, and show that its asymptotic behavior parallels that of the conventional counterpart, with the LZ78 code length playing the role of the empirical entropy. We also estimate the number of universal types for sequences of length n, and show that it is of the form exp((1+o(1))γ n/log n) for a well characterized constant γ. We describe algorithms for enumerating the sequences in a universal type class, and for drawing a sequence from the class with uniform probability. As an application, we consider the problem of universal simulation of individual sequences. A sequence drawn with uniform probability from the universal type class of x n is an optimal simulation of x n in a well defined mathematical sense.
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Fully Dynamic Secret Sharing Schemes

'.. We consider secret sharing schemes in which the dealer is able (after a preprocessing stage) to activate a particular access structure out of a given set and/or to allow the participants to reconstruct different secrets (in different time instants) by sending them the same broadcast message. In this ..'
Abstract - Cited by 19 (1 self) - Add to MetaCart
We consider secret sharing schemes in which the dealer is able (after a preprocessing stage) to activate a particular access structure out of a given set and/or to allow the participants to reconstruct different secrets (in different time instants) by sending them the same broadcast message. In this paper we establish a formal setting to study secret sharing schemes of this kind. The security of the schemes presented is unconditional, since they are not based on any computational assumption. We give bounds on the size of the shares held by participants, on the size of the broadcast message, and on the randomness needed in such schemes. 1 Introduction A secret sharing scheme is a method of dividing a secret s among a set P of participants in such a way that: if the participants in A ` P are qualified to know the secret then by pooling together their information they can reconstruct the secret s; but any set A of participants not qualified to know s has absolutely no information on the..
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Efficient Approximation of Product Distributions

- in Proceedings of the 24th Annual ACM Symposium on Theory of Computing, 1998
'.. We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random var ..'
Abstract - Cited by 19 (1 self) - Add to MetaCart
We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random variables. Preliminary version has appeared in the Proceedings of the 24th ACM Symp. on Theory of Computing (STOC), pages 10--16, 1992. y Dept. of Electrical Engineering--Systems, Tel--Aviv University, Ramat--Aviv, Tel--Aviv 69978, Israel. Email: guy@eng.tau.ac.il. z Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: oded@wisdom.weizmann.ac.il. Research partially supported by grant No. 89-00312 from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel. x International Computer Science Institute, Berkeley, CA 94704, USA. Email: luby@icsi.berkeley.edu. Research supported in part by National Science Founda..

📒 Independent Random Sampling Methodsby Luca Martino

Genre : Computers
Publisher : Springer
ISBN_10 : 9783319726342
Copyright Year : 2018-03-31
File Format : All Formats
File Download : 280
Price : FREE

E-BOOK EXCERPT

This book systematically addresses the design and analysis of efficient techniques for independent random sampling. Both general-purpose approaches, which can be used to generate samples from arbitrary probability distributions, and tailored techniques, designed to efficiently address common real-world practical problems, are introduced and discussed in detail. In turn, the monograph presents fundamental results and methodologies in the field, elaborating and developing them into the latest techniques. The theory and methods are illustrated with a varied collection of examples, which are discussed in detail in the text and supplemented with ready-to-run computer code. The main problem addressed in the book is how to generate independent random samples from an arbitrary probability distribution with the weakest possible constraints or assumptions in a form suitable for practical implementation. The authors review the fundamental results and methods in the field, address the latest methods, and emphasize the links and interplay between ostensibly diverse techniques.

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