Author

# Alfredo Viola

Bio: Alfredo Viola is an academic researcher from University of the Republic. The author has contributed to research in topics: Linear probing & Finite field. The author has an hindex of 8, co-authored 39 publications receiving 331 citations.

##### Papers

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TL;DR: In this article, moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy are presented for full tables and sparse tables with a fixed filling ratio strictly smaller than one.

Abstract: This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n
3/2
) , the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices $ -\frac{1}{3},\frac{2}{3} $ .) For sparse tables, the construction cost has expectation O(n) , standard deviation O ( $ \sqrt{n} $ ), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.

145 citations

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TL;DR: This paper studies the distribution of individual displacements for the standard and the Robin Hood linear probing hashing algorithms when the a table of size m has n elements.

Abstract: This paper studies the distribution of individual displacements for the standard and the Robin Hood linear probing hashing algorithms. When the a table of size m has n elements, the distribution of the search cost of a random element is studied for both algorithms. Specifically, exact distributions for fixed m and n are found as well as when the table is α-full, and α strictly smaller than 1. Moreover, for full tables, limit laws for both algorithms are derived.

23 citations

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TL;DR: This work considers open addressing hashing and implements it by using the Robin Hood strategy; that is, in case of collision, the element that has traveled the farthest can stay in the slot and virtually matches the performance of multiple-choice hash methods.

Abstract: We consider open addressing hashing and implement it by using the Robin Hood strategy; that is, in case of collision, the element that has traveled the farthest can stay in the slot. We hash $\sim \alpha n$ elements into a table of size n where each probe is independent and uniformly distributed over the table, and $\alpha < 1$ is a constant. Let $M_n$ be the maximum search time for any of the elements in the table. We show that with probability tending to one, $M_n \in [ \log_2 \log n + \sigma, \log_2 \log n + \tau ]$ for some constants $\sigma, \tau$ depending upon $\alpha$ only. This is an exponential improvement over the maximum search time in case of the standard FCFS (firstcome first served) collision strategy and virtually matches the performance of multiple-choice hash methods.

20 citations

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TL;DR: The results strongly suggest that a suitable implementation of ν-find could be the method of choice in a practical setting and show that proportion-from-s-like strategies are optimal when s→∞.

Abstract: Quickselect with median-of-3 is largely used in practice and its behavior is fairly well understood. However, the following natural adaptive variant, which we call proportion-from-3, had not been previously analyzed: “choose as pivot the smallest of the sample if the relative rank of the sought element is below 1/3, the largest if the relative rank is above 2/3, and the median if the relative rank is between 1/3 and 2/3.” We first analyze the average number of comparisons made when using proportion-from-2 and then for proportion-from-3. We also analyze ν-find, a generalization of proportion-from-3 with interval breakpoints at ν and 1-ν. We show that there exists an optimal value of ν and we also provide the range of values of ν where ν-find outperforms median-of-3. Then, we consider the average total cost of these strategies, which takes into account the cost of both comparisons and exchanges. Our results strongly suggest that a suitable implementation of ν-find could be the method of choice in a practical setting. We also study the behavior of proportion-from-s with s>3 and in particular we show that proportion-from-s-like strategies are optimal when s→∞.

19 citations

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TL;DR: A complete characterization of the first order correlation immune Boolean functions that includes the functions that are 1-resilient is presented, and it is conjectured that the exact complete enumeration for general n is intractable.

Abstract: This paper presents a complete characterization of the first order correlation immune Boolean functions that includes the functions that are 1-resilient. The approach consists in defining an equivalence relation on the full set of Boolean functions with a fixed number of variables. An equivalence class in this relation, called a first-order correlation class, provides a measure of the distance between the Boolean functions it contains and the correlation-immune Boolean functions. The key idea consists on manipulating only the equivalence classes instead of the set of Boolean functions. To achieve this goal, a class operator is introduced to construct a class with n variables from two classes of n - 1 variables. In particular, the class of 1-resilient functions on n variables is considered. An original and efficient method to enumerate all the Boolean functions in this class is proposed by performing a recursive decomposition of classes with less variables. A bottom up algorithm provides the exact number of 1-resilient Boolean functions with seven variables which is 23478015754788854439497622689296. A tight estimation of the number of 1-resilient functions with eight variables is obtained by performing a partial enumeration. It is conjectured that the exact complete enumeration for general n is intractable.

15 citations

##### Cited by

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TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations

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01 Jan 2009

TL;DR: This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.

Abstract: Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

3,616 citations

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01 Jan 2011

TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

3,554 citations

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TL;DR: This handbook is a very useful handbook for engineers, especially those working in signal processing, and provides real data bootstrap applications to illustrate the theory covered in the earlier chapters.

Abstract: tions. Bootstrap has found many applications in engineering field, including artificial neural networks, biomedical engineering, environmental engineering, image processing, and radar and sonar signal processing. Basic concepts of the bootstrap are summarized in each section as a step-by-step algorithm for ease of implementation. Most of the applications are taken from the signal processing literature. The principles of the bootstrap are introduced in Chapter 2. Both the nonparametric and parametric bootstrap procedures are explained. Babu and Singh (1984) have demonstrated that in general, these two procedures behave similarly for pivotal (Studentized) statistics. The fact that the bootstrap is not the solution for all of the problems has been known to statistics community for a long time; however, this fact is rarely touched on in the manuscripts meant for practitioners. It was first observed by Babu (1984) that the bootstrap does not work in the infinite variance case. Bootstrap Techniques for Signal Processing explains the limitations of bootstrap method with an example. I especially liked the presentation style. The basic results are stated without proofs; however, the application of each result is presented as a simple step-by-step process, easy for nonstatisticians to follow. The bootstrap procedures, such as moving block bootstrap for dependent data, along with applications to autoregressive models and for estimation of power spectral density, are also presented in Chapter 2. Signal detection in the presence of noise is generally formulated as a testing of hypothesis problem. Chapter 3 introduces principles of bootstrap hypothesis testing. The topics are introduced with interesting real life examples. Flow charts, typical in engineering literature, are used to aid explanations of the bootstrap hypothesis testing procedures. The bootstrap leads to second-order correction due to pivoting; this improvement in the results due to pivoting is also explained. In the second part of Chapter 3, signal processing is treated as a regression problem. The performance of the bootstrap for matched filters as well as constant false-alarm rate matched filters is also illustrated. Chapters 2 and 3 focus on estimation problems. Chapter 4 introduces bootstrap methods used in model selection. Due to the inherent structure of the subject matter, this chapter may be difficult for nonstatisticians to follow. Chapter 5 is the most impressive chapter in the book, especially from the standpoint of statisticians. It provides real data bootstrap applications to illustrate the theory covered in the earlier chapters. These include applications to optimal sensor placement for knock detection and land-mine detection. The authors also provide a MATLAB toolbox comprising frequently used routines. Overall, this is a very useful handbook for engineers, especially those working in signal processing.

1,292 citations