scholarly journals Distributional analysis of Robin Hood linear probing hashing with buckets

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Alfredo Viola
Keyword(s):  
Generating Functions ◽  
Bernoulli Numbers ◽  
Exact Distribution ◽  
Distributional Analysis ◽  
Asymptotic Results ◽  
Poisson Transform ◽  
Robin Hood ◽  
New Family ◽  
International Audience ◽  
The Cost

International audience This paper presents the first distributional analysis of a linear probing hashing scheme with buckets of size $b$. The exact distribution of the cost of successful searches for a $b \alpha$ -full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size $m$ and $n$ elements. A key element in the analysis is the use of a new family of numbers that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.

scholarly journals Distributional Analysis of the Parking Problem and Robin Hood Linear Probing Hashing with Buckets

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Alfredo Viola
Keyword(s):  
Bernoulli Numbers ◽  
Exact Distribution ◽  
Distributional Analysis ◽  
Asymptotic Results ◽  
Poisson Transform ◽  
Robin Hood ◽  
New Family ◽  
Parking Problem ◽  
International Audience ◽  
The Cost

International audience This paper presents the first distributional analysis of both, a parking problem and a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a b alpha-full table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size m and n elements. A key element in the analysis is the use of a new family of numbers, called Tuba Numbers, that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.

scholarly journals Enumeration of inscribed polyominos

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Alain Goupil ◽  
Hugo Cloutier ◽  
Fathallah Nouboud
Keyword(s):  
Generating Functions ◽  
Minimum Area ◽  
Lattice Trees ◽  
New Family ◽  
International Audience

International audience We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one. Nous introduisons une nouvelle famille de polyominos inscrits dans un rectangle de format donné pour lesquels des formules exactes et des séries génératrices sont présentées. Nous étudions en particulier les polyominos inscrits d'aire minimale et ceux d'aire minimale plus un. Ces résultats sont ensuite utilisés pour l'énumération de polyominos arbres inscrits dans un rectangle d'aire minimum plus un.

scholarly journals On the number of decomposable trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Stephan G. Wagner
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Limit Case ◽  
Asymptotic Results ◽  
Spanning Forest ◽  
International Audience ◽  
Generated Family ◽  
Fixed Tree

International audience A tree is called $k$-decomposable if it has a spanning forest whose components are all of size $k$. Analogously, a tree is called $T$-decomposable for a fixed tree $T$ if it has a spanning forest whose components are all isomorphic to $T$. In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of $k$-decomposable and $T$-decomposable trees from a so-called simply generated family of trees - we find that there is a surprisingly simple functional equation for the counting series of $k$-decomposable trees. In particular, we will study the limit case when $k$ goes to $\infty$. It turns out that the ratio of $k$-decomposable trees increases when $k$ becomes large.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals On a Class of Optimal Stopping Problems with Mixed Constraints

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Selection Strategy ◽  
Optimal Selection ◽  
Recursive Structure ◽  
Limiting Behavior ◽  
Mixed Constraints ◽  
Selection Strategies ◽  
International Audience ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Independent Identically Distributed

International audience Let X(1),X(2),...,X(n) be independent, identically distributed uniform random variables on [0, 1]. We can observe the outcomes sequentially and must select online at least r of them, and, moreover, in expectation at least mu >= r. Here mu need not be integer. We see X(k) as the cost of selecting item k and want to minimize the expected total cost under the described combined (r, mu)-constraint. We will see that an optimal selection strategy exists on the set S(n) of all selection strategies for which the decision at instant k may depend on the value X(k), on the number N(k) of selections up to time k and of the number n - k of forthcoming observations. Let sigma(r,mu)(n) be the corresponding S(n)-optimal selection strategy and v(r,mu)(n) its value. The main goal of this paper is to determine these and to understand the limiting behavior of v(r,mu)(n). After discussion of the specific character of this combination of two types of constraints we conclude that the S(n)-problem has a recursive structure and solve it in terms of a double recursion. Our interest will then focus on the limiting behavior of nv(r,mu)(n) as n -> infinity. This sequence converges and its limit allows for the interpretation of a normalized limiting cost L (r, mu) of the (r, mu)-constraint. Our main result is that L(r, mu) = g(r) ((mu - r)(2)/(2)) where g(r) is the r(th) iterate of the function g(x) = 1 + x + root 1 + 2x. Our motivation to study mixed-constraints problems is indicated by several examples of possible applications. We also shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies S n to the set of full-history-dependent and/or randomized strategies.

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

scholarly journals Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence: Extended abstract.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Order Term ◽  
Combinatorial Problem ◽  
Random Order ◽  
Priority Queues ◽  
Sorting Algorithm ◽  
Asymptotic Results ◽  
First Order ◽  
Asymptotically Normal ◽  
International Audience ◽  
Space Requirements

International audience We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order $n^{1/2}$, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order $n^{1/3}$. We also treat some variations, including priority queues and sock-sorting.

scholarly journals Enumerating alternating tree families

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Nous Obtenons ◽  
Asymptotic Results ◽  
Root Node ◽  
Ordered Trees ◽  
Exponential Generating Function ◽  
Random Node ◽  
Enumeration Problems ◽  
International Audience ◽  
Nous Examinons ◽  
Labelled Trees

International audience We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results. Nous étudions deux problèmes de dénombrement d'$\textit{arbres alternés haut-bas}$ : par définition, ce sont des arbres munis d'une racine et tels que, pour tout chemin partant de la racine, les valeurs $v_1,v_2,v_3,\ldots$ associées aux nœuds du chemin satisfont la chaîne d'inégalités $v_1 < v_2 > v_3 < v_4 > \cdots$. D'une part, nous considérons diverses familles d'arbres intéressantes du point de vue de l'analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et $d$-aires) et nous étudions pour chaque famille le nombre total $T_n$ d'arbres alternés haut-bas de taille $n$. Nous obtenons pour toutes les familles d'arbres considérées une caractérisation implicite de la fonction génératrice exponentielle $T(z)$. Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients $T_n$ de plusieurs familles d'arbres. D'autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l'influence de l'étiquetage alterné haut-bas sur l'allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d'un nœud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.

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