scholarly journals On the Number of 2-Protected Nodes in Tries and Suffix Trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Jeffrey Gaither ◽  
Yushi Homma ◽  
Mark Sellke ◽  
Mark Daniel Ward
Keyword(s):  
Suffix Trees ◽  
First Order ◽  
International Audience

International audience We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with $n$ leaves, the number of 2-protected nodes is approximately 0.803$n$, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe.

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 The First-Order Theory of Ordering Constraints over Feature Trees

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Martin Müller ◽  
Joachim Niehren ◽  
Ralf Treinen
Keyword(s):  
Finite Automata ◽  
Order Theory ◽  
Inclusion Problem ◽  
First Order ◽  
First Order Theory ◽  
Infinite Trees ◽  
Ordering Constraints ◽  
International Audience ◽  
Existential Quantification ◽  
Feature Trees

International audience The system FT< of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate the first-order theory of FT< and its fragments in detail, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT< is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We show that the entailment problem of FT< with existential quantification is PSPACE-complete. So far, this problem has been shown decidable, coNP-hard in case of finite trees, PSPACE-hard in case of arbitrary trees, and cubic time when restricted to quantifier-free entailment judgments. To show PSPACE-completeness, we show that the entailment problem of FT< with existential quantification is equivalent to the inclusion problem of non-deterministic finite automata. Available at http://www.ps.uni-saarland.de/Publications/documents/FTSubTheory_98.pdf

scholarly journals Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Alexandre Boudet
Keyword(s):  
Higher Order ◽  
Unification Algorithm ◽  
First Order ◽  
Equational Theories ◽  
Order Case ◽  
International Audience ◽  
Syntactic Theories

International audience We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.

scholarly journals Decomposable graphs and definitions with no quantifier alternation

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Oleg Pikhurko ◽  
Joel Spencer ◽  
Oleg Verbitsky
Keyword(s):  
The Other ◽  
Connected Components ◽  
Large Graphs ◽  
First Order ◽  
Minimum Number ◽  
International Audience ◽  
Decomposable Graphs ◽  
Number Of Iterations ◽  
Quantifier Alternation ◽  
Binary Logarithm

International audience Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.

scholarly journals Asymptotic behaviour of a non-commutative rational series with a nonnegative linear representation

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Philippe Dumas ◽  
Helger Lipmaa ◽  
Johan Wallén
Keyword(s):  
Asymptotic Behaviour ◽  
Analysis Of Algorithms ◽  
Analytic Number Theory ◽  
Linear Representation ◽  
Numeration System ◽  
First Order ◽  
Rational Series ◽  
International Audience ◽  
Nonnegative Coefficients ◽  
The Mean

Analysis of Algorithms International audience We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.

scholarly journals Analysis of the multiplicity matching parameter in suffix trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Mark Daniel Ward ◽  
Wojciech Szpankowski
Keyword(s):  
Suffix Tree ◽  
Recurrence Relations ◽  
Complex Analysis ◽  
Analysis Of Algorithms ◽  
Suffix Trees ◽  
Logarithmic Series Distribution ◽  
International Audience ◽  
Number Of Leaves ◽  
Branching Point ◽  
Logarithmic Series

International audience In a suffix tree, the multiplicity matching parameter (MMP) $M_n$ is the number of leaves in the subtree rooted at the branching point of the $(n+1)$st insertion. Equivalently, the MMP is the number of pointers into the database in the Lempel-Ziv '77 data compression algorithm. We prove that the MMP asymptotically follows the logarithmic series distribution plus some fluctuations. In the proof we compare the distribution of the MMP in suffix trees to its distribution in tries built over independent strings. Our results are derived by both probabilistic and analytic techniques of the analysis of algorithms. In particular, we utilize combinatorics on words, bivariate generating functions, pattern matching, recurrence relations, analytical poissonization and depoissonization, the Mellin transform, and complex analysis.

scholarly journals Deciding whether the ordering is necessary in a Presburger formula

2010 ◽  
Vol Vol. 12 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Christian Choffrut ◽  
Achille Frigeri
Keyword(s):  
First Order ◽  
International Audience

Automata, Logic and Semantics International audience We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.

scholarly journals Towards automated proofs of observational properties

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Narjes Berregeb ◽  
Riadh Robbana ◽  
Ashish Tiwari
Keyword(s):  
Classical Sense ◽  
First Order ◽  
Finite Set ◽  
International Audience

International audience Observational theories are a generalization of first-order theories where two objects are observationally equal if they cannot be distinguished by experiments with observable results. Such experiments, called contexts, are usually infinite. Therfore, we consider a special finite set of contexts, called cover-contexts, ''\emphcovering'' all the observable contexts. Then, we show that to prove that two objects are observationally equal, it is sufficient to prove that they are equal (in the classical sense) under these cover-contexts. We give methods based on rewriting techniques, for constructing such cover-contexts for interesting classes of observational specifications.

scholarly journals q-gram analysis and urn models

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Pierre Nicodème
Keyword(s):  
Numerical Simulations ◽  
Random Sequence ◽  
Bernoulli Model ◽  
Urn Models ◽  
Fixed Size ◽  
Border Effects ◽  
First Order ◽  
International Audience ◽  
Dependent Model ◽  
Independent Model

International audience Words of fixed size q are commonly referred to as $q$-grams. We consider the problem of $q$-gram filtration, a method commonly used to speed upsequence comparison. We are interested in the statistics of the number of $q$-grams common to two random texts (where multiplicities are not counted) in the non uniform Bernoulli model. In the exact and dependent model, when omitting border effects, a $q$-gramin a random sequence depends on the $q-1$ preceding $q$-grams. In an approximate and independent model, we draw randomly a $q$-gram at each position, independently of the others positions. Using ball and urn models, we analyze the independent model. Numerical simulations show that this model is an excellent first order approximationto the dependent model. We provide an algorithm to compute the moments.

scholarly journals Random graphs with bounded maximum degree: asymptotic structure and a logical limit law

2012 ◽  
Vol Vol. 14 no. 2 ◽  
Author(s):  
Vera Koponen
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Undirected Graphs ◽  
Asymptotic Structure ◽  
Typical Structure ◽  
Fixed Integer ◽  
First Order ◽  
International Audience

General International audience For any fixed integer R≥2 we characterise the typical structure of undirected graphs with vertices 1,...,n and maximum degree R, as n tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If R≥5 then also an unlabelled limit law holds.

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