scholarly journals Digital Trees and Memoryless Sources: from Arithmetics to Analysis

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Philippe Flajolet ◽  
Mathieu Roux ◽  
Brigitte Vallée
Keyword(s):  
Probabilistic Models ◽  
Asymptotic Form ◽  
Leader Election ◽  
Text Compression ◽  
Digital Trees ◽  
Arithmetic Properties ◽  
International Audience ◽  
Error Terms ◽  
Dynamic Hashing

International audience Digital trees, also known as $\textit{"tries''}$, are fundamental to a number of algorithmic schemes, including radix-based searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the $\textit{memoryless source}$, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters' expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous $\textit{Dirichlet series}$. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific $\textit{arithmetic properties}$, especially irrationality measures, of the sources under consideration.

Characterization of the Uncertainties in the Inspection Results of Ultrasonic Intelligent Pigs

2013 ◽  
Author(s):  
Mamdouh M. Salama ◽  
Bruce J. Nestleroth ◽  
Marc A. Maes ◽  
Chris Dash
Keyword(s):  
Probabilistic Models ◽  
Service Providers ◽  
Quantitative Risk Assessment ◽  
Life Extension ◽  
Internal Corrosion ◽  
Integrity Assessment ◽  
Diameter Pipe ◽  
Length Width ◽  
Pull Through

In-Line Inspections using magnetic flux leakage (MFL) and the Ultrasonic (UT) intelligent pigs are the most common tools used to assess the integrity of pipelines. But, both MFL and UT inspection results are subject to various sources of uncertainties which must be quantified and accounted for in the integrity assessment of the inspected pipeline. A series of pull-through tests (PTT) of seven MFL tools and two UT tools from five service providers was performed on a 12-inch diameter pipe containing pre-existing internal corrosion defects of various length, width, and depth, and located in a variety of circumferential and longitudinal positions. The results of these tests are used to quantify the detectability statistics and the sizing uncertainties of the different tools for future use in developing calibrated probabilistic models for reliability based inspection, quantitative risk assessment and life extension studies for pipelines. The results of the MFL tools were presented in 2012 OMAE conference and this paper presents the results of the two UT tools.

scholarly journals Lattice reduction in two dimensions: analyses under realistic probabilistic models

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Brigitte Vallée ◽  
Antonio Vera
Keyword(s):  
Dynamical Systems ◽  
Probabilistic Models ◽  
Two Dimensions ◽  
Lattice Reduction ◽  
Lll Algorithm ◽  
Transfer Operators ◽  
International Audience ◽  
Dimension 2

International audience The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm "inside'' the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals A probabilistic analysis of a leader election algorithm

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Hanene Mohamed
Keyword(s):  
Asymptotic Behavior ◽  
Probabilistic Analysis ◽  
Probabilistic Approach ◽  
Leader Election ◽  
Hitting Time ◽  
Elimination Process ◽  
International Audience ◽  
Election Algorithm ◽  
The Cost ◽  
Leader Election Algorithm

International audience A leader election algorithm is an elimination process that divides recursively into tow subgroups an initial group of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1. In this paper the biased case is analyzed. We are interested in the cost of the algorithm e. the number of operations needed until the algorithm stops. Using a probabilistic approach, the asymptotic behavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequences on [0,1].

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals An optimal cardinality estimation algorithm based on order statistics and its full analysis

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Jérémie Lumbroso
Keyword(s):  
Order Statistics ◽  
Estimation Algorithm ◽  
Fine Tuning ◽  
Simple Analysis ◽  
Relative Precision ◽  
Asymptotic Regime ◽  
Cardinality Estimation ◽  
International Audience ◽  
Full Analysis

International audience Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.

scholarly journals Self-complementing permutations of k-uniform hypergraphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Artur Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Unique Integer ◽  
International Audience ◽  
Sigma E

Graphs and Algorithms International audience A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.

scholarly journals Asymptotics of polygons in restricted geometries subject to a force

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Nicholas Beaton ◽  
Jeremy Eng ◽  
Christine Soteros
Keyword(s):  
Free Energy ◽  
Asymptotic Form ◽  
Restricted Geometry ◽  
International Audience

International audience We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z3. These polygons are subjected to a force f, parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. In this extended abstract we obtain and prove the asymptotic form of the free energy in the limit f → −∞. We conjecture that the f → −∞ asymptote is the same as the free energy of Hamiltonian polygons, which visit every vertex in a L × M × N box.

scholarly journals Solving equations over small unary algebras

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Przemyslaw Broniek
Keyword(s):  
Polynomial Time ◽  
Partial Characterization ◽  
Polynomial Equations ◽  
Solving Equations ◽  
International Audience ◽  
Np Complete ◽  
System Of Polynomial Equations

International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.

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