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On the asymptotic analysis of the Euclidean algorithm

G.H. Norton
1990 Journal of symbolic computation  
It would be interesting to see if these techniques could be generalized to the average analysis of a (terminating) minimal remainder Euclidean algorithm for the algebraic integers in a quadratic number  ...  Introduction For non-negative integers m, n let T(m, n) denote the number of divisions required to compute GCD(m, n) by the Euclidean algorithm.  ... 
doi:10.1016/s0747-7171(08)80036-3 fatcat:evi753nj2vf7lon3vuiekz56ki

Digits and continuants in euclidean algorithms. Ergodic versus tauberian theorems

Brigitte Vallée
2000 Journal de Théorie des Nombres de Bordeaux  
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms  ...  Average-case Analysis of Euclidean Algorithms We now come back to the precise analysis of the two Euclidean algorithms and we focus on two main quantities: the bit-complexity of Euclidean Algorithms and  ...  Main parameters for the analysis of Euclidean Algorithms.  ... 
doi:10.5802/jtnb.296 fatcat:w657vlij3ngnrok7ptp7j2brn4

A note on "Euclidean algorithms are Gaussian" by V. Baladi and B. Vallée

E. Cesaratto
2009 Journal of Number Theory  
Acknowledgments This work was done when the author spent one year in the GREYC laboratory with a post-doctoral position funded by CNRS.  ...  The author is very grateful to Viviane Baladi and Brigitte Vallée for many and valuable discussions. I also thank the referee for useful comments.  ...  Average-case analysis estimates the asymptotic expectation of C (when N → ∞), whereas distributional analysis determines the asymptotic distribution of C .  ... 
doi:10.1016/j.jnt.2009.02.018 fatcat:l6eyquwxmrbati2jz2tqnqxttu

Towards Asymptotically Optimal One-to-One PDP Algorithms for Capacity 2+ Vehicles [article]

Martin Olsen
2016 arXiv   pre-print
We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension d where n transportation requests are picked i.i.d. with a separate origin-destination pair for  ...  Secondly, we extend our algorithm to the classical single-vehicle PDP where the objective is to minimize the total distance traveled by the vehicle and present results indicating that the extended algorithm  ...  The plan computed by the algorithm is shown in Fig. 3 . Analysis of the PDP-C Algorithm We let SOL denote the value of the plan computed by the PDP-C algorithm.  ... 
arXiv:1610.09132v1 fatcat:rjzlv4gx6nf63poj2rl26ns22i

Analysis of fast versions of the Euclid Algorithm

Eda Cesaratto, Julien Clement, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée
2007 2007 Proceedings of the Fourth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
This work has been already presented in the ANALCO'07 conference, and a larger abstract already appeared in the proceedings of this conference [5] . A long paper on this subject is in preparation.  ...  Since 90, the Caen Group [16, 18, 17] has performed an average-case analysis of various parameters of a large class of Euclidean algorithms.  ...  On an input (A, B) of binary size n, the Interrupted Euclidean algorithm E [δ,δ+γ] starts at the index k of the execution of the Euclid Algorithm, as soon as the remainder A k has already lost δ n bits  ... 
doi:10.1137/1.9781611972979.14 dblp:conf/analco/CesarattoDLMV07 fatcat:gviclhfz7ze7vazxc3ms7vykt4

A Unifying Framework for the Analysis of a Class of Euclidean Algorithms [chapter]

Brigitte Vallée
2000 Lecture Notes in Computer Science  
We develop a general framework for the analysis of algorithms of a broad Euclidean type.  ...  These methods provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions.  ...  I wish to thank Pierre Ducos for earlier discussions of 1992, Charlie Lem ee and J er emie Bourdon for their master's theses closely related to this work, Thomas Prellberg for his explanations of the inducing  ... 
doi:10.1007/10719839_34 fatcat:zqomwaxgpnhd7gxp6nkcapavw4

On greedy geographic routing algorithms in sensing-covered networks

Guoliang Xing, Chenyang Lu, Robert Pless, Qingfeng Huang
2004 Proceedings of the 5th ACM international symposium on Mobile ad hoc networking and computing - MobiHoc '04  
In particular, we derive theoretical upper bounds on the network dilation of sensing-covered networks under greedy geographic routing algorithms.  ...  Our geometric analysis and simulation results demonstrate that existing greedy geographic routing algorithms can successfully find short routing paths based on local states in sensing-covered networks.  ...  The Euclidean dilation of R is defined similarly. DILATION ANALYSIS BASED ON DT In this section we study the dilation property of sensingcovered networks based on Delaunay Triangulation (DT).  ... 
doi:10.1145/989459.989465 dblp:conf/mobihoc/XingLPH04 fatcat:ggxqgori5be6fhjckey5xcdiiy

A new convergence analysis and perturbation resilience of some accelerated proximal forward–backward algorithms with errors

Daniel Reem, Alvaro De Pierro
2017 Inverse Problems  
This paper shows that inexact versions of a method of Beck and Teboulle (FISTA) preserve, in a Hilbert space setting, the same (non-asymptotic) rate of convergence under some assumptions on the decay rate  ...  A consequence of the analysis is that the accelerated method is perturbation resilient, making it suitable, in principle, for the superiorization methodology.  ...  ) New analysis of proximal algorithms 30 Nov 2015  ... 
doi:10.1088/1361-6420/33/4/044001 fatcat:boc2o5v2ibgv5iyvi4awgqysty

Vector Quantization based Power Allocation for Non - Ergodic Cognitive Radio Systems

Amrit Mukherjee, Department of Electronics and Communication Engineering, K L University, India, Amlan Datta, School of Electronics Engineering KIIT University, India
2016 Journal of Engineering Science and Technology Review  
On such type of systems, the power allocation for each corresponding nodes to be sensed by cognitive transmitter may not require all the time.  ...  This Letter addresses the problem of dynamic power allocation based on spectrum sensing outputs for non-ergodic systems.  ...  Analysis of Asymptotic Autocorrelation through Blind Source Separation The analysis of asymptotic autocorrelation through Blind source separation method has been implemented.  ... 
doi:10.25103/jestr.095.12 fatcat:to4rq3amtbdddpsmpz4ryylife

A Probabilistic Analysis of Christofides' Algorithm [chapter]

Markus Bläser, Konstatinos Panagiotou, B. V. Raghavendra Rao
2012 Lecture Notes in Computer Science  
As a first step towards obtaining an average case analysis of Christofides' algorithm, we provide a probabilistic analysis for the stochastic version of the algorithm for the Euclidean traveling salesman  ...  Our main result provides bounds for the length of the computed tour that hold almost surely. We also provide an experimental evaluation of Christofides's algorithm.  ...  Starting with Karp's work, there has been a lot of interest in the probabilistic analysis of heuristic algorithms for Euclidean optimization problems.  ... 
doi:10.1007/978-3-642-31155-0_20 fatcat:2ewyq64tbnca5dex34eukntwmy

On the practical importance of asymptotic optimality in certain heuristic algorithms

Harilaos N. Psaraftis
1984 Networks  
This article presents an informal discussion of the issue of asymptotic optimality of heuristics from the viewpoint of the operations research practitioner.  ...  Possible explanations of this phenomenon are offered and suggestions for further research toward a better understanding of this problem are presented.  ...  This article attempts to shed more light into those issues as follows: Section 2 focuses on an algorithm that has been recently developed for the routing of a fleet of vehicles to serve points on the Euclidean  ... 
doi:10.1002/net.3230140409 fatcat:yhljkes5avarlksq6g4j5cwkjy

(1 +i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

André Weilert
2000 Journal of symbolic computation  
Thus one can calculate the generator of a finite generated ideal as the GCD of the generators. the Euclidean algorithm and Lehmer-type GCD algorithms Since Euclid (cf.  ...  Thus one can calculate the GCD of two Gaussian integers using the ordinary Euclidean algorithm in its least remainder version.  ...  The author (1999; 2000) presented an asymptotically fast algorithm to compute the GCD in Z[i] based on the ideas of controlled Euclidean descent.  ... 
doi:10.1006/jsco.2000.0422 fatcat:3xtu4jbnmjgpfmee5r2qturs5i

Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem [chapter]

Anand Srivastav, Sören Werth
2007 Lecture Notes in Computer Science  
In the b-degree constrained Euclidean minimum spanning tree problem (bMST) we are given n points in [0; 1] d and a degree con-  ...  Weide [18] was the rst to distinguish the models in the probabilistic analysis of algorithms. We give a short overview of the theory of subadditive and superadditive Euclidean functionals.  ...  The work of Beardwood, Halton and Hammersley [4] motivated a large body of research on the probabilistic analysis of Euclidean optimization problems as minimum spanning tree, minimum perfect matching  ... 
doi:10.1007/978-3-540-77050-3_41 fatcat:bkvvm7yaofcczcslrtswkv5ufu

Asymptotically Optimal Algorithms for One-to-One Pickup and Delivery Problems With Applications to Transportation Systems

Kyle Treleaven, Marco Pavone, Emilio Frazzoli
2013 IEEE Transactions on Automatic Control  
Second, we asymptotically characterize the length of the optimal SCP tour.  ...  First, by embedding the problem within a stochastic framework, we present a novel algorithm for the SCP that: (i) is asymptotically optimal, i.e., it produces, almost surely, a solution approaching the  ...  CONCLUSION In this paper we have presented SPLICE, a class of asymptotically optimal, polynomial-time algorithms for the stochastic (Euclidean) SCP.  ... 
doi:10.1109/tac.2013.2259993 fatcat:nwc7ajvk2jhg3anmcwakvvjp5y

Sharp Estimates for the Main Parameters of the Euclid Algorithm [chapter]

Loïck Lhote, Brigitte Vallée
2006 Lecture Notes in Computer Science  
We perform a "dynamical analysis" which heavily uses the dynamical system underlying the Euclidean algorithm.  ...  We prove that the bit-complexity of the extended Euclid algorithm asymptotically follows a gaussian law, and we exhibit the speed of convergence towards the normal law.  ...  This result also plays a central rôle in the analysis of the so-called Interrupted Euclidean algorithm which stops as soon as the remainder v i is less than v δ 0 .  ... 
doi:10.1007/11682462_63 fatcat:vkqx6kq34nartc5gct3jcwlgte
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