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## Algorithms, Volume 3, Issue 3 (September 2010), Pages 216-328

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# Research

Open AccessArticle Algorithmic Solution of Stochastic Differential Equations
Algorithms 2010, 3(3), 216-223; doi:10.3390/a3030216
Received: 17 May 2010 / Revised: 15 June 2010 / Accepted: 29 June 2010 / Published: 1 July 2010
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Abstract
This brief note presents an algorithm to solve ordinary stochastic differential equations (SDEs). The algorithm is based on the joint solution of a system of two partial differential equations and provides strong solutions for finite-dimensional systems of SDEs driven by standard Wiener processes
This brief note presents an algorithm to solve ordinary stochastic differential equations (SDEs). The algorithm is based on the joint solution of a system of two partial differential equations and provides strong solutions for finite-dimensional systems of SDEs driven by standard Wiener processes and with adapted initial data. Several examples illustrate its use. Full article
Open AccessArticle Segment LLL Reduction of Lattice Bases Using Modular Arithmetic
Algorithms 2010, 3(3), 224-243; doi:10.3390/a3030224
Received: 28 May 2010 / Accepted: 29 June 2010 / Published: 12 July 2010
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Abstract
The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction
The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction algorithm that generates lattice basis satisfying a weaker condition than the LLL reduced basis with O(n) improvement than the LLL algorithm. In this paper we combine Storjohann’s modular arithmetic approach with the segment-LLL approach to further improve the worst case complexity of the segment-LLL algorithms by a factor of n0.5. Full article
Open AccessArticle An O(n)-Round Strategy for the Magnus-Derek Game
Algorithms 2010, 3(3), 244-254; doi:10.3390/a3030244
Received: 8 June 2010 / Revised: 6 July 2010 / Accepted: 8 July 2010 / Published: 15 July 2010
Cited by 2 | PDF Full-text (241 KB) | HTML Full-text | XML Full-text
Abstract
We analyze further the Magnus-Derek game, a two-player game played on a round table with n positions. The players jointly control the movement of a token. One player, Magnus, aims to maximize the number of positions visited while minimizing the number of rounds.
We analyze further the Magnus-Derek game, a two-player game played on a round table with n positions. The players jointly control the movement of a token. One player, Magnus, aims to maximize the number of positions visited while minimizing the number of rounds. The other player, Derek, attempts to minimize the number of visited positions. We present a new strategy for Magnus that succeeds in visiting the maximal number of positions in 3(n – 1) rounds, which is the optimal number of rounds up to a constant factor. Full article
Open AccessArticle Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries
Algorithms 2010, 3(3), 265-275; doi:10.3390/a3030265
Received: 5 May 2010 / Revised: 8 July 2010 / Accepted: 22 July 2010 / Published: 26 July 2010
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Abstract
The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise
The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries. Full article
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Open AccessArticle Univariate Cubic L1 Interpolating Splines: Analytical Results for Linearity, Convexity and Oscillation on 5-PointWindows
Algorithms 2010, 3(3), 276-293; doi:10.3390/a3030276
Received: 9 June 2010 / Revised: 11 July 2010 / Accepted: 20 July 2010 / Published: 30 July 2010
Cited by 7 | PDF Full-text (150 KB) | HTML Full-text | XML Full-text
Abstract
We analytically investigate univariate C1 continuous cubic L1 interpolating splines calculated by minimizing an L1 spline functional based on the second derivative on 5-point windows. Specifically, we link geometric properties of the data points in the windows with linearity, convexity
We analytically investigate univariate C1 continuous cubic L1 interpolating splines calculated by minimizing an L1 spline functional based on the second derivative on 5-point windows. Specifically, we link geometric properties of the data points in the windows with linearity, convexity and oscillation properties of the resulting L1 spline. These analytical results provide the basis for a computationally efficient algorithm for calculation of L1 splines on 5-point windows. Full article
Open AccessArticle Fluidsim: A Car Traffic Simulation Prototype Based on FluidDynamic
Algorithms 2010, 3(3), 294-310; doi:10.3390/a3030294
Received: 7 July 2010 / Accepted: 3 August 2010 / Published: 9 August 2010
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Abstract
We present a car traffic simulation prototype for complex networks, that is formed by a collection of roads and junctions. Traffic load evolution is described by a model based on fluid dynamic conservation laws, deduced from conservation of the number of cars. The
We present a car traffic simulation prototype for complex networks, that is formed by a collection of roads and junctions. Traffic load evolution is described by a model based on fluid dynamic conservation laws, deduced from conservation of the number of cars. The model contains some additional hypothesis in order to reproduce specific car traffic features such as route based car distribution at nodes and the presence of right-of-way at the crossroads. A complete implementation of this model is then presented, together with computational results on case studies. Full article
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Open AccessArticle Univariate Cubic L1 Interpolating Splines: Spline Functional, Window Size and Analysis-based Algorithm
Algorithms 2010, 3(3), 311-328; doi:10.3390/a3030311
Received: 11 July 2010 / Accepted: 10 August 2010 / Published: 20 August 2010
Cited by 9 | PDF Full-text (327 KB) | HTML Full-text | XML Full-text
Abstract
We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that
We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that second-derivative-based 5-point-window L1 splines preserve shape as well as or better than the other types of L1 splines. To calculate second-derivative-based 5-point-window L1 splines, we introduce an analysis-based, parallelizable algorithm. This algorithm is orders of magnitude faster than the previously widely used primal affine algorithm. Full article
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# Other

Open AccessObituary Ray Solomonoff (1926-2009)
Algorithms 2010, 3(3), 255-259; doi:10.3390/a30302555
Received: 12 March 2010 / Accepted: 14 March 2010 / Published: 20 July 2010
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Abstract
Ray Solomonoff was always inventive. As a child, he had a lab in his parent's cellar in Cleveland and a secret air hole to vent the smoke from his experiments. He gave his friend Marvin Minsky a so-called "Hurry" clock — a clock
Ray Solomonoff was always inventive. As a child, he had a lab in his parent's cellar in Cleveland and a secret air hole to vent the smoke from his experiments. He gave his friend Marvin Minsky a so-called "Hurry" clock — a clock labeled "HURRY" that ran very fast. Helped by a friend, he built a year round house in N.H. He put in thick insulation, enabling him to heat the house with two rows of light bulbs along the ceiling. I met Ray shortly after he finished this house, in 1969. I knew about foraging, so I showed him edible plants like Indian Cucumber Root. He was so happy: it was as if we found a fountain of champagne. [...] Full article
Open AccessObituary Ray Solomonoff, Founding Father of Algorithmic Information Theory
Algorithms 2010, 3(3), 260-264; doi:10.3390/a3030260
Received: 12 March 2010 / Accepted: 14 March 2010 / Published: 20 July 2010
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Abstract
Ray J. Solomonoff died on December 7, 2009, in Cambridge, Massachusetts, of complications of a stroke caused by an aneurism in his head. Ray was the first inventor of Algorithmic Information Theory which deals with the shortest effective description length of objects and