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Oscar Valero - One of the best experts on this subject based on the ideXlab platform.
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On fixed point theory in partially ordered sets and an application to Asymptotic Complexity of algorithms
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales. Serie A. Matemáticas, 2019Co-Authors: Asier Estevan, Juan-josé Miñana, Oscar ValeroAbstract:The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the Asymptotic Complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for Asymptotic Complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower Asymptotic bounds for the running time computing.
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A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to Asymptotic Complexity analysis of algorithms
Topology and its Applications, 2016Co-Authors: Zahra Mohammadi, Oscar ValeroAbstract:Abstract In this paper we continue the study of fixed point theory in partial quasi-metric spaces and its usefulness in Complexity analysis of algorithms. Concretely we prove two new fixed point results for monotone and continuous self-mappings in 0-complete partial quasi-metric spaces and, in addition, we show that the assumptions in the statement of such results cannot be weakened. Furthermore, as an application, we present a quantitative fixed point technique which is helpful for Asymptotic Complexity analysis of algorithms.
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New results on the Baire partial quasi-metric space, fixed point theory and AsymptoticComplexity analysis for recursive programs
Fixed Point Theory and Applications, 2014Co-Authors: Maryam A Alghamdi, Naseer Shahzad, Oscar ValeroAbstract:In Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012), a new mathematical fixed point technique, that uses the so-called Baire partial quasi-metric space, was introduced with the aim of providing the Asymptotic Complexity of a class of recursive algorithms. The aforementioned technique presents the advantage that requires less calculations than the quasi-metric original one given by Schellekens (Electron. Notes Theor. Comput. Sci. 1:211-232, 1995). In this paper we continue the study, started in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012), on the use of partial quasi-metric spaces for Asymptotic Complexity analysis of algorithms. Concretely, our main purpose is to prove that the Baire partial quasi-metric space is an appropriate mathematical framework for discussing via fixed point arguments the Asymptotic Complexity of a general class of recursive algorithms to which all the algorithms analyzed in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012) belong. The obtained results are illustrated by means of applying them to yield the Complexity of two celebrated recursive algorithms which don not belong to the class discussed in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012). MSC: 47H10; 54E50; 68Q15; 68Q25; 68W40
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Fixed point theorems in generalized metric spaces with applications to computer science
Fixed Point Theory and Applications, 2013Co-Authors: Maryam A Alghamdi, Naseer Shahzad, Oscar ValeroAbstract:In 1994, Matthews introduced the notion of a partial metric space in order to obtain a suitable mathematical tool for program verification (Matthews in Ann. N.Y. Acad. Sci. 728:183-197, 1994). He gave an application of this new structure to formulate a suitable test for lazy data flow deadlock in Kahn’s model of parallel computation by means of a partial metric version of the celebrated Banach fixed point theorem (Matthews in Theor. Comput. Sci. 151:195-205, 1995). In this paper, motivated by the utility of partial metrics in computer science, we discuss whether they are a suitable tool for Asymptotic Complexity analysis of algorithms. Concretely, we show that the Matthews fixed point theorem does not constitute, in principle, an appropriate implement for the aforementioned purpose. Inspired by the preceding fact, we prove two fixed point theorems which provide the mathematical basis for a new technique to carry out Asymptotic Complexity analysis of algorithms via partial metrics. Furthermore, in order to illustrate and to validate the developed theory, we apply our results to analyze the Asymptotic Complexity of two celebrated recursive algorithms.
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New results on the mathematical foundations of Asymptotic Complexity analysis of algorithms via Complexity spaces
International Journal of Computer Mathematics, 2012Co-Authors: Salvador Romaguera, Pedro Tirado, Oscar ValeroAbstract:Schellekens [ The Smyth completion: A common foundation for denotational semantics and Complexity analysis , Electron. Notes Theor. Comput. Sci. 1 1995, pp. 211–232.] introduced the theory of Complexity quasi-metric spaces as a part of the development of a topological foundation for the Asymptotic Complexity analysis of programs and algorithms in 1995. The applicability of this theory to the Asymptotic Complexity analysis of divide and conquer algorithms was also illustrated by Schellekens in the same paper. In particular, he gave a new formal proof, based on the use of the Banach fixed-point theorem, of the well-known fact that the Asymptotic upper bound of the average running time of computing of Mergesort belongs to the Asymptotic Complexity class of n log 2 n . Recently, Schellekens’ method has been shown to be useful in yielding Asymptotic upper bounds for a class of algorithms whose running time of computing leads to recurrence equations different from the divide and conquer ones reported in Cerda-Uguet et al. [ The Baire partial quasi-metric space: A mathematical tool for the Asymptotic Complexity analysis in Computer Science , Theory Comput. Syst. 50 2012, pp. 387–399.]. However, the variety of algorithms whose Complexity can be analysed with this approach is not much larger than that of algorithms that can be analysed with the original Schellekens method. In this paper, on the one hand, we extend Schellekens’ method in order to yield Asymptotic upper bounds for a certain class of recursive algorithms whose running time of computing cannot be discussed following the techniques given by Cerda-Uguet et al. and, on the other hand, we improve the original Schellekens method by introducing a new fixed-point technique for providing, contrary to the case of the method introduced by Cerda-Uguet et al. , lower Asymptotic bounds of the running time of computing of the aforementioned algorithms and those studied by Cerda-Uguet et al. We illustrate and validate the developed method by applying our results to provide the Asymptotic Complexity class Asymptotic upper and lower bounds of the celebrated algorithms Quicksort, Largetwo and Hanoi.
Erik Martensson - One of the best experts on this subject based on the ideXlab platform.
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the Asymptotic Complexity of coded bkw with sieving using increasing reduction factors
International Symposium on Information Theory, 2019Co-Authors: Erik MartenssonAbstract:The Learning with Errors problem (LWE) is one of the main candidates for post-quantum cryptography. At Asiacrypt 2017, coded-BKW with sieving, an algorithm combining the Blum-Kalai-Wasserman algorithm (BKW) with lattice sieving techniques, was proposed. In this paper, we improve that algorithm by using different reduction factors in different steps of the sieving part of the algorithm. In the Regev setting, where q = n2 and $\sigma = {n^{1.5}}/\left( {\sqrt {2\pi } \log _2^2n} \right)$, the Asymptotic Complexity is 20.8917n, improving the previously best Complexity of 20.8927n. When a quantum computer is assumed or the number of samples is limited, we get a similar level of improvement.
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The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors
arXiv: Cryptography and Security, 2019Co-Authors: Erik MartenssonAbstract:The Learning with Errors problem (LWE) is one of the main candidates for post-quantum cryptography. At Asiacrypt 2017, coded-BKW with sieving, an algorithm combining the Blum-Kalai-Wasserman algorithm (BKW) with lattice sieving techniques, was proposed. In this paper, we improve that algorithm by using different reduction factors in different steps of the sieving part of the algorithm. In the Regev setting, where $q = n^2$ and $\sigma = n^{1.5}/(\sqrt{2\pi}\log_2^2 n)$, the Asymptotic Complexity is $2^{0.8917n}$, improving the previously best Complexity of $2^{{0.8927n}}$. When a quantum computer is assumed or the number of samples is limited, we get a similar level of improvement.
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ISIT - The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors
2019 IEEE International Symposium on Information Theory (ISIT), 2019Co-Authors: Erik MartenssonAbstract:The Learning with Errors problem (LWE) is one of the main candidates for post-quantum cryptography. At Asiacrypt 2017, coded-BKW with sieving, an algorithm combining the Blum-Kalai-Wasserman algorithm (BKW) with lattice sieving techniques, was proposed. In this paper, we improve that algorithm by using different reduction factors in different steps of the sieving part of the algorithm. In the Regev setting, where q = n2 and $\sigma = {n^{1.5}}/\left( {\sqrt {2\pi } \log _2^2n} \right)$, the Asymptotic Complexity is 20.8917n, improving the previously best Complexity of 20.8927n. When a quantum computer is assumed or the number of samples is limited, we get a similar level of improvement.
Olivier Devillers - One of the best experts on this subject based on the ideXlab platform.
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Vertex Removal in Two Dimensional Delaunay Triangulation: Speed-up by Low Degrees Optimization
Computational Geometry, 2011Co-Authors: Olivier DevillersAbstract:The theoretical Complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O(d), O(d log d), or O(d²). In fact, the Asymptotic Complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3≤ d≤ 7, it improves the global behavior of the removal for random points by more than 45%.
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Vertex removal in two-dimensional Delaunay triangulation: Speed-up by low degrees optimization
Computational Geometry, 2011Co-Authors: Olivier DevillersAbstract:The theoretical Complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O(d), O(dlogd), or O(d^2). In fact, the Asymptotic Complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3=
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Delaunay Triangulation of Imprecise Points, Preprocess and Actually Get a Fast Query Time
Journal of Computational Geometry, 2011Co-Authors: Olivier DevillersAbstract:We propose a new algorithm to preprocess a set of n disjoint unit disks in O(n log n) expected time, allowing to compute the Delaunay triangulation of a set of n points, one from each disk, in O(n) expected time. Our algorithm has the same Asymptotic Complexity as previous ones for this problem, but our algorithm is much simpler and it runs faster in practice than a direct computation of the Delaunay triangulation.
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Vertex Removal in Two Dimensional Delaunay Triangulation: Asymptotic Complexity is Pointless
2009Co-Authors: Olivier DevillersAbstract:The theoretical Complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point with usual results O(d), O(d log d), or O(d²). In fact the Asymptotic Complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3 ≤ d ≤ 7, it improves the global behavior of the removal for random points by a factor of 2.
Michael Sagraloff - One of the best experts on this subject based on the ideXlab platform.
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A deterministic algorithm for isolating real roots of a real polynomial
Journal of Symbolic Computation, 2011Co-Authors: Kurt Mehlhorn, Michael SagraloffAbstract:We describe a bisection algorithm for root isolation of polynomials with real coefficients. It is assumed that the coefficients can be approximated with arbitrary precision; exact computation in the field of coefficients is not required. We refer to such coefficients as bitstream coefficients. The algorithm is simpler, deterministic and has better Asymptotic Complexity than the randomized algorithm of Eigenwillig et al. (2005). We also discuss a partial extension to multiple roots.
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ISSAC - Isolating real roots of real polynomials
Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009Co-Authors: Kurt Mehlhorn, Michael SagraloffAbstract:We describe a bisection algorithm for root isolation of polynomials with real coefficients. It is assumed that the coefficients can be approximated with arbitrary precision; exact computation in the field of coefficients is not required. We refer to such coefficients as bitstream coefficients. The algorithm is deterministic and has almost the same Asymptotic Complexity as the randomized algorithm of [12]. We also discuss a partial extension to multiple roots.
Maryam A Alghamdi - One of the best experts on this subject based on the ideXlab platform.
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New results on the Baire partial quasi-metric space, fixed point theory and AsymptoticComplexity analysis for recursive programs
Fixed Point Theory and Applications, 2014Co-Authors: Maryam A Alghamdi, Naseer Shahzad, Oscar ValeroAbstract:In Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012), a new mathematical fixed point technique, that uses the so-called Baire partial quasi-metric space, was introduced with the aim of providing the Asymptotic Complexity of a class of recursive algorithms. The aforementioned technique presents the advantage that requires less calculations than the quasi-metric original one given by Schellekens (Electron. Notes Theor. Comput. Sci. 1:211-232, 1995). In this paper we continue the study, started in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012), on the use of partial quasi-metric spaces for Asymptotic Complexity analysis of algorithms. Concretely, our main purpose is to prove that the Baire partial quasi-metric space is an appropriate mathematical framework for discussing via fixed point arguments the Asymptotic Complexity of a general class of recursive algorithms to which all the algorithms analyzed in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012) belong. The obtained results are illustrated by means of applying them to yield the Complexity of two celebrated recursive algorithms which don not belong to the class discussed in Cerda-Uguet et al. (Theory Comput. Syst. 50:387-399, 2012). MSC: 47H10; 54E50; 68Q15; 68Q25; 68W40
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Fixed point theorems in generalized metric spaces with applications to computer science
Fixed Point Theory and Applications, 2013Co-Authors: Maryam A Alghamdi, Naseer Shahzad, Oscar ValeroAbstract:In 1994, Matthews introduced the notion of a partial metric space in order to obtain a suitable mathematical tool for program verification (Matthews in Ann. N.Y. Acad. Sci. 728:183-197, 1994). He gave an application of this new structure to formulate a suitable test for lazy data flow deadlock in Kahn’s model of parallel computation by means of a partial metric version of the celebrated Banach fixed point theorem (Matthews in Theor. Comput. Sci. 151:195-205, 1995). In this paper, motivated by the utility of partial metrics in computer science, we discuss whether they are a suitable tool for Asymptotic Complexity analysis of algorithms. Concretely, we show that the Matthews fixed point theorem does not constitute, in principle, an appropriate implement for the aforementioned purpose. Inspired by the preceding fact, we prove two fixed point theorems which provide the mathematical basis for a new technique to carry out Asymptotic Complexity analysis of algorithms via partial metrics. Furthermore, in order to illustrate and to validate the developed theory, we apply our results to analyze the Asymptotic Complexity of two celebrated recursive algorithms.
