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Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Arbitrary precision real arithmetic: design and algorithms Valerie MenissierMorain
"... We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe ..."
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We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe how to produce a sequence representing the result of the application of this function to its arguments, according to the sequences representing these arguments. For each algorithm we prove that the resulting sequence is a valid representation of the exact real result. This arithmetic is the rst abritrary precision real arithmetic with mathematically proved algorithms. Resume Nous proposons une representation des nombres reels calculables ainsi que des algorithmes pour les fonctions elementaires usuelles pour cette representation. Un nombre reel est represente par une suite de nombres Badiques nis et pour chaque fonction classique (rationnelle, algebrique ou transcendante), nous montrons comment produire une suite representant le resultat a partir de suites representant les parametres. Pour chacun de ces algorithmes nous demontrons que la suite qui en resulte represente bien le resultat reel exact. Cette arithmetique est la premiere arithmetique reelle en precision arbitraire dotee d'un jeu complet d'algorithmes
Faster Language Edit Distance, Connection to Allpairs Shortest Paths and Related Problems
"... Given a context free language L(G) over alphabet Σ and a string s ∈ Σ∗, the language edit distance problem seeks the minimum number of edits (insertions, deletions and substitutions) required to convert s into a valid member of L(G). The wellknown dynamic programming algorithm solves this problem i ..."
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Given a context free language L(G) over alphabet Σ and a string s ∈ Σ∗, the language edit distance problem seeks the minimum number of edits (insertions, deletions and substitutions) required to convert s into a valid member of L(G). The wellknown dynamic programming algorithm solves this problem in O(n3) time (ignoring grammar size) where n is the string length [Aho, Peterson 1972, Myers 1985]. Despite its numerous applications in data management, machine learning, compiler optimization, computational biology, computer vision and linguistics, there is no algorithm known till date that computes or approximates language edit distance problem in true subcubic time. In this paper we give the first such algorithm that computes language edit distance almost optimally. For any arbitrary > 0, our algorithm runs in Õ ( n ω poly() ) time and returns an estimate within a multiplicative approximation factor of (1 + ) with high probability, where ω is the exponent of ordinary matrix multiplication of n dimensional square matrices. It also computes the edit script. We further solve the local alignment problem; for all substrings of s, we can estimate their language edit distance