Computation of highly regular nearby points

  • We call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/N finds a highly regular nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no short relation m~ of length less than /spl alpha//2 exists for points x~ within half the x'-distance from x. The integer relation m for x' is for random x up to an average factor 2/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x.

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Author:Carsten Rössner, Claus Peter SchnorrGND
Document Type:Preprint
Date of Publication (online):2005/07/13
Year of first Publication:1995
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2005/07/13
Tag:computational complexity; computational geometry; highly regular nearby points; inner product; integer vector; short integer relation
Preprint, später in: 3rd Israel Symposium on the Theory of Computing and Systems, 1995
Source:3rd Israel Symposium on the Theory of Computing and Systems, 1995 ,
Institutes:Informatik und Mathematik / Mathematik
Informatik und Mathematik / Informatik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht