Constructing $k$-radius sequences
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- by Simon R. Blackburn and James F. McKee PDF
- Math. Comp. 81 (2012), 2439-2459 Request permission
Abstract:
An $n$-ary $k$-radius sequence is a finite sequence of elements taken from an alphabet of size $n$ such that any two distinct elements of the alphabet occur within distance $k$ of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let $f_k(n)$ be the shortest length of any $k$-radius sequence. We improve on earlier estimates for $f_k(n)$ by using tilings and logarithms. The main result is that $f_k(n)\sim \frac {1}{k}\binom {n}{2}$ as $n\rightarrow \infty$ whenever there exists a tiling of $\mathbb {Z}^{\pi (k)}$ by a certain cluster of $k$ hypercubes. In particular this result holds for infinitely many $k$, including all $k\le 194$ and all $k$ such that $k+1$ or $2k+1$ is prime. For certain $k$, in particular when $2k+1$ is prime, we get a sharper error term using the theory of logarithms.References
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Additional Information
- Simon R. Blackburn
- Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
- Email: s.blackburn@rhul.ac.uk
- James F. McKee
- Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
- Email: james.mckee@rhul.ac.uk
- Received by editor(s): August 5, 2010
- Received by editor(s) in revised form: December 17, 2010
- Published electronically: August 2, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 2439-2459
- MSC (2010): Primary 94A55
- DOI: https://doi.org/10.1090/S0025-5718-2011-02510-X
- MathSciNet review: 2945165