Constructing $k$-radius sequences
Simon R. Blackburn,James McKee +1 more
TL;DR: It is shown that f_k(n) ~ n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists, which holds for infinitely many k, including all k < 195 and all k such that k+1 or 2k+1 is prime.
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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) ~ n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists. In particular this result holds for infinitely many k, including all k < 195 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.
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Citations
Splitter Sets and $k$ -Radius Sequences
TL;DR: This paper gives some new constructions of perfect splitter sets, as well as some nonexistence results on them, and obtain some new results on optimal conflict-avoiding codes.
26
On the Nonexistence of Perfect Splitter Sets
Tao Zhang,Gennian Ge +1 more
TL;DR: This paper proves some nonexistence results for nonsingular perfect splitter sets and gives some necessary conditions for the existence of purely singular perfectsplitter sets.
22
The existence of k-radius sequences
TL;DR: In this paper, it was shown that whenever k is fixed and n->~f"k(n)~1k n2", then the problem is solvable using a probabilistic argument.
18
Universal cycles for minimum coverings of pairs by triples, with application to 2-radius sequences
TL;DR: A new ordering, extending the notion of universal cycles of Chung et al. (1992), is proposed for the blocks of k-uniform set systems, and existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders.
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Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences
TL;DR: In this paper, a new ordering, extending the notion of universal cycles of Chung et al., is proposed for the blocks of $k$-uniform set systems, and the existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders.
13
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