693 research outputs found
On Greedy Algorithms for Binary de Bruijn Sequences
We propose a general greedy algorithm for binary de Bruijn sequences, called
Generalized Prefer-Opposite (GPO) Algorithm, and its modifications. By
identifying specific feedback functions and initial states, we demonstrate that
most previously-known greedy algorithms that generate binary de Bruijn
sequences are particular cases of our new algorithm
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
Efficient tilings of de Bruijn and Kautz graphs
Kautz and de Bruijn graphs have a high degree of connectivity which makes
them ideal candidates for massively parallel computer network topologies. In
order to realize a practical computer architecture based on these graphs, it is
useful to have a means of constructing a large-scale system from smaller,
simpler modules. In this paper we consider the mathematical problem of
uniformly tiling a de Bruijn or Kautz graph. This can be viewed as a
generalization of the graph bisection problem. We focus on the problem of graph
tilings by a set of identical subgraphs. Tiles should contain a maximal number
of internal edges so as to minimize the number of edges connecting distinct
tiles. We find necessary and sufficient conditions for the construction of
tilings. We derive a simple lower bound on the number of edges which must leave
each tile, and construct a class of tilings whose number of edges leaving each
tile agrees asymptotically in form with the lower bound to within a constant
factor. These tilings make possible the construction of large-scale computing
systems based on de Bruijn and Kautz graph topologies.Comment: 29 pages, 11 figure
Perfect Necklaces
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix
a finite alphabet. Recall that a word is a finite sequence of symbols in the
alphabet and a circular word, or necklace, is the equivalence class of a word
under rotations. For positive integers k and n, we call a necklace
(k,n)-perfect if each word of length k occurs exactly n times at positions
which are different modulo n for any convention on the starting point. We call
a necklace perfect if it is (k,k)-perfect for some k. We prove that every
arithmetic sequence with difference coprime with the alphabet size induces a
perfect necklace. In particular, the concatenation of all words of the same
length in lexicographic order yields a perfect necklace. For each k and n, we
give a closed formula for the number of (k,n)-perfect necklaces. Finally, we
prove that every infinite periodic sequence whose period coincides with some
(k,n)-perfect necklace for any n, passes all statistical tests of size up to k,
but not all larger tests. This last theorem motivated this work
Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees
Some of the most interesting de Bruijn sequences can be constructed in
seemingly unrelated ways. In particular, the "Granddaddy" and "Grandmama" can
be understood by joining necklace cycles into a tree using simple parent rules,
or by concatenating smaller strings (e.g., Lyndon words) in lexicographic
orders. These constructions are elegant, but their equivalences seem to come
out of thin air, and the community has had limited success in finding others of
the same ilk. We aim to demystify the connection between cycle-joining trees
and concatenation schemes by introducing "concatenation trees". These
structures combine binary trees and ordered trees, and traversals yield
concatenation schemes for their sequences.
In this work, we focus on the four simplest cycle-joining trees using the
pure cycling register (PCR): "Granddaddy" (PCR1), "Grandmama" (PCR2), "Granny"
(PCR3), and "Grandpa" (PCR4). In particular, we formally prove a previously
observed correspondence for PCR3 and we unravel the mystery behind PCR4. More
broadly, this work lays the foundation for translating cycle-joining trees to
known concatenation constructions for a variety of underlying feedback
functions including the complementing cycling register (CCR), pure summing
register (PSR), complementing summing register (CSR), and pure run-length
register (PRR)
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