Nonbinary Counterparts of the Prefer-Same and Prefer-Opposite de Bruijn Sequences (original) (raw)
Related papers
Efficient constructions of the Prefer-same and Prefer-opposite de Bruijn sequences
ArXiv, 2020
The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al.[AIEE Transactions 77 (1958)]. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length 2n2^n2n de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using O(n)O(n)O(n) time per bit and only O(n)O(n)O(n) space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.
Revisiting the Prefer-same and Prefer-opposite de Bruijn sequence constructions
Theoretical Computer Science, 2021
We present a simple greedy algorithm to construct the prefer-same de Bruijn sequence and prove that it is equivalent to the more complex algorithm first stated by Eldert et al. without proof [AIEE Transactions 77 (1958)], and later by Fredricksen [SIAM Review 24 (1982)]. Then we prove that the resulting sequence has the lexicographically largest run-length representation among all de Bruijn sequences. Furthermore, we prove that the sequence resulting from a prefer-opposite greedy construction has the lexicographically smallest run-length representation among all de Bruijn sequences.
Efficient construction of the Prefer-same de Bruijn sequence
2020
The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al. [AIEE Transactions 77 (1958)] in 1958. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length 2 de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using O(n) time per bit and only O(n) space. 2012 ACM Subject Classification Mathematics of computing→ Discrete mathematics
Spans of preference functions for de Bruijn sequences
Discrete Applied Mathematics, 2012
A nonbinary Ford sequence is a de Bruijn sequence generated by simple rules that determine the priorities of what symbols are to be tried first, given an initial word of size n which is the order of the sequence being generated. This set of rules is generalized by the concept of a preference function of span n − 1, which gives the priorities of what symbols to appear after a substring of size n − 1 is encountered. In this paper we characterize preference functions that generate full de Bruijn sequences. More significantly, We establish that any preference function that generates a de Bruijn sequence of order n also generates de Bruijn sequences of all orders higher than n, thus making the Ford sequence no special case. Consequently, we define the preference function complexity of a de Bruijn sequence to be the least possible span of a preference function that generates this de Bruijn sequence.
A recursive construction of nonbinary de Bruijn sequences
Designs, Codes and Cryptography, 2011
This paper presents a method to find new de Bruijn cycles based on ones of lesser order. This is done by mapping a de Bruijn cycle to several vertex disjoint cycles in a de Bruijn digraph of higher order and connecting these cycles into one full cycle.
A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence
ArXiv, 2021
We propose a novel construction for the well-known prefer-max De Bruijn sequence, based on the cycle joining technique. We further show that the construction implies known results from the literature in a straightforward manner. First, it implies the correctness of the onion theorem, stating that, effectively, the reverse of prefer-max is in fact an infinite De Bruijn sequence. Second, it implies the correctness of recently discovered shift rules for prefer-max, prefer-min, and their reversals. Lastly, it forms an alternative proof for the seminal FKM-theorem.
Generating (n, 2) De Bruijn Sequences with Some Balance and Uniformity Properties
ARS COMBINATORIA-WATERLOO …, 2004
This paper presents two new algorithms for generating (n,2) de Bruijn sequences which possess certain properties. The sequences generated by the proposed algorithms may be useful for experimenters to systematically investigate intertrial repetition effects. Characteristics are compared with those of randomly sampled (n,2) de Bruijn sequences.
An efficient shift rule for the prefer-max De Bruijn sequence
Discrete Mathematics
A shift rule for the prefer-max De Bruijn sequence is formulated, for all sequence orders, and over any finite alphabet. An efficient algorithm for this shift rule is presented, which has linear (in the sequence order) time and memory complexity.
On the distribution of de bruijn sequences of low complexity
Journal of Combinatorial Theory, Series A, 1985
The distribution y(c, n) of de Bruijn sequences of order n and linear complexity c is investigated. Some new results are proved on the distribution of de Bruijn sequences of low complexity, i.e., their complexity is between 2"-' + n and 2"-'+2"-2. It is proved that for n>5 and 2"-'+n<c<2"-'+2"-2, y(c,n)zO (mod 4). It is shown that for n 4 11, y(2"-' + n, n) > 0. It is also proved that ~(2"~' +2"-', n)>4?(2"-'-1, n-2) and we give a recursive method to generate de Bruijn sequences of complexity 2"-' + 2" *.
De Bruijn Sequences with varying Combs
Integers
For a given alphabet A and length n, a de Bruijn sequence corresponds to a string of length |A| n where every string of length n occurs as a consecutive substring (and we allow the ends to wrap around). We consider the relaxation wherein the letters of the substring are not consecutive but rather fixed by some pattern, called a comb. We give several constructions showing how to construct some sequences for combs, as well as give several ways to form combs without de Bruijn sequences.