arXiv subjects

Daniel Gabrić

Publications and source records attributed to Daniel Gabrić.

3 recordsLinked to original sources

arXiv2026

Construction of orientable sequences in $O(1)$-amortized time per bit

An orientable sequence of order $n$ is a cyclic binary sequence such that each length-$n$ substring appears at most once \emph{in either direction}. Maximal length orientable sequences are known only for $n\leq 7$, and a trivial upper bound on their length is $2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$. This paper presents the first efficient algorithm to construct orientable sequences that reach this upper bound, asymptotically; more specifically, our algorithm constructs orientable sequences via cycle-joining and a successor-rule approach requiring $O(n)$ time per bit and $O(n)$ space. This answers a longstanding open question from Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Applying a recent concatenation-tree framework, the same sequences can be generated in $O(1)$-amortized time per bit using $O(n^2)$ space. Our sequences are applied to find new longest-known (aperiodic) orientable sequences for $n\leq 20$.

cs.DS↗

arXiv2025

Las Vegas algorithms to generate universal cycles and de Bruijn sequences uniformly at random

We present practical algorithms for generating universal cycles uniformly at random. In particular, we consider universal cycles for shorthand permutations, subsets and multiset permutations, weak orders, and orientable sequences. Additionally, we consider de Bruijn sequences, weight-range de Bruin sequences, and de Bruijn sequences, with forbidden $0^z$ substring. Each algorithm, seeded with a random element from the given set, applies a random walk of an underlying Eulerian de Bruijn graph to obtain a random arborescence (spanning in-tree). Given the random arborescence and the de Bruijn graph, a corresponding random universal cycle can be generated in constant time per symbol. We present experimental results on the average cover time needed to compute a random arborescence for each object using a Las Vegas algorithm.

cs.DM↗

arXiv2024

Constructing $k$-ary Orientable Sequences with Asymptotically Optimal Length

An orientable sequence of order $n$ over an alphabet $\{0,1,\ldots, k{-}1\}$ is a cyclic sequence such that each length-$n$ substring appears at most once \emph{in either direction}. When $k= 2$, efficient algorithms are known to construct binary orientable sequences, with asymptotically optimal length, by applying the classic cycle-joining technique. The key to the construction is the definition of a parent rule to construct a cycle-joining tree of asymmetric bracelets. Unfortunately, the parent rule does not generalize to larger alphabets. Furthermore, unlike the binary case, a cycle-joining tree does not immediately lead to a simple successor-rule when $k \geq 3$ unless the tree has certain properties. In this paper, we derive a parent rule to derive a cycle-joining tree of $k$-ary asymmetric bracelets. This leads to a successor rule that constructs asymptotically optimal $k$-ary orientable sequences in $O(n)$ time per symbol using $O(n)$ space. In the special case when $n=2$, we provide a simple construction of $k$-ary orientable sequences of maximal length.

cs.DM↗