Recently, Cenzato et al. proposed a new text index, called the suffixient array, which is a subset of the suffix array and supports locating a single pattern occurrence or finding its maximal exact matches (MEMs), assuming random access to the input text T[1..n] is available. They show that, given the suffix array, the longest common prefix array, and the Burrows-Wheeler transform (BWT) of the reverse of T[1..n] over an alphabet {1,…,σ}, a suffixient array can be constructed in linear time. However, their construction algorithms require multiple scans of these arrays. When restricted to a single pass over the arrays, they present an alternative construction algorithm running in O(n + r log σ) time, where r is the number of runs in the BWT of the reversed text. In this paper, we present a new one-pass algorithm that constructs a suffixient array in linear time under the standard RAM model.

The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon factorization. Differently from the Lyndon words, the inverse Lyndon words may be bordered. The relationship between the two factorizations is related to the inverse lexicographic ordering, and has only been recently explored. More precisely, a main open question is how to get an inverse Lyndon factorization from a classical Lyndon factorization under the inverse lexicographic ordering, named CFLin. In this paper we reveal a strong connection between these two factorizations where the border plays a relevant role. More precisely, we show two main results. We say that a factorization has the border property if a nonempty border of a factor cannot be a prefix of the next factor. First we show that there exists a unique inverse Lyndon factorization having the border property. Then we show that this unique factorization with the border property is the so-called canonical inverse Lyndon factorization, named ICFL. By showing that ICFL is obtained by compacting factors of the Lyndon factorization over the inverse lexicographic ordering, we provide a linear time algorithm for computing ICFL from CFLin.

In application domains such as routing, network analysis, scheduling, and planning, directed graphs are widely used as both formal models and core data structures for the development of efficient algorithmic solutions. In these areas, graphs are often evolving in time: for example, connection links may fail due to temporary technical issues, meaning that edges of the graph cannot be traversed for some time interval and alternative paths have to be followed. In classical computation graphs have been implemented both explicitly through adjacency matrices/lists and symbolically as ordered binary decision diagrams. Moreover, ad-hoc visit procedures have been developed to deal with dynamically evolving graphs. Quantum computation, exploiting interference and entanglement, has provided an exponential speed-up for specific problems, e.g., database search and integer factorization. In the quantum framework everything must be represented and manipulated using reversible operators. This poses a challenge when one has to deal with traversals of dynamically evolving directed graphs. Graph traversals are not intrinsically reversible because of converging paths. In the case of dynamically evolving graphs also the creation/destruction of paths comes into play against reversibility. In this paper we propose a novel high level graph representation in quantum computation supporting dynamic connectivity typical of real-world network applications. Our procedure allows to encode any multigraph into a unitary matrix. We devise algorithms for computing the encoding that are optimal in terms of time and space and we show the effectiveness of the proposal with some examples. We describe how to react to edge/node failures in constant time. Furthermore, we present two methods to perform quantum random walks taking advantage of this encoding: with and without projectors. We implement and test our encoding obtaining that the theoretical bounds for the running time are confirmed by the empirical results and providing more details on the behavior of the algorithms over graphs of different densities.

We tackle the (classic) problem of minimizing (non)deterministic finite automata. The algorithm we put forward has the peculiarity of being incremental, i.e., the minimization proceeds by successive iterations, each producing a partially minimized automaton language-equivalent to the input one. Our algorithm builds upon Almeida et al. from 2014, fixing a minor mistake and generalizing it to the nondeterministic case. It relies on a coloring procedure of a graph associated to the automaton, keeping track of partial information. After dealing with the deterministic case, we extend this idea to the bisimulation-minimization of nondeterministic automata. The algorithms for both the deterministic and the nondeterministic cases run in time O(nm) for an automaton with n states and m transitions. The complexity for the deterministic case matches the complexity claimed by Almeida et al.. The nondeterministic case improves the fastest known incremental algorithm for this problem. We conclude introducing and using a notion of signature of a state, whose aim is to exploit pre-computed information potentially available, to speed-up the process. A signature is used to produce an initial partition of the automaton's states and can be easily integrated in both the incremental and the non-incremental algorithm.