We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the respective string length. We show that an arithmetically progressed permutation P coincides with the suffix array of a unary, binary, or ternary string. We further analyze the conditions of a given P under which we can find a uniquely defined string over either a binary or ternary alphabet having P as its suffix array. For the binary case, we show its connection to lower Christoffel words, balanced words, and Fibonacci words. In addition to solving the arithmetically progressed suffix array problem, we give the shape of the Burrows–Wheeler transform of those strings solving this problem. These results give rise to numerous future research directions.

In this note we consider the concept of alphabet ordering in the context of string factoring. We propose a greedy algorithm that produces Lyndon factorizations with small numbers of factors which can be modified to produce large numbers of factors. For the technique we introduce the Exponent Parikh vector. Applications and research directions derived from circ-UMFFs are discussed.

A degenerate or indeterminate string on an alphabet Σ is a sequence of non-empty subsets of Σ. Given a degenerate string t of length n and its Burrows–Wheeler transform we present a new method for searching for a degenerate pattern of length m in t running in O ( m n ) time on a constant size alphabet Σ. Furthermore, it is a hybrid pattern matching technique that works on both regular and degenerate strings. A degenerate string is said to be conservative if its number of non-solid letters is upper-bounded by a fixed positive constant q ; in this case we show that the search time complexity is O ( q m 2 ) for counting the number of occurrences and O ( q m 2 + occ ) for reporting the found occurrences where occ is the number of occurrences of the pattern in t . Experimental results show that our method performs well in practice.

In this paper we focus on a total (but non-lexicographic) ordering of strings called V-order. We devise a new linear-time algorithm for computing the V-comparison of two finite strings. In comparison with the previous algorithm in the literature, our algorithm is both conceptually simpler, based on recording letter positions in increasing order, and more straightforward to implement, requiring only linked lists.

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAMparallel algorithmwas described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary.