We study the impact that string reversal can have on several repetitiveness measures. First, we exhibit an infinite family of strings where the number, r, of runs in the run-length encoding of the Burrows-Wheeler transform (BWT) can increase additively by Θ(n) when reversing the string. This substantially improves the known Ω(log n) lower-bound for the additive sensitivity of r and it is asymptotically tight. We generalize our result to other variants of the BWT, including the variant with an appended end-of-string symbol and the bijective BWT. We show that an analogous result holds for the size z of the Lempel-Ziv 77 (LZ) parsing of the text, and also for some of its variants, including the non-overlapping LZ parsing, and the LZ-end parsing. Moreover, we describe a family of strings for which the ratio z(w^R)/z(w) approaches 3 from below as |w| → ∞. We also show an asymptotically tight lower-bound of Θ(n) for the additive sensitivity of the size v of the smallest lexicographic parsing to string reversal. Finally, we show that the multiplicative sensitivity of v to reversing the string is Θ(log n), and this lower-bound is also tight. Overall, our results expose the limitations of repetitiveness measures that are widely used in practice, against string reversal - a simple and natural data transformation.

A morphism is a mapping that transforms words through letter-wise substitution, where each symbol is consistently replaced by a fixed word. In the field of combinatorics on words, one topic that has attracted considerable attention is the characterization of morphisms that preserve specific properties, such as overlap-freeness, square-freeness, lexicographic order, and primitivity. Continuing this direction, we initiate the study on occurrence-preserving morphisms, which address the following fundamental question: given a morphism ϕ, two words u and v, and k ≥ 1, under what conditions does the number of occurrences of u in v equal the number of occurrences of ϕ^k(u) in ϕ^k(v)? To answer this question, we introduce the notion of interference-free morphisms, examine their properties, and uncover a connection to recognizable morphisms. We then present a precise characterization of occurrence-preserving morphisms in terms of interference-freeness. As applications of our characterization, we first show that there exists a bijection between the starting positions of the occurrences of u in v and those of ϕ^k(u) in ϕ^k(v). We then apply the characterization to the Fibonacci and Thue-Morse words to identify their minimal unique substrings (MUSs). Finally, we exploit the connection between MUSs and net occurrences to simplify existing proofs on net occurrences in these words.

Random access to highly compressed strings - represented by straight-line programs or Lempel-Ziv parses, for example - is a well-studied topic. Random access to such strings in strongly sublogarithmic time is impossible in the worst case, but previous authors have shown how to support faster access to specific characters and their neighbourhoods. In this paper we explore whether, since better compression can impede access, we can support faster access to less compressible substrings of highly compressed strings. We first show how, given a run-length compressed straight-line program (RLSLP) of size g_{rl} or a block tree of size L, we can build an O (g_{rl})-space or an O (L)-space data structure, respectively, that supports access to any character in time logarithmic in the length of the longest repeated substring containing that character. That is, the more "incongruous" a character is with respect to the characters around, the faster we can support access to it. We then prove a similar but more powerful and sophisticated result for parsings in which phrases' sources do not overlap much larger phrases, with the query time depending also on the number of phrases we must copy from their sources to obtain the queried character.

Abstract The problem of detecting and measuring the repetitiveness of one-dimensional strings has been extensively studied in data compression and text indexing. Our understanding of these issues has been significantly improved by the introduction of the notion of string attractor (Kempa and Prezza 2018) and by the results showing the relationship between attractors and other measures of compressibility. When the input data are structured in a non-linear way, as in two-dimensional strings, inherent redundancy often offers an even richer source for compression. However, systematic studies on repetitiveness measures for two-dimensional strings are still scarce. In this paper, we extend to two or more dimensions the main measures of complexity introduced for one-dimensional strings. We distinguish between the measures $$\delta $$ δ and $$\gamma $$ γ , defined in terms of the substrings of the input, and the measures g , $$g_{rl}$$ g rl , and b , which are based on copy-paste mechanisms. We study the properties and mutual relationships between these two classes and we show that the two classes become incomparable for d -dimensional inputs as soon as $$d\geqslant 2$$ d ⩾ 2 . Moreover, we show that our grammar-based representation of a d -dimensional string of size N enables direct access to any symbol in $$O(\log N)$$ O ( log N ) time. We also compare our measures for two-dimensional strings with the 2D Block Tree data structure (Brisaboa et al., Comput. J. 67 (1), 391–406, 2024) and provide some insights for the design of future effective two-dimensional compressors. A preliminary version of this paper appeared in the proceedings of the conference SPIRE 2024.

A suffixient set is a novel combinatorial object that captures the essential information of repetitive strings in a way that, provided with a random access mechanism, supports various forms of pattern matching. In this paper, we study the size χ of the smallest suffixient set as a repetitiveness measure. First, we study its sensitivity to various string operations. We show that χ cannot increase by more than 2 after appending or prepending a character to the string. As a consequence, we are able to give simple linear-time online algorithms to compute smallest suffixient sets. We also show that, although reversing the string can increase χ by an arbitrary O ( n ) value, it always holds χ ( T )/ χ ( T R ) ≤ 2. We also prove lower and upper bounds for the additive or multiplicative increase of χ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as Ω ( n ) for all those operations. Secondly, we place χ among known repetitiveness measures. In particular, we show χ ≤ 2 r (where r is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where χ = o ( v ) (where v is the size of the smallext lexicographic parse of the string), and that χ is uncomparable to almost all reachable measures based on copy-paste mechanisms. In passing, we give precise bounds for χ for some relevant string families, for example χ ≤ σ + 2 on episturmian words over alphabets of size σ (e.g., χ ≤ 4 on Fibonacci strings, for which we precisely characterize the only two smallest suffixient sets).

We study how the application of morphisms affects the number r of equal-letter runs in the Burrows–Wheeler Transform (BWT). This parameter has emerged as a key repetitiveness measure in compressed indexing. We focus on the notion of BWT-run sensitivity after application of morphisms. For binary alphabets, we characterize the class of injective morphisms that preserve the number of BWT-runs up to a bounded additive increase by showing that it coincides with the known class of primitivity-preserving morphisms, which are those that map primitive words to primitive words. We further prove that deciding whether a given binary morphism has bounded BWT-run sensitivity is possible in polynomial time with respect to the total length of the images of the two letters. Additionally, we explore new structural and combinatorial properties of synchronizing and recognizable morphisms. These results establish new connections between BWT-based compressibility, code theory, and symbolic dynamics.

Abstract A bit catastrophe, loosely defined, is when a change in just one character of a string causes a significant change in the size of the compressed string. We study this phenomenon for the Burrows-Wheeler Transform (BWT), a string transform at the heart of several of the most popular compressors and aligners today. The parameter determining the size of the compressed data is the number of equal-letter runs of the BWT, commonly denoted r . We exhibit infinite families of strings in which insertion, deletion, resp. substitution of one character increases r from constant to $$\Theta (\log n)$$ Θ ( log n ) , where n is the length of the string. These strings can be interpreted both as examples for an increase by a multiplicative or an additive $$\Theta (\log n)$$ Θ ( log n ) -factor. As regards the multiplicative factor, they attain the upper bound given by Akagi, Funakoshi, and Inenaga [Inf & Comput. 2023] of $$\mathcal{O}(\log n \log r)$$ O ( log n log r ) , since here $$r=\mathcal{O}(1)$$ r = O ( 1 ) . We then give examples of strings in which insertion, deletion, resp. substitution of a character increases r by a $$\Theta (\sqrt{n})$$ Θ ( n ) additive factor. These strings significantly improve the best known lower bound for an additive factor of $$\Omega (\log n)$$ Ω ( log n ) [Giuliani et al., SOFSEM 2021].