Novel Results on the Number of Runs of the Burrows-Wheeler-Transform

Abstract

The Burrows-Wheeler-Transform (BWT), a reversible string transformation, is one of the fundamental components of many current data structures in string processing. It is central in data compression, as well as in efficient query algorithms for sequence data, such as webpages, genomic and other biological sequences, or indeed any textual data. The BWT lends itself well to compression because its number of equal-letterruns (usually referred to as r) is often considerably lower than that of the original string; in particular, it is well suited for strings with many repeated factors. In fact, much attention has been paid to the r parameter as measure of repetitiveness, especially to evaluate the performance in terms of both space and time of compressed indexing data structures.In this paper, we investigate ρ(v), the ratio of r and of the number of runs of the BWT of the reverse of v. Kempa and Kociumaka [FOCS 2020] gave the first non-trivial upper bound as ρ(v) = O(log 2 (n)), for any string v of length n. However, nothing is known about the tightness of this upper bound. We present infinite families of binary strings for which ρ(v) = Θ(log n) holds, thus giving the first non-trivial lower bound on ρ(n), the maximum over all strings of length n.Our results suggest that r is not an ideal measure of the repetitiveness of the string, since the number of repeated factors is invariant between the string and its reverse. We believe that there is a more intricate relationship between the number of runs of the BWT and the string's combinatorial properties.

Sensitivity of string compressors and repetitiveness measures

“…We strengthen our upper bound results by presenting matching lower bounds on the worst-case sensitivity for all these major versions of the Lempel-Ziv 77 factorizations. This contrasts with the previously known related results such that the size z 78 of the Lempel-Ziv 78 factorization can increase by a factor of Ω(n 3/4 ) [Lagarde and Perifel, 2018], and the number r of runs in the Burrows-Wheeler transform can increase by a factor of Ω(log n) [Giuliani et al, 2021] when a character is prepended to an input string of length n. We also study the worst-case sensitivity of several grammar compression algorithms including Bisection, AVL-grammar, GCIS (Grammar Compression by Induced Sorting), and CDAWG (Compact Directed Acyclic Word Graph). Further, we extend the notion of the worst-case sensitivity to string repetitiveness measures such as the smallest string attractor size γ and the substring complexity δ, and present matching upper and lower bounds of the worst-case multiplicative sensitivity for γ and δ.…”

“…The recent work by Giuliani et al [20], however, shows that the number r of runs in the BWT of a string of length n can grow by a multiplicative factor of Ω(log n) when a single character is prepended to the input string. The other work by Lagarde and Perifel [33] shows that the size of the dictionary of LZ78, which is equal to the number of factors in the respective LZ78 factorization, can grow by a multiplicative factor of Ω(n 1/4 ), again when a single character is prepended to the input string.…”

“…All the results reported in this article and in the related work are summarized in Table 1. insertion -Ω(log n) [20] In addition to the afore-mentioned multiplicative sensitivity, we also present the worstcase additive sensitivity which is defined as max T ∈Σ n {C(T ) − C(T ) : ed(T, T ) = 1} for all the string compressors/repetitiveness measures C dealt in this paper. We remark that the additive sensitivity allows one to observe and evaluate more details in the changes of the output sizes, as summarized in Table 2.…”

Logarithmic equal-letter runs for BWT of purely morphic words

“…It is important to note that the performance in terms of both space and time of text compressors and compressed indexing data structures applied on a text w can be evaluated by using r bwt (w) [11]. An upper bound on the number of clusters produced by BW T has been provided in [13], in particular, it has been proved that r bwt(w) = O(z(w) log 2 n) where z(w) is the number of phrases in the LZ77 factorization of w and n is the length of w. The ratio between r bwt (w) and the number of clusters in the BW T of the reverse of w has been studied in [12]. A recent comparative survey illustrating the properties of r bwt (w) and other repetitiveness measures can be found in [19].…”

Counting Lyndon Subsequences

“…Lyndon factors enjoy a rich class of algorithmic and stringology applications including: counting and finding the maximal repetitions (a.k.a. runs) in a string [2] and in a trie [8], constant-space pattern matching [3], comparison of the sizes of run-length Burrows-Wheeler Transform of a sting and its reverse [4], substring minimal suffix queries [1], the shortest common superstring problem [7], and grammar-compressed self-index (Lyndon-SLP) [9].…”