A Separation of $$\gamma $$ and b via Thue–Morse Words

On Stricter Reachable Repetitiveness Measures*

“…In asymptotic terms, it holds δ ≤ γ ≤ b ≤ c ≤ z ≤ g and, except for c ≤ z, there are string families where each measure is asymptotically smaller than the next. The recent result by Bannai et al [2], showing that there exists a string family where γ = o(b), establishes a clear separation between unreachable lower bounds (δ,γ) and reachable measures (b and the larger ones).…”

“…Storer and Szymanski [19] introduced bidirectional macro schemes, which allow sources appear to the left or to the right of their phrases, as long as circular dependencies are avoided. We follow the definition by Bannai et al [2].…”

“…The measure γ is defined as the cardinality of the smallest of such attractors for a given string w [1, n], and it always holds that γ = O(b). Further, a string family where γ = o(b) exists [2].…”

Computing NP-hard Repetitiveness Measures via MAX-SAT

“…Namely, they showed that for any input text, the size of the smallest string attractor is a lower bound for the output sizes of all known dictionary compressors. Since then, relations between these various repetitiveness measures have been heavily investigated [2,4,15,18,20,31,33].…”

“…Thus, any efficient dictionary compression algorithm can (most likely) merely compute approximations of γ, b, or g. Although for any text, the relation δ ≤ γ ≤ b ≤ z ≤ g is known, where δ [20] and z [24] are repetitiveness measures known to be computable in linear time (cf. [7,Lemma 5.7] for δ and [8] for z), the gap between the measures can be quite large; string families giving a logarithmic factor gap are known for each pair of measures [2,31]. Since the sizes of some recent data structures such as [7,34], depend on these repetitiveness measures, their exact sizes are crucial knowledge.…”