The paper explores repetitiveness measures for two-dimensional strings, which are important for compressing and indexing two-dimensional data like images and matrices.
The key highlights and insights are:
The authors introduce new definitions of the repetitiveness measures δ2D and γ2D, which use rectangular substrings instead of the previous square-based definitions. They show that these new measures can have significantly different values compared to the square-based measures, even for one-dimensional strings.
The paper generalizes straight-line programs (SLPs) and run-length SLPs to the two-dimensional setting, and introduces a new repetitiveness measure g2D based on 2D-SLPs. While computing g2D is NP-hard, the authors show that it is possible to access any cell of the 2D string in logarithmic time using linear space.
The authors also introduce 2D macro schemes as an extension of bidirectional macro schemes to two dimensions. They show that the relationships between the measures b, grl, and g are preserved in the 2D setting, but the relationship between δ, γ, and b can be different compared to the one-dimensional case.
Specifically, the paper shows that there exist 2D string families where b can be asymptotically smaller than γ, and where δ˝ (the square-based extension of δ) can be asymptotically larger than b. This highlights that measures defined for one-dimensional strings may not directly translate to the two-dimensional setting.
The results in this paper provide a foundation for understanding repetitiveness in two-dimensional data and can inform the design of efficient compression and indexing techniques for such data.
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