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Efficient Repetitiveness Measures for Two-Dimensional Strings


Core Concepts
This paper introduces extensions of repetitiveness measures, such as δ and γ, to two-dimensional strings. It proposes new definitions of these measures that use rectangular substrings instead of square substrings, and shows that these new measures can exhibit significant gaps compared to previous square-based definitions. The paper also generalizes straight-line programs and macro schemes to the two-dimensional setting, and analyzes the relationships between these measures.
Abstract
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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Key Insights Distilled From

by Giuseppe Rom... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07030.pdf
Exploring Repetitiveness Measures for Two-Dimensional Strings

Deeper Inquiries

What are some real-world applications where the proposed two-dimensional repetitiveness measures could be particularly useful

The proposed two-dimensional repetitiveness measures could be particularly useful in various real-world applications where structured data is represented in a two-dimensional format. One application could be in image compression, where images are represented as matrices of pixels. By analyzing the repetitiveness of substructures within the image matrix, efficient compression algorithms can be developed to reduce the storage space required for storing images. This can be especially beneficial in scenarios where large collections of images need to be stored or transmitted over networks with limited bandwidth. Another application could be in bioinformatics, where genomic data is often represented as two-dimensional matrices. By measuring the repetitiveness of certain patterns or sequences within the genomic data, researchers can identify common motifs or regions of interest that may have biological significance. This can aid in tasks such as genome assembly, sequence alignment, and identifying genetic variations. Furthermore, in the field of computer vision, the two-dimensional repetitiveness measures can be applied to analyze patterns in video data, such as identifying recurring sequences or frames. This can be useful for tasks like video summarization, object tracking, and anomaly detection in surveillance footage. Overall, the proposed two-dimensional repetitiveness measures have the potential to enhance data compression, pattern recognition, and analysis tasks in various domains where data is structured in a two-dimensional format.

How do the relationships between the different repetitiveness measures change as the dimensionality of the data increases beyond two dimensions

As the dimensionality of the data increases beyond two dimensions, the relationships between the different repetitiveness measures may undergo significant changes. In higher-dimensional data settings, the complexity of identifying and measuring repetitiveness patterns can increase due to the higher number of dimensions and the increased diversity of substructures within the data. For example, in three-dimensional data, such as volumetric medical imaging data, the relationships between measures like δ, γ, b, grl, and g may exhibit more intricate dependencies. The trade-offs between these measures in terms of computational complexity, space efficiency, and information content may vary in higher-dimensional spaces compared to the two-dimensional case. Additionally, as the dimensionality increases, the challenges of defining and computing repetitiveness measures become more pronounced. The interactions between different dimensions, the arrangement of data points, and the presence of higher-order patterns can impact the effectiveness and applicability of existing measures. Therefore, exploring the relationships between repetitiveness measures in multi-dimensional data settings requires careful consideration and specialized methodologies.

Are there any other generalizations of one-dimensional string concepts that could be explored in the context of multi-dimensional data

In the context of multi-dimensional data, there are several other generalizations of one-dimensional string concepts that could be explored to analyze and measure repetitiveness. Some potential generalizations include: N-Dimensional Macro Schemes: Extending the concept of macro schemes to N-dimensional data, where N is greater than 2. This would involve factorizing the data into higher-dimensional substructures and identifying repetitive patterns across multiple dimensions. N-Dimensional Run-Length Encoding: Generalizing run-length encoding techniques to N-dimensional data, allowing for efficient representation of repetitive sequences in multi-dimensional arrays or tensors. N-Dimensional Context-Free Grammars: Developing context-free grammars for N-dimensional data to capture complex repetitive structures and patterns in higher-dimensional spaces. N-Dimensional Entropy Measures: Exploring entropy measures tailored for N-dimensional data to quantify the amount of repetitiveness and information content in multi-dimensional datasets. By exploring these generalizations and adapting existing string concepts to multi-dimensional settings, researchers can gain deeper insights into the repetitiveness of complex data structures and develop more effective compression and analysis techniques for high-dimensional data.
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