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Computing Generalized Ranks of Persistence Modules via Unfolding to Zigzag Modules

Core Concepts
Efficiently compute the generalized rank of persistence modules by unfolding to zigzag modules.
The article discusses a method to compute the generalized rank of persistence modules by unfolding them into zigzag modules. It introduces the concept of s-completeness and provides an algorithm, GENRANK, to determine the s-complete decomposition of a module. The algorithm iteratively converts full interval modules in a direct decomposition to s-complete modules until all convertible modules are identified. The process involves checking foldability and adding limit modules to make intervals complete. Theoretical propositions support the algorithm's validity and efficiency.
P-indexed persistence module M defined over poset P. Rank computation for 2-parameter persistence modules. Linear time algorithm for degree-1 homology in graphs. Efficiency comparison with other algorithms.

Deeper Inquiries

How does the concept of s-completeness impact the efficiency of computing generalized ranks

The concept of s-completeness plays a crucial role in the efficiency of computing generalized ranks. By identifying s-complete interval modules in a direct decomposition, we can focus on converting only those full interval modules that are convertible and have foldable complements. This targeted approach reduces unnecessary computations and iterations, leading to a more efficient algorithm. Additionally, by ensuring that each converted module is s-complete, we guarantee that the resulting decomposition aligns with the rank of the original persistence module MZZ.

What are the practical implications of using zigzag modules for rank computation in real-world applications

Using zigzag modules for rank computation offers several practical implications in real-world applications. Firstly, it provides a structured framework for unfolding complex poset-indexed persistence modules into simpler zigzag structures, enabling easier analysis and manipulation of data. Secondly, the ability to decompose zigzag modules into interval modules allows for efficient computation of generalized ranks through algorithms designed specifically for barcode extraction from these structures. This approach streamlines the process of determining generalized ranks in topological data analysis tasks involving multi-parameter persistence modules.

How can this approach be extended to handle more complex parameterizations in persistence modules

To extend this approach to handle more complex parameterizations in persistence modules, one could explore adapting the unfolding technique to accommodate higher-dimensional posets or non-linear parameter spaces. By developing methods to unfold such intricate structures into appropriate zigzag paths or surfaces, it would be possible to apply similar algorithms for computing generalized ranks efficiently across diverse parameterizations. Additionally, incorporating advanced techniques from algebraic topology and computational geometry could enhance the scalability and versatility of this method when dealing with high-dimensional or nonlinear datasets commonly encountered in real-world applications like biological data analysis or geometric modeling.