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Title: Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications
The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a 𝐙²-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If the bifiltration and I have at most t simplices and points respectively, this computation takes O(t^ω) time where ω ∈ [2,2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.  more » « less
Award ID(s):
2049010
NSF-PAR ID:
10348047
Author(s) / Creator(s):
; ;
Editor(s):
Xavier Goaoc; Michael Kerber
Date Published:
Journal Name:
Leibniz international proceedings in informatics
Volume:
224
ISSN:
1868-8969
Page Range / eLocation ID:
34:1--34:17
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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