Year: 2026
Type: preprint
Abstract: We study dynamic algorithms for maintaining fundamental algebraic properties of matrices, specifically, rank, basis, and full-rank submatrices, with applications to maximum matching on dynamic graphs. Prior dynamic algorithms for rank achieve subquadratic update times but scale with the matrix dimension n, and could not always maintain the corresponding objects such as a basis or maximum full-rank submatrix. We present the first dynamic rank algorithms whose update time scales with the matrix rank r, achieving Õ(r^1.405) time per entry-update and Õ(r^1.528 + z) per column-update, where z is the number of changed entries. This extends to Õ(|M|^1.405) edge-update time to maintain the size |M| of a maximum matching. We also give dynamic algorithms for maintaining a column-basis subject to column-updates and a maximum full-rank submatrix subject to entry-updates.
Source: arXiv (Cornell University)
Institutions: Georgia Institute of Technology, Columbia University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7160920698
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