Dynamic Rank, Basis, and Matching
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909031067549696 |
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| author | Brand, Jan van den Kumar, Vishal Zhang, Daniel J. |
| author_facet | Brand, Jan van den Kumar, Vishal Zhang, Daniel J. |
| contents | 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 $\tilde O(r^{1.405})$ time per entry-update and $\tilde O(r^{1.528}+ z)$ per column-update, where $z$ is the number of changed entries. This extends to $\tilde O(|M|^{1.405})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamic Rank, Basis, and Matching Brand, Jan van den Kumar, Vishal Zhang, Daniel J. Data Structures and Algorithms 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 $\tilde O(r^{1.405})$ time per entry-update and $\tilde O(r^{1.528}+ z)$ per column-update, where $z$ is the number of changed entries. This extends to $\tilde O(|M|^{1.405})$ 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. |
| title | Dynamic Rank, Basis, and Matching |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2605.09917 |