Fast algorithms for the characteristic polynomial
Theoretical Computer Science
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
Lower bounds for dynamic algebraic problems
Information and Computation
Design of Dynamic Data Structures
Design of Dynamic Data Structures
Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Deterministic network coding by matrix completion
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Faster dynamic matchings and vertex connectivity
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Dynamic Normal Forms and Dynamic Characteristic Polynomial
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Dynamic normal forms and dynamic characteristic polynomial
Theoretical Computer Science
Fast matrix rank algorithms and applications
Journal of the ACM (JACM)
| Hi-index | 5.23 |
We consider maintaining information about the rank of a matrix under changes of the entries. For nxn matrices, we show an upper bound of O(n^1^.^5^7^5) arithmetic operations and a lower bound of @W(n) arithmetic operations per element change. The upper bound is valid when changing up to O(n^0^.^5^7^5) entries in a single column of the matrix. We also give an algorithm that maintains the rank using O(n^2) arithmetic operations per rank one update. These bounds appear to be the first nontrivial bounds for the problem. The upper bounds are valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound for element updates uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions.