Dynamic Matrix Rank with Partial Lookahead

Abstract

We consider the problem of maintaining information about the rank of a matrix M under changes to its entries. For an n×n matrix M, we show an amortized upper bound of O(n ω−1) arithmetic operations per change for this problem, where ω<2.373 is the exponent for matrix multiplication, under the assumption that there is a lookahead of up to Θ(n) locations. That is, we know up to the next Θ(n) locations (i 1,j 1),(i 2,j 2),…, whose entries are going to change, in advance; however we do not know the new entries in these locations in advance. We get the new entries in these locations in a dynamic manner. The dynamic matrix rank problem was first studied by Frandsen and Frandsen who showed an upper bound of O(n 1.575) and a lower bound of Ω(n) for this problem and later Sankowski showed an upper bound of O(n 1.495) for this problem when allowing randomization and a small probability of error. These algorithms do not assume any lookahead. For the dynamic matrix rank problem with lookahead, Sankowski and Mucha showed a randomized algorithm (with a small probability of error) that is more efficient than these algorithms.