Algebraic Graph Algorithms

Piotr Sankowski

Abstract: The aim of this paper is to survey the results on dynamic algebraic algorithms, with main interest in matrix functions such as, determinant, inverse, rank and characteristic polynomial. First of all we summary the papers that in dynamic setup these problems can be solved faster than evaluating everything from scratch. The static complexity of these problem equals the matrix multiplication complexity, whereas the presented solutions work in subquadratic or quadratic (characteristic polynomial) time in the worst case. The dynamic matrix computations can be used to solve the following graphs problems in dynamic setup: computing transitive closure, computing shortest paths lengths, computing maximum matching size and computing vertex connectivity. For all of these problem the dynamic approach lead to the first known subquadratic algorithms. Astonishingly, the dynamic matrix algorithms can be used to obtain efficient static algorithms for the perfect matching problem as well. Using the O(n 2) algorithms for the dynamic matrix inverse, one can obtain a very simple randomized algorithm for computing perfect matchings in O(n 3) time. When the fast matrix multiplication is used, the complexity of this algorithm can be improved to O(n ω) time, where ω is the exponent of the best known matrix multiplication algorithm. Since ω< 2.38, this algorithm breaks through the O(n 2.5) barrier for the matching problem. The interplay between algebraic algorithms and graphs problems can be explored even further in order to obtain O(Wn ω) time algorithms for single source shortest paths problem and weighted bipartite matching problem.