Fully dynamic transitive closure: breaking through the O(n/sup 2/) barrier

We introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive closure, in particular we devise a deterministic algorithm for general directed graphs that achieves O(n/sup 2/) amortized time for updates, while preserving unit worst-case cost for queries. In case of deletions only, our algorithm performs updates faster in O(n) amortized time. Our matrix-based approach yields an algorithm for directed acyclic graphs which breaks through the O(n/sup 2/) barrier on the single-operation complexity of fully dynamic transitive closure. We can answer queries in O(n/sup /spl epsiv//) time and perform updates in O(n/sup /spl omega/(1,/spl epsiv/,1)-/spl epsiv//+n/sup 1+/spl epsiv//) time, for any /spl epsiv//spl isin/[0,1], where /spl omega/(1,/spl epsiv/,1) is the exponent of the multiplication of an n/spl times/n/sup /spl epsiv// matrix by an n/sup /spl epsiv///spl times/n matrix. The current best bounds on /spl omega/(1,/spl epsiv/,1) imply an O(n/sup 0.575/) query time and an O(n/sup 1.575/) update time. Our subquadratic algorithm is randomized, and has one-side error.