### Trade-offs for fully dynamic transitive closure on DAGs: breaking through the $O(n^{2}$ barrierTrade-offs for fully dynamic transitive closure on DAGs: breaking through the $O(n^{2}$ barrier

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 Author Demetrescu, Camil ♦ Italiano, Giuseppe F. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2005 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Dynamic graph algorithms ♦ Transitive closure Abstract We present an algorithm for directed acyclic graphs that breaks through the $O(n^{2})$ barrier on the single-operation complexity of fully dynamic transitive closure, where $\textit{n}$ is the number of edges in the graph. We can answer queries in $O(n^{ε})$ worst-case time and perform updates in $O(n^{ω(1,ε,1)™ε}+n^{1+ε})$ worst-case time, for any ε∈[0,1], where ω(1,ε,1) is the exponent of the multiplication of an $\textit{n}$ × $n^{ε}$ matrix by an $n^{ε}$ × $\textit{n}$ matrix. The current best bounds on ω(1,ε,1) imply an $O(n^{0.575})$ query time and an $O(n^{1.575})$ update time in the worst case. Our subquadratic algorithm is randomized, and has one-sided error. As an application of this result, we show how to solve single-source reachability in $O(n^{1.575})$ time per update and constant time per query. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2005-03-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 52 Issue Number 2 Page Count 10 Starting Page 147 Ending Page 156

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Source: ACM Digital Library