Fully Dynamic Maximal Matching in O (log n) Update Time

Abstract

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes $O( \log n)$ expected amortized time for each edge update where $n$ is the number of vertices in the graph. While there is a trivial $O(n)$ algorithm for edge update, the previous best known result for this problem was due to Ivkovi\'c and Llyod\cite{llyod}. For a graph with $n$ vertices and $m$ edges, they give an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sub linear only for a sparse graph. %To the best of our knowledge this %is the first polylog update time for maximal matching that implies an % exponential improvement from the previous results. For the related problem of maximum matching, Onak and Rubinfeld \cite{onak} designed a randomized data structure that achieves $O(\log^2 n)$ expected amortized time for each update for maintaining a $c$-approximate maximum matching for some large constant $c$. In contrast, we can maintain a factor two approximate maximum matching in $O(\log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes $O(\log n )$expected amortized time per update.