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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 for a graph with $n$ vertices and $m$ edges is $O( {(n+ m)}^{0.7072})$ 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 Rubinfield \cite{onak} designed a randomized data structure that achieves $O(\log^2 n)$ 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 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 time per update.

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