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# Hall-type theorems for fast almost dynamic matching and applications

A dynamic set of size up to K is a set in which elements can be inserted anddeleted and which at any moment in its history has at most K elements. Indynamic matching in a bipartite graph, each element, when it is inserted in adynamic subset of left nodes, makes a request to be matched with one of itsneighbors, and the request has to be satisfied on-the-fly without knowingfuture insertions and deletions and without revoking past matchings. Weconsider a relaxation of dynamic matching in which each matching can survive atmost T insertions, and a right node can be assigned to more than one node ofthe dynamic set. We show that a bipartite graph satisfying the condition inHall Marriage Theorem up to K has fast T-surviving dynamic matching for dynamicsets of size up to K, in which every right node can be assigned to at mostO(log(KT)) left nodes. Fast matching means that each matching is done in timepoly(log N, log T, D), where N is the number of left nodes, and D is the leftdegree. We obtain a similar result for epsilon-rich matching, in which a leftnode needs to be assigned (1-epsilon) fraction of its neighbors. By takingO(log (KT)) clones of the right set, one obtains T-surviving dynamic standardmatching (with no sharing of right nodes). We construct explicit bipartitegraphs admitting T-surviving dynamic matching up to K with small left degree Dand small right set R, and similarly for $\epsilon$-rich matching.Specifically, D and |R|/K are polynomial in log N and log T, and for$\epsilon$-rich the dependency is quasipolynomial. Previous constructions, bothnon-explicit and explicit, did not require the T-surviving restriction, but hadonly slow matching algorithms running in time exponential in K log N. We givetwo applications. The first one is in the area of non-blocking networks, andthe second one is about one-probe storage schemes for dynamic sets.

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