Hall-type theorems for fast almost dynamic matching and applications
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A dynamic set of size up to K is a set in which elements can be inserted and deleted and which at any moment in its history has at most K elements. In dynamic matching in a bipartite graph, each element, when it is inserted in a dynamic subset of left nodes, makes a request to be matched with one of its neighbors, and the request has to be satisfied on-the-fly without knowing future insertions and deletions and without revoking past matchings. We consider a relaxation of dynamic matching in which each matching can survive at most T insertions, and a right node can be assigned to more than one node of the dynamic set. We show that a bipartite graph satisfying the condition in Hall Marriage Theorem up to K has fast T-surviving dynamic matching for dynamic sets of size up to K, in which every right node can be assigned to at most O(log(KT)) left nodes. Fast matching means that each matching is done in time poly(log N, log T, D), where N is the number of left nodes, and D is the left degree. We obtain a similar result for epsilon-rich matching, in which a left node needs to be assigned (1-epsilon) fraction of its neighbors. By taking O(log (KT)) clones of the right set, one obtains T-surviving dynamic standard matching (with no sharing of right nodes). We construct explicit bipartite graphs admitting T-surviving dynamic matching up to K with small left degree D and small right set R, and similarly for ϵ-rich matching. Specifically, D and |R|/K are polynomial in log N and log T, and for ϵ-rich the dependency is quasipolynomial. Previous constructions, both non-explicit and explicit, did not require the T-surviving restriction, but had only slow matching algorithms running in time exponential in K log N. We give two applications. The first one is in the area of non-blocking networks, and the second one is about one-probe storage schemes for dynamic sets.
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