Online matching in lossless expanders

Bauwens and Zimand [BZ 2019] have shown that lossless expanders have an interesting online matching property. The result appears in an implicit form in [BZ 2019]. We present an explicit version of this property which is directly amenable to typical applications, prove it in a self-contained manner that clarifies the role of some parameters, and give two applications. A $(K, \epsilon)$ lossless expander is a bipartite graph such that any subset $S$ of size at most $K$ of nodes on the left side of the bipartition has at least $(1-\epsilon) D |S|$ neighbors, where $D$ is the left degree.The main result is that any such graph, after a slight modification, admits $(1-O(\epsilon)D, 1)$ online matching up to size $K$. This means that for any sequence $S=(x_1, \ldots, x_K)$ of nodes on the left side of the bipartition, one can assign in an online manner to each node $x_i$ in $S$ a set $A_i$ consisting of $(1-O(\epsilon))$ fraction of its neighbors so that the sets $A_1, \ldots, A_K$ are pairwise disjoint. "Online manner" refers to the fact that, for every $i$, the set of nodes assigned to $x_i$ only depends on the nodes assigned to $x_1, \ldots, x_{i-1}$. The first application concerns storage schemes for representing a set $S$, so that a membership query "Is $x \in S$?" can be answered probabilistically by reading a single bit. All the previous one-probe storage schemes were for a static set $S$. We show that a lossless expander can be used to construct a one-probe storage scheme for dynamic sets, i.e., sets in which elements can be inserted and deleted without affecting the representation of other elements. The second application is about non-blocking networks.