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We study connections between expansion in bipartite graphs and efficient online matching modeled via several games. In the basic game, an opponent switches {\em on} and {\em off} nodes on the left side and, at any moment, at most $K$ nodes may be on. Each time a node is switched on, it must be irrevocably matched with one of its neighbors. A bipartite graph has $e$-expansion up to $K$ if every set $S$ of at most $K$ left nodes has at least $e\#S$ neighbors. If all left nodes have degree $D$ and $e$ is close to $D$, then the graph is a lossless expander. We show that lossless expanders allow for a polynomial time strategy in the above game, and, furthermore, with a slight modification, they allow a strategy running in time $O(D \log N)$, where $N$ is the number of left nodes. Using this game and a few related variants, we derive applications in data structures and switching networks. Namely, (a) 1-query bitprobe storage schemes for dynamic sets (previous schemes work only for static sets),(b) explicit space- and time-efficient storage schemes for static and dynamic sets with non-adaptive access to memory (the first fully dynamic dictionary with non-adaptive probing using almost optimal space), and (c) non-explicit constant depth non-blocking $N$-connectors with poly$(\log N)$ time path finding algorithms whose size is optimal within a factor of $O(\log N)$ (previous connectors are double-exponentially slower)., Comment: The exposition has been improved and a few minor issues have been fixed

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., Comment: Abstract shortened to meet arxiv requirements

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any standard Turing machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a O$(\log |x|)$-short program for $x$. We also show that there exists a computable function that maps every $x$ to a list of size $|x|^2$ containing a O$(1)$-short program for $x$. This is essentially optimal because we prove that for each such function there is a $c$ and infinitely many $x$ for which the list has size at least $c|x|^2$. Finally we show that for some standard machines, computable functions generating lists with $0$-short programs, must have infinitely often list sizes proportional to $2^{|x|}$.