Online matching games in bipartite expanders and applications

Abstract

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).