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).

翻译：我们研究二分图扩展性与通过多种博弈建模的高效在线匹配之间的联系。在基础博弈中，对手在左侧节点上进行{\em 开启}和{\em 关闭}操作，且任意时刻最多有$K$个节点处于开启状态。每当节点被开启时，必须立即与其某个邻居节点进行不可撤销的匹配。若对于任意最多包含$K$个左侧节点的集合$S$，其邻居节点数至少为$e\#S$，则称该二分图具有$K$以下$e$扩展性。若所有左侧节点的度均为$D$且$e$接近$D$，则该图构成无损扩展器。我们证明无损扩展器允许在上述博弈中采用多项式时间策略，并且经过轻微修改后，可进一步实现$O(D \log N)$时间复杂度的策略，其中$N$为左侧节点总数。通过该博弈及若干相关变体，我们推导出在数据结构和交换网络中的应用，具体包括：(a) 动态集合的1-查询比特探针存储方案（现有方案仅适用于静态集合），(b) 具有非自适应内存访问的静态与动态集合的显式空间-时间高效存储方案（首个采用近乎最优空间且支持非自适应探测的全动态字典），以及(c) 非显式常数深度无阻塞$N$连接器，其路径查找算法时间复杂度为多对数级$\text{poly}(\log N)$，且规模在$O(\log N)$因子内达到最优（现有连接器的速度存在双指数级差距）。