We show that in bipartite graphs a large expansion factor implies very fast dynamic matching. Coupled with known constructions of lossless expanders, this gives a solution to the main open problem in a classical paper of Feldman, Friedman, and Pippenger (SIAM J. Discret. Math., 1(2):158-173, 1988). Application 1: storing sets. We construct 1-query bitprobes that store a dynamic subset $S$ of an $N$ element set. A membership query reads a single bit, whose location is computed in time poly$(\log N, \log (1/\varepsilon))$ time and is correct with probability $1-\epsilon$. Elements can be inserted and removed efficiently in time quasipoly$(\log N)$. Previous constructions were static: membership queries have the same parameters, but each update requires the recomputation of the whole data structure, which takes time poly$(\# S \log N)$. Moreover, the size of our scheme is smaller than the best known constructions for static sets. Application 2: switching networks. We construct explicit constant depth $N$-connectors of essentially minimum size in which the path-finding algorithm runs in time quasipoly$(\log N)$. In the non-explicit construction in Feldman, Friedman and Pippenger (SIAM J. Discret. Math., 1(2):158-173, 1988). and in the explicit construction of Wigderson and Zuckerman (Combinatorica, 19(1):125-138, 1999) the runtime is exponential in $N$.

翻译：我们证明，在二分图中，大的扩展因子能够实现极快的动态匹配。结合已知的无损扩展器构造，这解决了Feldman、Friedman和Pippenger经典论文（SIAM J. Discret. Math., 1(2):158-173, 1988）中的主要开放问题。应用一：集合存储。我们构造了存储N元集合动态子集S的单查询比特探针。成员查询仅需读取一个比特，其位置可在poly(log N, log(1/ε))时间内计算，并以1-ε的概率正确。元素插入和删除可在quasipoly(log N)时间内高效完成。先前的构造是静态的：成员查询具有相同参数，但每次更新需重新计算整个数据结构，耗时为poly(#S log N)。此外，我们方案的大小优于已知的最佳静态集合构造。应用二：交换网络。我们构造了具有最小规模的显式恒定深度N-连接器，其寻路算法运行时间为quasipoly(log N)。在Feldman、Friedman和Pippenger（SIAM J. Discret. Math., 1(2):158-173, 1988）的非显式构造以及Wigderson和Zuckerman（Combinatorica, 19(1):125-138, 1999）的显式构造中，运行时间均为关于N的指数级。