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，大的扩张因子意味着极快的动态匹配。结合已知的无损扩张器构造，这解决了Feldman、Friedman和Pippenger（SIAM J. Discret. Math., 1(2):158-173, 1988）经典论文中的主要开放问题。应用1：集合存储。我们构造了1查询位探针（1-query bitprobe）来存储一个含$N$元素集合的动态子集$S$。成员查询仅需读取一个比特，该比特的位置在时间poly$(\log N, \log (1/\varepsilon))$内计算得到，并以概率$1-\epsilon$正确。元素可在时间quasipoly$(\log N)$内高效插入和删除。先前构造是静态的：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poly$(\# S \log N)$。此外，我们的方案大小小于已知最优的静态集构造。应用2：交换网络。我们构造了显式恒定深度、几乎最小尺寸的$N$连接器，其中路径寻找算法运行时间为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$N$的指数级。