Online matching games in bipartite expanders: applications to bitprobes and dictionaries with non-adaptive probing

Abstract

A dictionary is a data structure which stores pairs (key, value). The operation query on input key returns value if the dictionary contains a pair (key, value) and nil otherwise. A dynamic dictionary also supports the insert and delete operations. The model with cells of bitsize $n + m + 1$ is used, where $n$ and $m$ are the bitsizes of keys and values. We give 2 dictionaries that use $(1+o(1))K$ cells to store $K$ pairs and in which query makes \emph{non-adaptive probes} to the data structure: a static one in which query makes $\smash{\widetilde O(n^2)}$ probes, and a dynamic one in which it makes $\smash{\widetilde O(n^2 \log K)}$ probes. Also, for the first time, a dictionary is given in which all 3 operations are non-adaptive, but the size is larger: $O(nK)$ cells. The constructions are explicit and, consequently, all operations run in time $\poly(n,m)$. 1-bitprobes storage schemes store a set $S \subseteq \{0,1\}^n$ and answer a membership query by reading a single bit. We introduce 1-bitprobes that are {\em dynamic}. Such a scheme is given whose size is smaller than in all previous constructions which are static. The \textsf{query} operation has runtime $n^{O(1)}$. The operations delete and insert have runtime $n^{O(\log n)}$. The proofs rely on a strategy for an online matching game played on a given bipartite graph. An opponent switches left nodes {\em on} and {\em off}. The strategy needs to assign an irrevocable match to a node when it is turned {\em on}. Such games were introduced in the companion paper~\cite{companion-Hall}. The applications in this paper rely on efficient strategies assuming that each node is turned {\em off} after a polynomial number of rounds. We present efficient strategies for (lossless) expanders.