二分展开图中的在线匹配博弈：在非自适应探测的位探针和字典中的应用

中文摘要 AI 辅助

字典是一种存储键值对（key, value）的数据结构。查询操作在输入键时，如果字典包含该键值对则返回对应的值，否则返回nil。动态字典还支持插入和删除操作。本文采用单元大小为n+m+1比特的模型，其中n和m分别是键和值的比特大小。我们给出了两个使用(1+o(1))K个单元存储K个键值对的字典，其中查询操作对数据结构进行非自适应探测：一个是静态字典，查询进行O~(n^2)次探测；另一个是动态字典，查询进行O~(n^2 log K)次探测。此外，我们首次给出了一个所有三个操作（查询、插入、删除）都是非自适应的字典，但其大小更大：O(nK)个单元。这些构造是显式的，因此所有操作都在poly(n,m)时间内运行。1比特探针存储方案存储一个集合S⊆{0,1}^n，并通过读取单个比特来回答成员查询。我们引入了动态的1比特探针方案。我们给出的这种方案的大小比所有之前的静态构造都要小。查询操作的运行时间为n^{O(1)}。删除和插入操作的运行时间为n^{O(log n)}。证明依赖于在给定二分图上进行的在线匹配博弈的策略。对手会打开和关闭左节点。策略需要在节点被打开时为其分配一个不可撤销的匹配。这类博弈在伴随论文中引入。本文的应用依赖于高效策略，假设每个节点在多轮（多项式轮数）后被关闭。我们为（无损）展开图提出了高效策略。

英文摘要

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.

发表机构

• National Research University Higher School of Economics, Faculty of Computer Science（高等经济大学计算机学院）
• Department of Computer and Information Sciences, Towson University（汤森大学计算机与信息科学系）

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