无损扩展器中的在线匹配
Online matching in lossless expanders
摘要：
Bauwens和Zimand [BZ 2019]已经证明无损扩展器具有有趣的在线匹配特性。结果以隐式形式出现在[BZ 2019]中。我们提出了一个明确的版本，这个属性是直接服从典型的应用程序，证明它在一个独立的方式，澄清了一些参数的作用，并给出两个应用程序。一个$(K, ε)$无损扩展器是一个二分图，使得在二分图的左侧的节点的大小最多为$K$的任何子集$S$至少有$(1-ε) D |S|$邻居，其中$D$是左度。主要结果是，任何这样的图，经过轻微的修改，允许$(1-O(ε)D, 1)$在线匹配到大小为$K$。这意味着，对于二分区左侧的节点的任何序列$S=(x_1, \ldots, x_K)$，可以以在线方式向$S$中的每个节点$x_i$分配由其邻居的$(1-O(ε))$分数组成的集合$A_i$，使得集合$A_1, \ldots, A_K$成对不相交。“在线方式”是指对于每个$i$，分配给$x_i$的节点集合仅取决于分配给$x_1, \ldots, x_{i-1}$的节点。第一应用涉及用于表示集合$S$的存储方案，使得成员资格查询“Is $x \in S$？“可以通过读取一个比特来概率地回答。所有以前的单探针存储方案都是针对静态集合$S$的。我们表明，一个无损扩展器可以用来构建一个单探针存储方案的动态集，即，可以在其中插入和删除元素而不影响其他元素的表示的集合。第二个应用是关于非阻塞网络。
英文摘要：
Bauwens and Zimand [BZ 2019] have shown that lossless expanders have an interesting online matching property. The result appears in an implicit form in [BZ 2019]. We present an explicit version of this property which is directly amenable to typical applications, prove it in a self-contained manner that clarifies the role of some parameters, and give two applications. A $(K, ε)$ lossless expander is a bipartite graph such that any subset $S$ of size at most $K$ of nodes on the left side of the bipartition has at least $(1-ε) D |S|$ neighbors, where $D$ is the left degree.The main result is that any such graph, after a slight modification, admits $(1-O(ε)D, 1)$ online matching up to size $K$. This means that for any sequence $S=(x_1, \ldots, x_K)$ of nodes on the left side of the bipartition, one can assign in an online manner to each node $x_i$ in $S$ a set $A_i$ consisting of $(1-O(ε))$ fraction of its neighbors so that the sets $A_1, \ldots, A_K$ are pairwise disjoint. "Online manner" refers to the fact that, for every $i$, the set of nodes assigned to $x_i$ only depends on the nodes assigned to $x_1, \ldots, x_{i-1}$. The first application concerns storage schemes for representing a set $S$, so that a membership query "Is $x \in S$?" can be answered probabilistically by reading a single bit. All the previous one-probe storage schemes were for a static set $S$. We show that a lossless expander can be used to construct a one-probe storage scheme for dynamic sets, i.e., sets in which elements can be inserted and deleted without affecting the representation of other elements. The second application is about non-blocking networks.
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