具有单侧误差的伪随机图和位探测方案
Pseudo-random graphs and bit probe schemes with one-sided error
摘要：
我们研究概率位探测方案的成员资格问题。给定一个集合A，它最多有n个来自大小为m的宇宙的元素，我们组织这样一个结构，查询类型为"x在A中吗？"可以很快得到答复。H. Buhrman、P.B. Miltersen、J. Radhakrishnan和S. Venkatesh提出了一种基于扩展器的比特探测方案。他们的方案需要$O(n\log m)$位的空间，并且只需要从存储器中读取一个随机选择的位来回答查询。答案是正确的概率很高，有双侧错误。在本文中，我们表明，对于同样的问题，存在一个位探测计划，单边错误，需要空间的$O(n\log^2 m+\poly(\log m))$位。与Buhrman、Miltersen、Radhakrishnan和Venkatesh的模型的不同之处在于，我们考虑的是带有辅助字的位探测方案。这意味着，在我们的方案中，存储器被分成两个不同大小的部分：$O(n\log^2 m)$位的主存储器和$\log^{O(1)}m$位的短字，该短字为存储的集合A预先计算一次并被"缓存"。要回答一个问题"x在A中吗？"我们允许从主存储器中读取整个缓存的字和仅一位。对于一些合理的参数值，我们的空间界限比任何没有缓存数据的方案都要好。
英文摘要：
We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type "Is x in A?" can be answered very quickly. H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe scheme based on expanders. Their scheme needs space of $O(n\log m)$ bits, and requires to read only one randomly chosen bit from the memory to answer a query. The answer is correct with high probability with two-sided errors. In this paper we show that for the same problem there exists a bit-probe scheme with one-sided error that needs space of $O(n\log^2 m+\poly(\log m))$ bits. The difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh is that we consider a bit-probe scheme with an auxiliary word. This means that in our scheme the memory is split into two parts of different size: the main storage of $O(n\log^2 m)$ bits and a short word of $\log^{O(1)}m$ bits that is pre-computed once for the stored set A and `cached'. To answer a query "Is x in A?" we allow to read the whole cached word and only one bit from the main storage. For some reasonable values of parameters our space bound is better than what can be achieved by any scheme without cached data.
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