Pseudo-random graphs and bit probe schemes with one-sided error

Abstract

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 x ε 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(nlogm) bits, and requires to read only one randomly chosen bit from the memory to answer a query. The answer is correct with probability 2/3 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(nlog2 m + poly(logm)) 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(nlog2 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 x A? we allow to read the whole cached word and only one bit from the main storage. For some reasonable values of parameters (e.g., for poly(logm) n m) our space bound is better than what can be achieved by any scheme without cached data (the lower bound ω was proven in [11]). We obtain a slightly weaker result (space of size n 1 + δ poly(logm) bits and two bit probes for every query) for a scheme that is effectively encodable. Our construction is based on the idea of naive derandomization, which is of independent interest. First we prove that a random combinatorial object (a graph) has the required properties, and then show that such a graph can be obtained as an outcome of a pseudo-random generator. Thus, a suitable graph can be specified by a short seed of a PRG, and we can put an appropriate value of the seed into the cache memory of the scheme. © 2011 Springer-Verlag Berlin Heidelberg.