Set Membership with Non-Adaptive Bit Probes

We consider the non-adaptive bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form “Is x in S?” by non-adaptively probing the bit vector at t places. Let sN (m,n, t) be the minimum number of bits of storage needed for such a scheme. Buhrman, Miltersen, Radhakrishnan, and Srinivasan [4] and Alon and Feige [1] investigated sN (m,n, t) for various ranges of the parameter t. We show the following. General upper bound (t ≥ 5 and odd): For odd t ≥ 5, sN (m,n, t) = O(tm 2 t−1n1− 2 t−1 lg 2m n ). This improves on a result of Buhrman et al. that states for odd t ≥ 5, sN (m,n, t) = O(m 4 t+1n). For small values of t (odd t ≥ 3 and t ≤ 1 10 lg lgm) and n ≤ m 1− ( > 0), we obtain adaptive schemes that use a little less space: O(exp(e2t)m 2 t+1n1− 2 t+1 lgm). Three probes (t = 3) lower bound: We show that sN (m,n, 3) = Ω( √ mn) for n ≥ n0 for some constant n0. This improves on a result of Alon and Feige that states that for n ≥ 16 lgm, sN (m,n, 3) = Ω( √ mn lgm ). The complexity of the non-adaptive scheme might, in principle, depend on the function that is used to determine the answer based on the three bits read (one may assume that all queries use the same function). Let sfN (m,n, 3) be the minimum number of bits of storage required in a three-probe non-adaptive scheme where the function f : {0, 1}3 → {0, 1} is used to answer the queries. We show that for large class of functions f (including the majority function on three bits), we in fact have sN (m,n, 3) = Ω(m1− 1 cn ) for n ≥ 4 and some c > 0. In particular, three-probe non-adaptive schemes that use such query functions f do not give any asymptotic savings over the trivial characteristic vector when n ≥ logm. 1998 ACM Subject Classification E.1 Data Structures, E.4 Coding and Information Theory