Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes

Abstract

We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ? S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that s(m,n,t) = ?((2^t-1)m^{1/(t - min{2?log n?, n-3/2})}) for n ? 2,t ? ?log n?+1 . In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ? ?(nlog n), n ? 2, we design fully-explicit schemes that show that s(m,n,t) = ?((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}), asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n. In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n?, for n ? n?, s_N(m,n,3) ? ?{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ? ?(?m)