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arXiv · 2610.19706

Revisiting Non-Adaptive Set Membership Schemes: Removing the Logarithmic Factor

Mirza Galib Anwarul Husain Baig·Aman Kumar·Pawan Kumar Saw·Vikash Tripathi
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Abstract

We study non-adaptive bit-probe schemes for static set membership. Each query reads a fixed number of bits and answers by strict majority. We revisit the random bipartite-graph construction of Garg and Radhakrishnan [STACS, 2017]. The main observation is that a slightly stronger expansion property already bounds the number of exceptional elements. This makes a separate probabilistic estimate for exceptional elements unnecessary. A detailed union-bound calculation then gives, for every fixed $t\ge5$, a logarithm-free existence bound of $O_t(m^{1/c}n^{1-1/c})$, where $c=\lfloor(t-1)/2\rfloor$. For odd $t$, this removes the logarithmic factor from the general upper bound of Garg and Radhakrishnan [STACS, 2017].

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Mirza Galib Anwarul Husain Baig, Aman Kumar, Pawan Kumar Saw, Vikash Tripathi. 2026-10-14. Revisiting Non-Adaptive Set Membership Schemes: Removing the Logarithmic Factor. https://arxiv.org/abs/2610.19706

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