Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation

Abstract

We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers emph{most} queries correctly with high probability and for the remaining queries, the decoder with high probability either answers correctly or declares ``don\u27t know.\u27\u27 Furthermore, if there is no noise on the data structure, it answers emph{all} queries correctly with high probability. Our model is the common generalization of an error-correcting data structure model proposed recently by de~Wolf, and the notion of ``relaxed locally decodable codes\u27\u27 developed in the PCP literature. We measure the efficiency of a data structure in terms of its emph{length} (the number of bits in its representation), and query-answering time, measured by the number of emph{bit-probes} to the (possibly corrupted) representation. We obtain results for the following two data structure problems: begin{itemize} item (Membership) Store a subset $S$ of size at most $s$ from a universe of size $n$ such that membership queries can be answered efficiently, i.e., decide if a given element from the universe is in $S$. \ We construct an error-correcting data structure for this problem with length nearly linear in $slog n$ that answers membership queries with $O(1)$ bit-probes. This nearly matches the asymptotically optimal parameters for the noiseless case: length $O(slog n)$ and one bit-probe, due to Buhrman, Miltersen, Radhakrishnan, and Venkatesh. item (Univariate polynomial evaluation) Store a univariate polynomial $g$ of degree $deg(g)leq s$ over the integers modulo $n$ such that evaluation queries can be answered efficiently, i.e., we can evaluate the output of $g$ on a given integer modulo $n$. \ We construct an error-correcting data structure for this problem with length nearly linear in $slog n$ that answers evaluation queries with $polylog scdotlog^{1+o(1)}n$ bit-probes. This nearly matches the parameters of the best-known noiseless construction, due to Kedlaya and Umans. end{itemize