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't know.'' 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'' 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 $s\log n$ that answers membership queries with $O(1)$ bit-probes. This nearly matches the asymptotically optimal parameters for the noiseless case: length $O(s\log n)$ 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 $s\log n$ that answers evaluation queries with $\polylog s\cdot\log^{1+o(1)}n$ bit-probes. This nearly matches the parameters of the best-known noiseless construction, due to Kedlaya and Umans. \end{itemize}
@InProceedings{chen_et_al:LIPIcs.STACS.2010.2455, author = {Chen, Victor and Grigorescu, Elena and de Wolf, Ronald}, title = {{Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {203--214}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2455}, URN = {urn:nbn:de:0030-drops-24558}, doi = {10.4230/LIPIcs.STACS.2010.2455}, annote = {Keywords: Data Structures, Error-Correcting Codes, Membership, Polynomial Evaluation} }
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