{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6674","name":"Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation","abstract":"We construct efficient data structures that are resilient against\r\na constant fraction of adversarial noise. Our model requires that\r\nthe 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.\r\n\r\nWe 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: \r\n\\begin{itemize}\r\n\\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 $S$. \\\\\r\n 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. \r\n\\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 $n$. \\\\\r\n We construct an error-correcting data structure for this problem\r\nwith length nearly linear in $s\\log n$ that answers evaluation queries\r\nwith $\\polylog s\\cdot\\log^{1+o(1)}n$ bit-probes. This nearly matches\r\nthe parameters of the best-known noiseless construction, due to Kedlaya and Umans.\r\n\\end{itemize}","keywords":["Data Structures","Error-Correcting Codes","Membership","Polynomial Evaluation"],"author":[{"@type":"Person","name":"Chen, Victor","givenName":"Victor","familyName":"Chen"},{"@type":"Person","name":"Grigorescu, Elena","givenName":"Elena","familyName":"Grigorescu"},{"@type":"Person","name":"de Wolf, Ronald","givenName":"Ronald","familyName":"de Wolf"}],"position":19,"pageStart":203,"pageEnd":214,"dateCreated":"2010-03-09","datePublished":"2010-03-09","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chen, Victor","givenName":"Victor","familyName":"Chen"},{"@type":"Person","name":"Grigorescu, Elena","givenName":"Elena","familyName":"Grigorescu"},{"@type":"Person","name":"de Wolf, Ronald","givenName":"Ronald","familyName":"de Wolf"}],"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2010.2455","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6208","volumeNumber":5,"name":"27th International Symposium on Theoretical Aspects of Computer Science","dateCreated":"2010-03-09","datePublished":"2010-03-09","editor":[{"@type":"Person","name":"Marion, Jean-Yves","givenName":"Jean-Yves","familyName":"Marion"},{"@type":"Person","name":"Schwentick, Thomas","givenName":"Thomas","familyName":"Schwentick"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6674","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6208"}}}