Document

**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

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}

Victor Chen, Elena Grigorescu, and Ronald de Wolf. Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 203-214, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@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} }

Document

**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for {}``triangle freeness'': A function $f:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}$ satisfies this property if $f(x),f(y),f(x+y)$ do not all equal $1$, for any pair $x,y\in\mathbb{F}_{2}^{n}$.
Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points $v_{1},\ldots,v_{k}\in\mathbb{F}_{2}^{k}$ and $f:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}$ satisfies the property that if for all linear maps $L:\mathbb{F}_{2}^{k}\to\mathbb{F}_{2}^{n}$ it is the case that $f(L(v_{1})),\ldots,f(L(v_{k}))$ do not all equal $1$. We show that this property is testable if the underlying matroid specified by $v_{1},\ldots,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties.
Our techniques extend those of Green and in particular we establish a link between the notion of {}``1-complexity linear systems'' of Green and Tao, and graphic matroids, to derive the results.

Arnab Bhattacharyya, Victor Chen, Madhu Sudan, and Ning Xie. Testing Linear-Invariant Non-Linear Properties. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 135-146, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bhattacharyya_et_al:LIPIcs.STACS.2009.1823, author = {Bhattacharyya, Arnab and Chen, Victor and Sudan, Madhu and Xie, Ning}, title = {{Testing Linear-Invariant Non-Linear Properties}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {135--146}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1823}, URN = {urn:nbn:de:0030-drops-18235}, doi = {10.4230/LIPIcs.STACS.2009.1823}, annote = {Keywords: } }

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