# A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy

@inproceedings{DallAgnol2020AST,
title={A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy},
author={Marcel Dall'Agnol and Tom Gur and Oded Lachish},
booktitle={Electron. Colloquium Comput. Complex.},
year={2020}
}
• Published in
Electron. Colloquium Comput…
10 October 2020
• Computer Science, Mathematics
We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 \log^2 q)}$ sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits…
2 Citations

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## References

SHOWING 1-10 OF 59 REFERENCES
Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability
• Computer Science, Mathematics
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
It is shown that every non-adaptive property testing algorithm making a constant number of queries, over a fixed alphabet, can be converted to a sample-based testing algorithm whose average number of query is a fixed, smaller than 1, power of n.
Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation
• Computer Science
Electron. Colloquium Comput. Complex.
• 2009
The 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 is applied.
Interactive proofs of proximity : Delegating computation in sublinear time
• Computer Science
• 2013
It is shown that there exist natural languages in NC, for which the sum of queries and communication in any constant-round interactive proof of proximity must be polynomially related to n, and that there are much better IPPs for specific functions, such as bipartiteness on random or wellmixing graphs, and the majority function.
Relaxed Locally Correctable Codes with Improved Parameters
• Computer Science
Electron. Colloquium Comput. Complex.
• 2020
This work improves the parameters of [BGH+06] by constructing an O(q)-query RLDC that encodes a message of length using a codeword of block length $n = O(k^{1+1/\sqrt{q}})$.
Universal Locally Verifiable Codes and 3-Round Interactive Proofs of Proximity for CSP
• Computer Science
Electron. Colloquium Comput. Complex.
• 2016
Two Query PCP with Sub-Constant Error
• Computer Science, Mathematics
2008 49th Annual IEEE Symposium on Foundations of Computer Science
• 2008
This work shows that the NP-Complete language 3Sat has a PCPverifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1), and improves many of the hardness of approximation results that are proved using the parallel repetition theorem.
On the Power of Relaxed Local Decoding Algorithms
• Computer Science
SODA
• 2020
Using algorithmic and combinatorial techniques, it is proved that codes with blocklength n = k 1+ o (1) cannot be relaxed decoded with O (1)-query algorithms.
Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding
• Computer Science
SIAM J. Comput.
• 2006
The main technical contribution is a construction of a “length-efficient” robust PCP of proximity, which does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al.
Locally testable codes and PCPs of almost-linear length
• Computer Science
JACM
• 2006
The novel techniques in use include a random projection of certain codewords and PCP-oracles that preserves local-testability, an adaptation of PCP constructions to obtain “linear PCP” for proving conjunctions of linear conditions, and design of PCPs with some new soundness properties.
Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity
• Computer Science
SODA
• 2020
This work constructs an O(1)-query relaxed LCC with nearlylinear block length n = k, for an arbitrarily small constant α > 0, which significantly narrows the gap between the lower bound which states that there are no O( 1)-query relax LCCs with block lengthn = k.