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

TL;DR

This paper introduces efficient data structures for membership and polynomial evaluation that are resilient to adversarial noise, ensuring high-probability correct responses or safe 'don't know' answers, generalizing existing models.

Contribution

It presents novel error-correcting data structures for membership and polynomial evaluation, combining efficiency with robustness against adversarial noise.

Findings

Data structures are efficient in size and query time.

They can correct a constant fraction of adversarial errors.

High probability of correct responses or safe 'don't know' answers.

Abstract

We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers 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 all queries correctly with high probability. Our model is the common generalization of a 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 length, measured by the number of bits in its representation, and query-answering time, measured by the number of bit-probes to the (possibly corrupted) representation. In this work, we study two data structure problems: membership and polynomial evaluation. We show that these two problems have constructions that are simultaneously efficient and error-correcting.