Error-Correcting Data Structures

TL;DR

This paper explores error-correcting data structures that encode objects efficiently to allow accurate queries despite adversarial errors, establishing bounds and relationships with locally decodable codes.

Contribution

It introduces a generalized model combining data structures and error-correcting codes, providing bounds and insights into their space-query time tradeoffs.

Findings

Optimal error-correcting data structures for Membership problem are characterized.

Bounds on space and query time tradeoffs are established.

Connections to locally decodable codes are demonstrated.

Abstract

We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been corrupted by a constant fraction of errors. This new model is the common generalization of (static) data structures and locally decodable error-correcting codes. The main issue is the tradeoff between the space used by the data structure and the time (number of probes) needed to answer a query about the encoded object. We prove a number of upper and lower bounds on various natural error-correcting data structure problems. In particular, we show that the optimal length of error-correcting data structures for the Membership problem (where we want to store subsets of size s from a universe of size n) is closely related to the optimal length of locally decodable codes for s-bit strings.