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

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.

💡 Research Summary

The paper introduces a new model for error‑correcting data structures that relaxes the stringent requirement of answering every query correctly in the presence of noise. In this model a data structure for a function f is characterized by four parameters (t, δ, ε, λ): t is the number of bit‑probes per query, δ is the fraction of adversarial bit‑flips tolerated in the stored representation, ε bounds the error probability for each query, and λ bounds the fraction of “bad” queries for which the decoder is allowed to output a special symbol ⊥ (“don’t know”). When λ = 0 the model coincides with de Wolf’s error‑correcting data structures, which are essentially locally decodable codes (LDCs). However, known constant‑probe LDCs have super‑polynomial length, making them impractical for most data‑structure tasks.

The authors observe that many data‑structure problems admit “relaxed” solutions: it is sufficient that the decoder answers correctly on most queries, while a small fraction may be declared unknown. This aligns with the notion of relaxed locally decodable codes (RLDCs) introduced by Ben‑Sasson et al., which achieve constant‑probe decoding with near‑linear length. By composing RLDCs with existing noise‑free data structures that have pseudo‑random probe patterns, the authors obtain error‑correcting structures that retain the optimal space‑time trade‑offs of the noiseless case.

Two canonical problems are studied:

1. Membership Queries – Store a set S ⊆