Certificates in Data Structures

TL;DR

This paper introduces and analyzes certificates in static data structures within the cell-probe model, establishing lower bounds that match or surpass existing bounds for various problems, notably improving the state of the art for approximate near neighbor in Hamming space.

Contribution

It extends the richness lemma to certificates, deriving new lower bounds for static data structures, including the first t=Omega(d) bound for ANN in Hamming space.

Findings

Certificates in data structures can be used to derive lower bounds.

The richness lemma applies to certificates with improved parameters.

Achieved the first t=Omega(d) lower bound for ANN in Hamming space.

Abstract

We study certificates in static data structures. In the cell-probe model, certificates are the cell probes which can uniquely identify the answer to the query. As a natural notion of nondeterministic cell probes, lower bounds for certificates in data structures immediately imply deterministic cell-probe lower bounds. In spite of this extra power brought by nondeterminism, we prove that two widely used tools for cell-probe lower bounds: richness lemma of Miltersen et al. and direct-sum richness lemma of Patrascu and Thorup, both hold for certificates in data structures with even better parameters. Applying these lemmas and adopting existing reductions, we obtain certificate lower bounds for a variety of static data structure problems. These certificate lower bounds are at least as good as the highest known cell-probe lower bounds for the respective problems. In particular, for approximate near neighbor (ANN) problem in Hamming distance, our lower bound improves the state of the art. When the space is strictly linear, our lower bound for ANN in d-dimensional Hamming space becomes t=Omega(d), which along with the recent breakthrough for polynomial evaluation of Larsen, are the only two t=Omega(d) lower bounds ever proved for any problems in the cell-probe model.