Certificates in Data Structures

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. [10] and direct-sum richness lemma of Pǎtraşcu and Thorup [17], 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 = Ω(d), which along with the recent breakthrough for polynomial evaluation of Larsen [8], are the only two t = Ω(d) lower bounds ever proved for any problems in the cell-probe model.