Lower bound techniques for data structures

Abstract

We describe new techniques for proving lower bounds on data-structure problems, with the following broad consequences: * the first [omega](lg n) lower bound for any dynamic problem, improving on a bound that had been standing since 1989; * for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show [omega](lg n/ lg lg n) bounds when the space is O(n - polylog n). Using these techniques, we analyze a variety of central data-structure problems, and obtain improved lower bounds for the following: * the partial-sums problem (a fundamental application of augmented binary search trees); * the predecessor problem (which is equivalent to IP lookup in Internet routers); * dynamic trees and dynamic connectivity; * orthogonal range stabbing. * orthogonal range counting, and orthogonal range reporting; * the partial match problem (searching with wild-cards); * (1 + [epsilon])-approximate near neighbor on the hypercube; * approximate nearest neighbor in the l[infinity] metric. Our new techniques lead to surprisingly non-technical proofs. For several problems, we obtain simpler proofs for bounds that were already known.