Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication

Abstract

This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party communication model that enables accelerated, error-preserving simulations of dynamic data structures. We use this technique to prove an Ω(n(log n/log log n)2) cell-probe lower bound for the dynamic 2D weighted orthogonal range counting problem (2D-ORC) with n/poly log n updates and n queries, that holds even for data structures with exp(-Ω(n)) success probability. This result not only proves the highest amortized lower bound to date, but is also tight in the strongest possible sense, as a matching upper bound can be obtained by a deterministic data structure with worst-case operational time. This is the first demonstration of a sharp threshold phenomenon for dynamic data structures. Our broader motivation is that cell-probe lower bounds for exponentially small success facilitate reductions from dynamic to static data structures. As a proof-of-concept, we show that a slightly strengthened version of our lower bound would imply an Ω((log n/log log n)2) lower bound for the static 3D-ORC problem with O(n logO(1) n) space. Such result would give a near quadratic improvement over the highest known static cell-probe lower bound, and break the long standing Ω(log n) barrier for static data structures.