On Dynamic Bit-Probe Complexity

This work present several advances in the understanding of dynamic data structures in the bit-probe model: • We improve the lower bound record for dynamic language membership problems to Ω(( lg n lg lg n ) ). Surpassing Ω(lg n) was listed as the first open problem in a survey by Miltersen. • We prove a bound of Ω( lg n lg lg lg n ) for maintaining partial sums in Z/2Z. Previously, the known bounds were Ω( lg n lg lg n ) and O(lg n). • We prove a surprising and tight upper bound of O( lg n lg lg n ) for the greater-than problem, and several predecessor-type problems. We use this to obtain the same upper bound for dynamic word and prefix problems in group-free monoids. We also obtain new lower bounds for the partial-sums problem in the cell-probe and externalmemory models. Our lower bounds are based on a surprising improvement of the classic chronogram technique of Fredman and Saks [1989], which makes it possible to prove logarithmic lower bounds by this approach. Before the work of M. Pǎtraşcu and Demaine [2004], this was the only known technique for dynamic lower bounds, and surpassing Ω( lg n lg lg n ) was a central open problem in cell-probe complexity.