Lower bounds for union-split-find related problems on random access machines

Abstract

We prove Ω(√log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the union-split-find problem, dynamic prefix problems and one-dimensional range query problems. The proof techniques include a general technique using perfect hashing for reducing static data structure problems (with a restriction of the size of the structure) into partially dynamic data structure problems (with no such restriction), thus providing a way to transfer lower bounds. We use a generalization of a method due to Ajtai for proving the lower bounds on the static problems, but describe the proof in terms of communication complexity, revealing a striking similarity to the proof used by Karchmer and Wigderson for proving lower bounds on the monotone circuit depth of connectivity.