UNIFYING THE LANDSCAPE OF CELL-PROBE LOWER BOUNDS
We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for (i) high-dimensional problems, where the goal is to show large space lower bounds; (ii) constant-dimensional geometric problems, where the goal is to bound the query time for space O(n . polylogn); and (iii) dynamic problems, where we are looking for a trade-off between query and update time. (In the last case, our bounds are slightly weaker than the originals, losing a lg lg n factor.) Our reductions also imply the following new results: (i) an Omega(lg n/lg lg n) bound for four-dimensional range reporting, given space O(n . polylogn) (this is quite timely, since a recent result [Y. Nekrich, in Proceedings of the 23rd ACM Symposium on Computational Geometry (SoCG), 2007, pp. 344-353] solved three-dimensional reporting in O(lg(2) lg n) time, raising the prospect that higher dimensions could also be easy); (ii) a tight space lower bound for the partial match problem, for constant query time; and (iii) the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness.
lower bounds data structures cell-probe complexity range queries