Unifying the Landscape of Cell-Probe Lower Bounds

Abstract

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 $O(n\cdot\mathrm{polylog}n)$; 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 $O(n\cdot\mathrm{polylog}n)$ (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.