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:• high-dimensional problems, where the goal is to show large space lower bounds.• constant-dimensional geometric problems, where the goal is to bound the query time for space O(n · polylogn).• dynamic problems, where we are looking for a trade-off between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lg lg n factor.)Our reductions also imply the following new results:• an Ω(lg n/ lg lg n) bound for 4-dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lg n) time, raising the prospect that higher dimensions could also be easy.• a tight space lower bound for the partial match problem, for constant query time.• the first lower bound for reachability oracles.In the process, we prove optimal randomized lower bounds for lopsided set disjointness. * The conference version of this paper appeared in FOCS'08 under the title (Data) Structures. † mip@alum.mit.edu. AT&T Labs. Parts of this work were done while the author was at MIT and IBM Almaden Research Center.

Equivalence between Priority Queues and Sorting in External Memory

“…Thus, a sorting lower bound (without any restrictions) has been considered to be more hopeful in the I/O model (with B not too small) than in the RAM model, and it was posed as a major open problem in [1]. Thus, our result provides a way to approach a sorting lower bound via that of priority queues, while data structure lower bounds have been considered (relatively) easier to obtain than (concrete) algorithm lower bounds (except in restricted computation models), as witnessed by the many recent strong cell probe lower bounds for data structures, such as [10,9] among many others. However, our result does not offer any new bounds for priority queues because we do not know of a better sorting algorithm than the comparison-based ones in the I/O model.…”

Optimal Planar Orthogonal Skyline Counting Queries

“…Our lower bound follows from a reduction of reachability in butterfly graphs to two-sided skyline counting queries, extending reductions by Pǎtraşcu [12] for two-dimensional rectangle stabbing and range counting queries. Our upper bounds are achieved by constructing a balanced search tree of degree Θ(lg ε n) over the points sorted by x-coordinate.…”

“…We prove our lower bound by reduction from the problem known as reachability oracles in the butterfly graph [12]. A butterfly graph of degree B and depth d is a directed graph with d + 1 layers, each having B d nodes ordered from left to right (see Figure 1).…”

“…Theorem 1 (Pǎtraşcu [12], Section 5) Any cell probe data structure answering reachability queries in subgraphs of the butterfly graph with degree B and depth d, having space usage S words of w bits, must have query time t = Ω(d), provided B = Ω(w 2 ) and lg B = Ω(lg Sd/N ). Here N denotes the number of non-sink nodes in the butterfly graph.…”

The cell probe complexity of dynamic range counting

“…This lower bound holds for regular counting (without weights), and even when just the parity of the number of points in the range is to be returned. In [7] he reproved this bound using an elegant reduction from the communication game known as lop-sided set disjointness. Subsequently Jørgensen and Larsen [3] proved a matching bound for the strongly related problems of range selection and range median.…”