The Cell Probe Complexity of Dynamic Range Counting
Abstract
In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem, namely a lower bound of $t_q=\Omega((\lg n/\lg(wt_u))^2)$. Here $n$ is the number of update operations, $w$ the cell size, $t_q$ the query time and $t_u$ the update time. In the most natural setting of cell size $w=\Theta(\lg n)$, this gives a lower bound of $t_q=\Omega((\lg n/\lg \lg n)^2)$ for any polylogarithmic update time. This bound is almost a quadratic improvement over the highest previous lower bound of $\Omega(\lg n)$, due to Pǎtrașcu and Demaine [SICOMP'06]. We prove the lower bound for the fundamental problem of weighted orthogonal range counting. In this problem, we are to support insertions of twodimensional points, each assigned a $\Theta(\lg n)$bit integer weight. A query to this problem is specified by a point $q=(x,y)$, and the goal is to report the sum of the weights assigned to the points dominated by $q$, where a point $(x',y')$ is dominated by $q$ if $x' \leq x$ and $y' \leq y$. In addition to being the highest cell probe lower bound to date, the lower bound is also tight for data structures with update time $t_u = \Omega(\lg^{2+\eps}n)$, where $\eps>0$ is an arbitrarily small constant.
 Publication:

arXiv eprints
 Pub Date:
 May 2011
 arXiv:
 arXiv:1105.5933
 Bibcode:
 2011arXiv1105.5933G
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity
 EPrint:
 This is an updated version of the paper which has been submitted to Journal of the ACM by invitation. The new version contains a new section which introduces an artificial problem for which it is significantly easier to apply the new lower bound technique