## On Range Searching in the Group Model and Combinatorial Discrepancy

Research output: Contribution to book/anthology/report/proceedingArticle in proceedingsResearchpeer-review

### DOI

• Department of Computer Science
In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that \$t_ut_q = Omega(disc^2/lg n)\$ where \$t_u\$ is the worst case update time, \$t_q\$ the worst case query time and \$disc\$ is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and near-tight lower bounds for all of the basic range searching problems. We list a few of them in the following:begin{itemize}item For half space range searching in \$d\$-dimensional space, we get a lower bound of \$t_u t_q = Omega(n^{1-1/d}/lg n)\$. This comes within a \$lg n lg lg n\$ factor of the best known upper bound. item For orthogonal range searching in \$d\$-dimensional space, we get a lower bound of \$t_u t_q = Omega(lg^{d-2+mu(d)}n)\$, where \$mu(d)>0\$ is some small but strictly positive function of \$d\$.item For ball range searching in \$d\$-dimensional space, we get a lower bound of \$t_u t_q = Omega(n^{1-1/d}/lg n)\$.end{itemize}We note that the previous highest lower bound for any explicit problem, due to P{v a}tra{c s}cu [STOC'07], states that \$t_q =Omega((lg n/lg(lg n+t_u))^2)\$, which does however hold for a less restrictive class of data structures. Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axis-aligned rectangles in dimensions \$d geq 3\$.
Original language English 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS) 8 IEEE Computer Society Press 2011 542-549 978-1-4577-1843-4 https://doi.org/10.1109/FOCS.2011.14 Published - 2011 52nd Annual IEEE Symposium on Foundations of Computer Science. FOCS 2011 - Palm Springs, CA, United StatesDuration: 22 Oct 2011 → 25 Oct 2011

### Conference

Conference 52nd Annual IEEE Symposium on Foundations of Computer Science. FOCS 2011 United States Palm Springs, CA 22/10/2011 → 25/10/2011

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