A New Lower Bound for Semigroup Orthogonal Range Searching
We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [Yao, 1982]: using m units of space, the query time Q(n) must be Omega(alpha(m,n) + n/(m-n+1)) where alpha(*,*) is the inverse Ackermann’s function, a very slowly growing function. In d dimensions, Bernard Chazelle [Chazelle, 1990] proved that the query time must be Q(n) = Omega((log_beta n)^{d-1}) where beta = 2m/n. Chazelle’s lower bound is known to be tight for when space consumption is "high" i.e., m = Omega(n log^{d+epsilon}n).
We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for "low space": we prove that we must have m Q(n) = Omega(n (log n log log n)^{d-1}). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.
Data Structures
Range Searching
Lower bounds
Theory of computation~Randomness, geometry and discrete structures
Theory of computation~Computational geometry
3:1-3:14
Regular Paper
A full version of the paper is available at [Peyman Afshani, 2019], https://arxiv.org/abs/1903.07967.
Peyman
Afshani
Peyman Afshani
Aarhus University, Denmark
supported by DFF (Det Frie Forskningsräd) of Danish Council for Indepndent Reserach under grant ID DFF-7014-00404.
10.4230/LIPIcs.SoCG.2019.3
Peyman Afshani. A New Lower Bound for Semigroup Orthogonal Range Searching, 2019. URL: http://arxiv.org/abs/1903.07967.
http://arxiv.org/abs/1903.07967
Peyman Afshani, Lars Arge, and Kasper Green Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In Symposium on Computational Geometry (SoCG), pages 323-332, 2012. URL: http://dx.doi.org/10.1145/2261250.2261299.
http://dx.doi.org/10.1145/2261250.2261299
Peyman Afshani and Anne Drimel. On the complexity of range searching among curves. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 898-917, 2017.
Pankaj K. Agarwal. Range searching. In J. E. Goodman, J. O'Rourke, and C. Toth, editors, Handbook of Discrete and Computational Geometry. CRC Press, Inc., 2016.
N. Alon and B. Schieber. OPTIMAL PREPROCESSING FOR ANSWERING ON-LINE PRODUCT QUERIES. Technical Report 71/87, Tel-Aviv University, 1987.
Sunil Arya, Theocharis Malamatos, and David M. Mount. On the Importance of Idempotence. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 564-573, 2006.
Jon Louis Bentley. Decomposable searching problems. Information Processing Letters (IPL), 8(5):244-251, 1979.
Jon Louis Bentley. Multidimensional divide-and-conquer. Communications of the ACM (CACM), 23(4):214-229, 1980.
Bernard Chazelle. Lower bounds for orthogonal range searching: part II. The arithmetic model. Journal of the ACM (JACM), 37(3):439-463, 1990.
Michael L. Fredman. The inherent complexity of dynamic data structures which accommodate range queries. In Proc. 21stProceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 191-199, October 1980.
Michael L. Fredman. A Lower Bound on the Complexity of Orthogonal Range Queries. Journal of the ACM (JACM), 28(4):696-705, 1981.
Michael L. Fredman. Lower Bounds on the Complexity of Some Optimal Data Structures. SIAM Journal of Computing, 10(1):1-10, 1981.
Robert Endre Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. Journal of Computer and System Sciences (JCSS), 18(2):110-127, 1979.
Andrew C. Yao. Space-time Tradeoff for Answering Range Queries (Extended Abstract). In Proceedings of ACM Symposium on Theory of Computing (STOC), STOC '82, pages 128-136. ACM, 1982.
Peyman Afshani
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode