Abstract
Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set \(\mathcal {B}\) of axis aligned boxes, whose computation generalizes that of the Klee’s Measure and of the Maximum Depth. In the worst case over instances of fixed input size n, we describe an algorithm of complexity within \(\mathcal {O}(n^\frac{d+1}{2}\log n)\), using space within \(\mathcal {O}(n\log n)\), mixing two techniques previously used to compute the Klee’s Measure. We refine this result and previous results on the Klee’s Measure and the Maximum Depth for various measures of difficulty of the input, such as the profile of the input and the degeneracy of the intersection graph formed by the boxes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Khamis, M.A., Ngo, H.Q., Ré, C., Rudra, A.: Joins via geometric resolutions: worst-case and beyond. In: Proceedings of the 34th ACM Symposium on Principles of Database Systems (PODS), Melbourne, Victoria, Australia, 31 May–4 June, 2015, pp. 213–228 (2015)
Afshani, P.: Fast computation of output-sensitive maxima in a word RAM. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Portland, Oregon, USA, January 5–7, 2014, pp. 1414–1423. SIAM (2014)
Afshani, P., Arge, L., Larsen, K.G.: Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In: Symposuim on Computational Geometry (SoCG), Chapel Hill, NC, USA, June 17–20, 2012, pp. 323–332 (2012)
Afshani, P., Barbay, J., Chan, T.M.: Instance-optimal geometric algorithms. J. ACM (JACM) 64(1), 3:1–3:38 (2017)
Barbay, J., Pérez-Lantero, P., Rojas-Ledesma, J.: Depth distribution in high dimension. ArXiv e-prints (2017)
Bentley, J.L.: Algorithms for Klee’s rectangle problems. Unpublished notes (1977)
Bringmann, K.: An improved algorithm for Klee’s measure problem on fat boxes. Comput. Geom. Theor. Appl. 45(5–6), 225–233 (2012)
Bringmann, K.: Bringing order to special cases of Klee’s measure problem. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 207–218. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40313-2_20
Chan, T.M.: A (slightly) faster algorithm for Klee’s Measure Problem. In: Proceedings of the 24th ACM Symposium on Computational Geometry (SoCG), College Park, MD, USA, June 9–11, 2008, pp. 94–100 (2008)
Chan, T.M.: Klee’s measure problem made easy. In: 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Berkeley, CA, USA, 26–29 October, 2013, pp. 410–419 (2013)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), New York, USA, pp. 1–6. ACM (1987)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
d’Amore, F., Nguyen, V.H., Roos, T., Widmayer, P.: On optimal cuts of hyperrectangles. Computing 55(3), 191–206 (1995)
Edelsbrunner, H.: A new approach to rectangle intersections part I. J. Comput. Math. (JCM) 13(3–4), 209–219 (1983)
Fredman, M.L., Weide, B.W.: On the complexity of computing the measure of \(\cup ^{n}_1 [a_i; b_i]\). Commun. ACM (CACM) 21(7), 540–544 (1978)
Gall, F.L.: Powers of tensors and fast matrix multiplication. In: International Symposium on Symbolic and Algebraic Computation (ISSAC), Kobe, Japan, July 23–25, 2014, pp. 296–303. ACM (2014)
Kirkpatrick, D.G., Seidel, R.: Output-size sensitive algorithms for finding maximal vectors. In: Proceedings of the First Annual Symposium on Computational Geometry (SoCG), Baltimore, Maryland, USA, June 5–7, 1985, pp. 89–96 (1985)
Klee, V.: Can the measure of \(\cup ^{n}_1 [a_i; b_i]\) be computed in less than O(n log n) steps? Am. Math. Mon. (AMM) 84(4), 284–285 (1977)
Lick, D.R., White, A.T.: k-Degenerate graphs. Can. J. Math. (CJM) 22, 1082–1096 (1970)
Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM (JACM) 30(3), 417–427 (1983)
Moffat, A., Petersson, O.: An overview of adaptive sorting. Aust. Comput. J. (ACJ) 24(2), 70–77 (1992)
Overmars, M.H., Yap, C.: New upper bounds in Klee’s measure problem. SIAM J. Comput. (SICOMP) 20(6), 1034–1045 (1991)
Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13(4), 354–356 (1969)
Yildiz, H., Hershberger, J., Suri, S.: A discrete and dynamic version of Klee’s measure problem. In: Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG), Toronto, Ontario, Canada, August 10–12, 2011 (2011)
Funding
All authors were supported by the Millennium Nucleus “Information and Coordination in Networks” ICM/FIC RC130003. Jérémy Barbay and Pablo Pérez-Lantero were supported by the projects CONICYT Fondecyt/Regular nos 1170366 and 1160543 (Chile) respectively, while Javiel Rojas-Ledesma was supported by CONICYT-PCHA/Doctorado Nacional/2013-63130209 (Chile).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Barbay, J., Pérez-Lantero, P., Rojas-Ledesma, J. (2017). Depth Distribution in High Dimensions. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-62389-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62388-7
Online ISBN: 978-3-319-62389-4
eBook Packages: Computer ScienceComputer Science (R0)