Abstract
We consider the problem of preprocessing N points in 2D, each endowed with a priority, to answer the following queries: given a axis-parallel rectangle, determine the point with the largest priority in the rectangle. Using the ideas of the effective entropy of range maxima queries and succinct indices for range maxima queries, we obtain a structure that uses O(N) words and answers the above query in O(logN loglogN) time. This a direct improvement of Chazelle’s result from 1985 [10] for this problem – Chazelle required O(N/ε) words to answer queries in O((logN)1 + ε) time for any constant ε > 0.
Work done while Farzan was employed by, and Raman was visiting, MPI.
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References
Afshani, P., Arge, L., Larsen, K.D.: Orthogonal range reporting: query lower bounds, optimal structures in 3-D, and higher-dimensional improvements. In: Snoeyink, J., de Berg, M., Mitchell, J.S.B., Rote, G., Teillaud, M. (eds.) Symposium on Computational Geometry, pp. 240–246. ACM (2010)
Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multi-labeled trees. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 680–689. SIAM, Philadelphia (2007)
Bender, M.A., Cole, R., Raman, R.: Exponential Structures for Efficient Cache-Oblivious Algorithms. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 195–207. Springer, Heidelberg (2002)
Bose, P., Chen, E.Y., He, M., Maheshwari, A., Morin, P.: Succinct geometric indexes supporting point location queries. In: Mathieu, C. (ed.) SODA, pp. 635–644. SIAM, Philadelphia (2009)
Bose, P., He, M., Maheshwari, A., Morin, P.: Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 98–109. Springer, Heidelberg (2009)
Brodal, G.S., Davoodi, P., Rao, S.S.: On Space Efficient Two Dimensional Range Minimum Data Structures. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 171–182. Springer, Heidelberg (2010)
Brodal, G.S., Jørgensen, A.G.: Data Structures for Range Median Queries. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 822–831. Springer, Heidelberg (2009)
Chan, T.M., Larsen, K.G., Pătraşcu, M.: Orthogonal range searching on the RAM, revisited. In: Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SoCG 2011, pp. 1–10. ACM, New York (2011), http://doi.acm.org/10.1145/1998196.1998198
Chan, T.M., Patrascu, M.: Transdichotomous results in computational geometry, I: Point location in sublogarithmic time. SIAM J. Comput. 39(2), 703–729 (2009)
Chazelle, B.: A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput. 17(3), 427–462 (1988), prel. vers. FOCS 1985 (1985)
Farzan, A., Munro, J.I., Raman, R.: Succinct indices for range queries with applications to orthogonal range maxima. Tech. Rep. CS-TR-12-001, U. Leicester (April 2012), http://arxiv.org/abs/1204.4835
Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proc. 16th Annual ACM Symposium on Theory of Computing, pp. 135–143. ACM (1984)
Golin, M.J., Iacono, J., Krizanc, D., Raman, R., Rao, S.S.: Encoding 2D Range Maximum Queries. In: Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 180–189. Springer, Heidelberg (2011)
JáJá, J., Mortensen, C.W., Shi, Q.: Space-Efficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 558–568. Springer, Heidelberg (2004)
Karpinski, M., Nekrich, Y.: Space Efficient Multi-Dimensional Range Reporting. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 215–224. Springer, Heidelberg (2009)
Makris, C., Tsakalidis, A.K.: Algorithms for three-dimensional dominance searching in linear space. Inf. Process. Lett. 66(6), 277–283 (1998)
Mehta, D.P., Sahni, S. (eds.): Handbook of Data Structures and Applications. Chapman & Hall/CRC (2009)
Nekrich, Y.: Orthogonal range searching in linear and almost-linear space. Comput. Geom. 42(4), 342–351 (2009)
Rahman, N., Raman, R.: Rank and select operations on binary strings. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, Springer (2008)
Yuan, H., Atallah, M.J.: Data structures for range minimum queries in multidimensional arrays. In: Charikar, M. (ed.) SODA 2010, pp. 150–160. SIAM, Philadelphia (2010)
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Farzan, A., Munro, J.I., Raman, R. (2012). Succinct Indices for Range Queries with Applications to Orthogonal Range Maxima. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_28
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