Abstract
Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a color. While the answer to an orthogonal point reporting query contains all points in the query range Q, the answer to a color reporting query contains only distinct colors of points in Q. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal O(k + 1) time, where k is the number of colors in the answer and N is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model.
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References
Alstrup, S., Brodal, G.S., Rauhe, T.: Optimal static range reporting in one dimension. In: Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 476–482 (2001)
Arge, L., Samoladas, V., Vitter, J.S.: On two-dimensional indexability and optimal range search indexing. In: Proc. 18th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), pp. 346–357 (1999)
Arge, L., Vitter, J.S.: Optimal external memory interval management. SIAM J. Comput. 32(6), 1488–1508 (2003)
Beame, P., Fich, F.E.: Optimal bounds for the predecessor problem and related problems. J. Comput. Syst. Sci. 65(1), 38–72 (2002)
Chan, T.M., Durocher, S., Skala, M., Wilkinson, B.T.: Linear-space data structures for range minority query in arrays. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 295–306. Springer, Heidelberg (2012)
Chazelle, B.: Filtering search: a new approach to query-answering. SIAM J. Comput. 15(3), 703–724 (1986)
Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci. 48(3), 533–551 (1994)
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 (STOC 1984), pp. 135–143 (1984)
Gupta, P., Janardan, R., Smid, M.H.M.: Further results on generalized intersection searching problems: counting, reporting, and dynamization. Journal of Algorithms 19(2), 282–317 (1995)
Janardan, R., Lopez, M.A.: Generalized intersection searching problems. International Journal of Computational Geometry and Applications 3(1), 39–69 (1993)
Larsen, K.G., Pagh, R.: I/O-efficient data structures for colored range and prefix reporting. In: Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 583–592 (2012)
Larsen, K.G., van Walderveen, F.: Near-optimal range reporting structures for categorical data. In: Proc. 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (to appear, 2013)
McCreight, E.M.: Priority search trees. SIAM J. Comput. 14(2), 257–276 (1985)
Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. J. Comput. Syst. Sci. 57(1), 37–49 (1998)
Mortensen, C.W.: Generalized static orthogonal range searching in less space. Technical report, IT University Technical Report Series 2003-33 (2003)
Mortensen, C.W., Pagh, R., Patrascu, M.: On dynamic range reporting in one dimension. In: Proc. 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 104–111 (2005)
Muthukrishnan, S.: Efficient algorithms for document retrieval problems. In: Proc. 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 657–666 (2002)
Nekrich, Y.: Space-efficient range reporting for categorical data. In: Proc. 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS), pp. 113–120 (2012)
Nekrich, Y., Vitter, J.S.: Optimal color range reporting in one dimension. CoRR, abs/1306.5029 (2013)
Shi, Q., JáJá, J.: Optimal and near-optimal algorithms for generalized intersection reporting on pointer machines. Inf. Process. Lett. 95(3), 382–388 (2005)
Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46(3), 362–394 (1999)
van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Math. Sys. Theory 10, 99–127 (1977)
Willard, D.E.: Examining computational geometry, van Emde Boas trees, and hashing from the perspective of the fusion tree. SIAM J. Comput. 29(3), 1030–1049 (2000)
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Nekrich, Y., Vitter, J.S. (2013). Optimal Color Range Reporting in One Dimension. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_63
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DOI: https://doi.org/10.1007/978-3-642-40450-4_63
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