Abstract
The two dimensional range minimum query problem is to preprocess a static two dimensional m by n array A of size N = m · n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using O(N/c) bits additional space requires Ω(c) query time, for any c where 1 ≤ c ≤ N. This lower bound holds for any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits additional space which can be preprocessed in O(N) time and achieves O(clog2 c) query time. For c = O(1), this is the first O(1) query time algorithm using optimal O(N) bits additional space. For the case where queries can not probe A, we give a data structure of size O(N· min {m,logn}) bits with O(1) query time, assuming m ≤ n. This leaves a gap to the lower bound of Ω(Nlogm) bits for this version of the problem.
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References
Aho, A., Hopcroft, J., Ullman, J.: On finding lowest common ancestors in trees. In: Proc. 5th Annual ACM Symposium on Theory of Computing, pp. 253–265. ACM Press, New York (1973)
Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest common ancestors: a survey and a new distributed algorithm. In: Proc. 14th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 258–264. ACM, New York (2002)
Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 286–294. Springer, Heidelberg (2007)
Atallah, M.J., Yuan, H.: Data structures for range minimum queries in multidimensional arrays. In: Proc. 20th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 150–160. SIAM, Philadelphia (2010)
Bender, M., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)
Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. Journal of Algorithms 57(2), 75–94 (2005)
Bentley, J.L.: Decomposable searching problems. Information Processing Letters 8(5), 244–251 (1979)
Berkman, O., Galil, Z., Schieber, B., Vishkin, U.: Highly parallelizable problems. In: Proc. 21st Ann. ACM Symposium on Theory of Computing, pp. 309–319. ACM, New York (1989)
Chazelle, B., Rosenberg, B.: Computing partial sums in multidimensional arrays. In: Proc. 5th Annual Symposium on Computational Geometry, pp. 131–139. ACM, New York (1989)
Demaine, E.D., Landau, G.M., Weimann, O.: On cartesian trees and range minimum queries. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 341–353. Springer, Heidelberg (2009)
Fischer, J.: Optimal succinctness for range minimum queries. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 158–169. Springer, Heidelberg (2010)
Fischer, J., Heun, V.: A new succinct representation of rmq-information and improvements in the enhanced suffix array. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 459–470. Springer, Heidelberg (2007)
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, New York (1984)
Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing 13(2), 338–355 (1984)
Miltersen, P.B.: Cell probe complexity - a survey. In: Advances in Data Structures Workshop (FSTTCS) (1999)
Sadakane, K.: Succinct data structures for flexible text retrieval systems. Journal of Discrete Algorithms 5(1), 12–22 (2007)
Schieber, B., Vishkin, U.: On finding lowest common ancestors: simplification and parallelization. SIAM Journal on Computing 17(6), 1253–1262 (1988)
Vuillemin, J.: A unifying look at data structures. Communications of the ACM 23(4), 229–239 (1980)
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Brodal, G.S., Davoodi, P., Rao, S.S. (2010). On Space Efficient Two Dimensional Range Minimum Data Structures. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15781-3_15
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DOI: https://doi.org/10.1007/978-3-642-15781-3_15
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