Abstract
Given an array of n elements from a total order, we propose encodings that support various range queries (range minimum, range maximum and their variants), and previous and next smaller/larger value queries. When query time is not of concern, we obtain a \(4.088n + o(n)\)-bit encoding that supports all these queries. For the case when we need to support all these queries in constant time, we give an encoding that takes \(4.585n + o(n)\) bits, where n is the length of input array. We first extend the original DFUDS [Algorithmica, 2005] encoding of the colored 2d-Min (Max) heap that supports the queries in constant time. Then, we combine the extended DFUDS of 2d-Min heap and 2d-Max heap using the Min-Max encoding of Gawrychowski and Nicholson [arXiv, 2014] with some modifications. We also obtain encodings that take lesser space and support a subset of these queries.
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References
Arroyuelo, D., Cánovas, R., Navarro, G., Sadakane, K.: Succinct trees in practice. In: ALENEX 2010, Austin, Texas, USA, January 16, 2010, pp. 84–97 (2010)
Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Proceedings of the LATIN 2000, pp. 88–94 (2000)
Benoit, D., Demaine, E.D., Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Representing trees of higher degree. Algorithmica 43(4), 275–292 (2005)
Farzan, A., Munro, J.I.: A uniform paradigm to succinctly encode various families of trees. Algorithmica 68(1), 16–40 (2014)
Fischer, J.: Combined data structure for previous- and next-smaller-values. Theor. Comput. Sci. 412(22), 2451–2456 (2011)
Fischer, J., Heun, V.: Finding range minima in the middle: Approximations and applications. Mathematics in Computer Science 3(1), 17–30 (2010)
Fischer, J., Heun, V.: Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM Journal on Computing 40(2), 465–492 (2011)
Gawrychowski, P., Nicholson, P.K.: Optimal encodings for range min-max and top-k (2014). CoRR abs/1411.6581, http://arxiv.org/abs/1411.6581
Golin, M., Iacono, J., Krizanc, D., Raman, R., Rao, S.S.: Encoding 2D range maximum queries. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 180–189. Springer, Heidelberg (2011)
Jansson, J., Sadakane, K., Sung, W.K.: Ultra-succinct representation of ordered trees with applications. J. Comput. Syst. Sci. 78(2), 619–631 (2012)
Miltersen, P.B.: Cell probe complexity - a survey. In: FSTTCS (1999)
Munro, J.I., Raman, V., Rao, S.S.: Space efficient suffix trees. J. Algorithms 39(2), 205–222 (2001)
Munro, J.I., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM Journal on Computing 31(3), 762–776 (2001)
Raman, R., Raman, V., Satti, S.R.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Transactions on Algorithms 3(4) (2007). Article 43
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Jo, S., Satti, S.R. (2015). Simultaneous Encodings for Range and Next/Previous Larger/Smaller Value Queries. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_51
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DOI: https://doi.org/10.1007/978-3-319-21398-9_51
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