We consider the problem of performing predecessor searches in a bounded universe while achieving query times that depend on the distribution of queries. We obtain several data structures with various properties: in particular, we give data structures that achieve expected query times logarithmic in the entropy of the distribution of queries but with space bounded in terms of universe size, as well as data structures that use only linear space but with query times that are higher (but still sublinear) functions of the entropy. For these structures, the distribution is assumed to be known. We also consider individual query times on universe elements with general weights, as well as the case when the distribution is not known in advance.
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In this paper, we define \(\log x = \log _2 (x + 2)\).
As will be apparent from our results, in bounded universes this lower bound does not hold, and one can achieve query times below it.
Andersson, A., Thorup, M.: Dynamic ordered sets with exponential search trees. J. ACM, 54(3):Article 13 (2007)
Bădoiu, M., Cole, R., Demaine, E.D., Iacono, J.: A unified access bound on comparison-based dynamic dictionaries. Theor. Comput. Sci. 382(2), 86–96 (2007)
Beame, P., Fich, F.E.: Optimal bounds for the predecessor problem and related problems. J. Comput. Syst. Sci. 65(1), 38–72 (2002)
Belazzougui, D., Kaporis, A.C., Spirakis,P.G.: Random input helps searching predecessors. arXiv:1104.4353 (2011)
Bent, S.W., Sleator, D.D., Tarjan, R.E.: Biased search trees. SIAM J. Comput. 14(3), 545–568 (1985)
Bose, P., Howat, J., Morin, A.: Distribution-sensitive dictionary with low space overhead. In Proceedings of the 11th International Symposium on Algorithms and Data Structures (WADS 2009), LNCS 5664, pp. 110–118 (2009)
Bose, P., Douïeb, K., Dujmović, V., Howat, J., Morin, P.: Fast local searches and updates in bounded universes. In Proceedings of the 22nd Canadian Conference on Computational Geometry (CCCG 2010), pp. 261–264 (2010)
Brodal, G.S., Markis, C., Sioutas, S., Tsakalidis, A., Tsichlas, K.: Optimal solutions for the temporal precedence problem. Algorithmica 33(4), 494–510 (2002)
Johnson, D.B.: A priority queue in which initialization and queue operations take \(O(\log \log D)\) time. Theory Comput. Syst. 15(1), 295–309 (1981)
Kaporis, A., Makris, C., Sioutas, S., Tsakalidis, A., Tsichlas, K., Zaroliagis, C.: Improved bounds for finger search on a RAM. In ESA ’03: Proceedings of the 11th Annual European Symposium on Algorithms, LNCS 2832, pp. 325–336 (2003)
Knuth, D.E.: Optimum binary search trees. Acta Inf. 1(1), 14–25 (1971)
Mehlhorn, K.: Nearly optimal binary search trees. Acta Inf. 5(4), 287–295 (1975)
Mehlhorn, K., Näher, S.: Bounded ordered dictionaries in \(O(\log \log N)\) time and \(O(n)\) space. Inf. Process. Lett. 35(4), 183–189 (1990)
Pătraşcu, M., Thorup, M.: Time–space trade-offs for predecessor search. In STOC ’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 232–240 (2006)
Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)
van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett. 6(3), 80–82 (1977)
Willard, D.E.: Log-logarithmic worst-case range queries are possible in space \(\varTheta (\log \log N)\). Inf. Process. Lett. 17(2), 81–84 (1983)
Research partially supported by the Danish Council for Independent Research, Natural Sciences.
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Bose, P., Fagerberg, R., Howat, J. et al. Biased Predecessor Search. Algorithmica 76, 1097–1105 (2016). https://doi.org/10.1007/s00453-016-0146-7
- Data structures
- Predecessor search
- Biased search trees