Abstract
A static dictionary is a data structure for storing a subset S of a finite universe U so that membership queries can be answered efficiently. We explore space efficient structures to also find the rank of an element if found. We first give a representation of a static dictionary that takes n lg m + O(lg lg m) bits of space and supports membership and rank (of an element present in S) queries in constant time, where n = |S| and m = |U|. Using our structure we also give a representation of a m-ary cardinal tree with n nodes using n⌈lgm⌉ + 2n + o(n) bits of space that supports the tree navigational operations in O(1) time, when m is o(2lg n/ lg lg n). For arbitrary m, we give a structure that takes the same space and supports all the navigational operations, except finding the child labeled i (for any i), in O(1) time. Finding the child labeled i in this structure takes O(lg lg lg m) time.
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References
M. Ajtai, “A lower bound for finding predecessors in Yao’s cell probe model”, Combinatorica 8 (1988) 235–247.
D. Benoit, “Compact Tree Representations”, Master’s Thesis, Department of Computer Science, University of Waterloo, Canada (1998).
D. Benoit, E. D. Demaine, J. I. Munro and V. Raman “Representing Trees of Higher Degree”, The Proceedings of the 6th International Workshop on Algorithms and Data Structures, Springer Verlag Lecture Notes in Computer Science 1663 (1999) 169–180.
A. Brodnik and J. I. Munro, “Membership in constant time and almost minimum space”, to appear in SIAM Journal on Computing.
D. R. Clark, “Compact Pat Trees”, Ph.D. Thesis, University of Waterloo, 1996.
D. R. Clark and J. I. Munro, “Efficient Suffix Trees on Secondary Storage”, Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms (1996) 383–391.
A. Fiat, M. Noar, J. P. Schmidt and A. Siegel, “Non-oblivious hashing”, Journal of the Association for Computing Machinery, 39(4) (1992) 764–782.
M. L. Fredman, J. Komlós and E. Szemerédi, “Storing a sparse table with O(1) access time”, Journal of the Association for Computing Machinery, 31 (1984) 538–544.
G. Jacobson, “Space-efficient Static Trees and Graphs”, Proceedings of the IEEE Symposium on Foundations of Computer Science (1989) 549–554.
J. I. Munro, “Tables”, Proceedings of the 16th FST & TCS conference, Springer Verlag Lecture Notes in Computer Science 1180 (1996) 37–42.
J. I. Munro and V. Raman, “Succinct representation of balanced parentheses, static trees and planar graphs”, Proceedings of the IEEE Symposium on Foundations of Computer Science (1997) 118–126.
Rasmus Pagh, “Low redundancy in dictionaries with O(1) worst case lookup time”, to appear in Proceedings of the International Colloquium on Automata, Languages and Programming (1999).
J. P. Schmidt and A. Siegel, “The spatial complexity of oblivious k-probe hash functions”, SIAM Journal on Computing 19(5) (1990) 775–786.
D. E. Willard, “Log-Logarithmic worst case range queries are possible in space _(n)”, Information Processing Letters 17 (1983) 81–89.
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© 1999 Springer-Verlag Berlin Heidelberg
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Raman, V., Rao, S.S. (1999). Static Dictionaries Supporting Rank. In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_3
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DOI: https://doi.org/10.1007/3-540-46632-0_3
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