Abstract
We consider the problem of defining a linear order on a set of combinatorial objects so that the following two operations can be performed efficiently: (1) determine the rank of an object in the linear order (ranking); (2) compute an object from its rank (unranking). Typical applications of such an ordering include testing a program on a random or selected set of input instances and searching for counterexamples of a conjecture involving structured objects. We reduce the problem of finding such a linear order to the problem of constructing a special mapping on the set of combinatorial objects called regular reduction. We demonstrate the power of regular reductions by developing O(n) time procedures for ranking and unranking B-trees on n leaves after O(n 2) time preprocessing; the best previous ordering algorithm, although also running in linear time, required exponential time and space preprocessing. Our new paradigm also yields improved ranking and unranking algorithms for binary trees of bounded height and for height-balanced trees (see [7]).
Work funded in part by the European Commission under a research fellowship of the Human Capital and Mobility (HCM) programme.
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References
A.V. Aho, J.E. Hopcroft, J.D. Ullman, The design and analysis of computer algorithms, Addison-Wesley, 1974.
G.M. Adel'son-Vel'skii and Y.M. Landis, An algorithm for the organization of information, Doklad. Akad. Nauk SSSR, 146 (1962), pp. 263–266; Soviet math. Dokl., 3 (1962), pp. 1259–1262.
C. Berge, Principles of combinatorics, Academic Press, New York, 1971.
R. Bayer and E. McCreight, Organization and maintenance of large ordered indexes, Acta Inform. 1 (1972), pp. 173–189.
U.I. Gupta, D.T. Lee and C.K. Wong, Ranking and unranking of B-trees, Journal of Algorithms 4(1983), pp. 51–60.
U.I. Gupta, D.T. Lee and C.K. Wong, Ranking and unranking of 2–3 trees, SIAM J. Comput. 11(1982), pp. 582–590.
P. Kelsen, Ranking and unranking trees using regular reductions, Technical Report 93-37, Department of Computer Science, University of British Columbia, 1993.
D.E. Knuth, The art of computer programming, vol. 3: Sorting and Searching, Addison-Wesley, Reading, MA, 1973.
L. Li, Ranking and unranking of AVL-trees, SIAM J. Comput. 15 (1986), pp. 1025–1035.
C.C Lee, D.T. Lee and C.K. Wong, Generating binary trees of bounded height, Acta Informatica 23, 529–544 (1986).
F. Ruskey and T.C. Hu, Generating binary trees lexicographically, SIAM J. Comput. 6 (1977), pp. 745–758.
F. Ruskey and D. Roelants van Baronaigien, Fast recursive algorithms for generating combinatorial objects, Congressus Numerantium, vol. 41 (1984), pp. 53–62.
H. Wilf, A unified setting for sequencing, ranking and selection algorithms for combinatorial objects, Advances in Mathematics, 24 (1977), pp.281–291.
S.G. Williamson, On the ordering, ranking and random generation of basic combinatorial sets, Lecture Notes in Mathematics, 579, Springer-Verlag, Berlin, 1976.
S. Zaks, Lexicographic generation of ordered trees, Theoretical Computer Science 10 (1980), pp, 63–82.
S. Zaks, Generating trees and other combinatorial objects lexicographically, SIAM J. Comput. 8 (1979), pp. 73–81.
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© 1996 Springer-Verlag Berlin Heidelberg
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Kelsen, P. (1996). Ranking and unranking trees using regular reductions. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_47
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DOI: https://doi.org/10.1007/3-540-60922-9_47
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