Abstract
We develop two probabilistic methods that allow us to analyze the maximum data structure size encountered during a sequence of insertions and deletions in data structures such as priority queues, dictionaries, linear lists, and symbol tables, and in sweepline structures for geometry and VLSI applications. The notion of the “maximum” is basic to issues of resource preallocation. We apply our methods to combinatorial models of file histories and probabilistic models, as well as to a non-Markovian process (algorithm) for processing sweepline information in an efficient way, called “hashing with lazy deletion” (HwLD). We derive expressions for the expected maximum data structure size that are asymptotically exact, that is, correct up to lower-order terms; in several cases of interest the expected value of the maximum size is asymptotically equal to the maximum expected size. At a high level, our first method isolates the primary contribution to the maximum and bounds the lesser effects. In our second technique we relate the continuous-time probabilistic model to its discrete analog—the maximum slot occupancy in hashing.
This work was done while the author was at Ecole Normale Supérieure, LIENS, 45, rue d'Ulm, 75230 Paris Cedex 05, France.
Support was provided in part by an NSF research grant and by an NSF Presidential Young Investigator Award with matching funds from IBM.
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References
W. Feller. An Introduction to Probability Theory and Its Applications. Volume 1. Wiley, New York (third edition 1968).
G. Fayolle. Personal communication (1988).
P. Flajolet. “Analyse d'algorithmes de manipulation d'arbres et de fichiers,” Cahiers du bureau universitaire de recherche opérationnelle, 34–35 (1981), 1–209.
P. Flajolet, J. Françon, and J. Vuillemin. “Computing Integrated Costs of Operations with Applications to Dictionaries,” Proceedings of the 11th Annual ACM Symposium on Theory of Computing, Atlanta (April–May 1979), 49–61.
P. Flajolet, J. Françon, and J. Vuillemin. “Sequence of Operations Analysis for Dynamic Data Structures,” Journal of Algorithms, 1(2) (1980), 111–141. A shortened version appeared in Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, Puerto Rico (October 1979), 183–195.
S. Karlin and J. L. McGregor. “The Differential Equations of Birth-and-Death Processes, and the Stieltjes Moment Problem,” Trans. of the American Mathematical Society, 85 (1957), 489–546.
L. Kleinrock. Queueing Systems. Volume I: Theory. Wiley & Sons, New York (1975).
V. F. Kolchin, B. A. Sevast'yanov, and V. P. Chistyakov. Random Allocations. V. H. Winston & Sons, Washington (1978).
C. M. Mathieu and J. S. Vitter. “Maximum Queue Size and Hashing with Lazy Deletion,” Proceedings of the 20th Annual Symposium on the Interface of Computing Science and Statistics, Reston, VA (April 1988).
J. Morrison, L. A. Shepp, and C. J. Van Wyk. “A Queueing Analysis of Hashing with Lazy Deletion,” SIAM Journal on Computing 16, 6 (December 1987), 1155–1164.
T. Ottmann and D. Wood. “Space-Economical Plane-Sweep Algorithms,” Computer Vision, Graphics, and Image Processing, 34 (1986), 35–51.
T. G. Szymanski and C. J. Van Wyk. “Space-Efficient Algorithms for VLSI Artwork Analysis,” Proceedings of the 20th IEEE Design Automation Conference (1983), 743–749.
C. J. Van Wyk and J. S. Vitter. “The Complexity of Hashing with Lazy Deletion,” Algorithmica, 1(1) (1986), 17–29.
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© 1989 Springer-Verlag Berlin Heidelberg
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Kenyon-Mathieu, C.M., Vitter, J.S. (1989). General methods for the analysis of the maximum size of dynamic data structures. In: Ausiello, G., Dezani-Ciancaglini, M., Della Rocca, S.R. (eds) Automata, Languages and Programming. ICALP 1989. Lecture Notes in Computer Science, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035778
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DOI: https://doi.org/10.1007/BFb0035778
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