Abstract
In this paper, we introduce an implicit data structure which represents a forest-structured partial order to efficiently perform, with respect to time and space, the following operations: 1) testing the relation among two elements (checking whether the two elements are related) and 2) given an elementu, searching for its father. The first operation can be performed in constant time, while the second one requires polylog time (logarithmic in the case of bounded degree). The data structure represents the order relation by referring only to internal nodes of the forest, thus achieving in many cases a significant saving in space occupation. Finally, the algorithm is shown to be optimal in a restricted computation model by deriving a lower bound on the space complexity within such a model.
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H. Alt, K. Mehlhorn and J. I. Munro,Partial match retrieval in implicit data structures, Inf. Proc. Lett., 19, 1984, pp. 61–66.
M. D. Atkinson, J. R. Sack, N. Santoro and T. Strothotte,Min-max heaps and generalized priority queues, Comm. A.C.M., 29, 1986, pp. 996–1000.
A. Borodin, F. E. Fich, F. Meyer auf der Heide, E. Upfal and A. Wigderson,A tradeoff between search and update time for the implicit dictionary problem, Theoret. Comp. Sci., 58, 1988, pp. 57–68.
B. Dushnik and E. Miller,Partially ordered sets, Amer. J. Math., 63, 1941, pp. 600–610.
G. N. Frederickson,Implicit data structures for the dictionary problem, J. Ass. Comp. Mach., 30, 1983, pp. 80–94.
G. Gambosi, E. Nardelli and M. Talamo,A pointer-free data structure for merging heaps and min-max heaps, Theoret. Comp. Sci., 84, 1991, pp. 107–126.
G. Gambosi, J. Nešetril and M. Talamo,Posets, boolean representation and quick path searching, Proc. 14th ICALP Conf., Lect. Not. in Comp. Sci., 267, Springer Verlag, 1987, pp. 404–424.
G. Gambosi, J. Nešetril and M. Talamo,On locally presented posets, Theoret. Comp. Sci., 70, 1990, pp. 251–260.
G. Jacobson,Space-efficient static trees and graphs, Proc. 30th Symp. on Found. of Comp. Sci., 1989, pp. 549–554.
S. Kannan, M. Naor and S. Rudich,Implicit representation of graphs, Proc. 20th Symp. on Theory of Computing, 1988, pp. 334–343.
F. Meyer auf der Heide,On genuinely time bounded computations, Proc. 6th STACS, Lect. Not. in Comp. Sci., 349, Springer-Verlag, 1989, pp. 1–16.
J. I. Munro,An implicit data structure supporting insertion, deletion and search in O (log2 n)time, J. Comp. Syst. Sci., 33, 1986, pp. 66–74.
J. I. Munro and H. Suwanda,Implicit data structures for fast search and update, J. Comp. Syst. Sci., 21, 1980, pp. 236–250.
J. R. Sack and T. Strothotte,An algorithm for merging heaps, Acta Inf., 22, 1985, pp. 171–186.
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Work partially performed in the framework of Esprit BRA project 3075 “Algorithms and Complexity” and of Italian MURST 40% project “Algoritmi e Strutture di Calcolo” and partially supported by the National Research Council (CNR) Project “Sistemi Informatici e Calcolo Parallelo”.
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Gambosi, G., Protasi, M. & Talamo, M. An efficient implicit data structure for relation testing and searching in partially ordered sets. BIT 33, 29–45 (1993). https://doi.org/10.1007/BF01990341
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DOI: https://doi.org/10.1007/BF01990341