Fringe analysis of synchronized parallel algorithms on 2–3 trees

  • R. Baeza-Yates
  • J. Gabarró
  • X. Messeguer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1518)

Abstract

We are interested in the fringe analysis of synchronized parallel insertion algorithms on 2–3 trees, namely the algorithm of W. Paul, U. Vishkin and H. Wagener (PVW). This algorithm inserts keys into a tree of size n with parallel time O(log n + log k).

Fringe analysis studies the distribution of the bottom subtrees and it is still an open problem for parallel algorithms on search trees. To tackle this problem we introduce a new kind of algorithms whose two extreme cases seems to upper and lower bounds the performance of the PVW algorithm.

We extend the fringe analysis to parallel algorithms and we get a rich mathematical structure giving new interpretations even in the sequential case. The process of insertions is modeled by a Markov chain and the coefficients of the transition matrix are related with the expected local behavior of our algorithm. Finally, we show that this matrix has a power expansion over (n+1)-1 where the coefficients are the binomial transform of the expected local behavior. This expansion shows that the parallel case can be approximated by iterating the sequential case.

Keywords

Fringe analysis Parallel algorithms 2–3 trees Binomial transform 

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References

  1. [BY95]
    R.A. Baeza-Yates. Fringe analysis revisited. ACM Computing Surveys, 27(1):109–119, 1995.CrossRefGoogle Scholar
  2. [EZG+82]_B. Eisenbarth, N. Ziviani, G.H. Gonnet, K. Mehlhorn, and D. Wood. The theory of fringe analysis and its application to 2–3 trees and B-trees. Information and Control, 55(1-3):125–174, 1982.MATHCrossRefMathSciNetGoogle Scholar
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    P.V. Poblete, J.I. Munro, and T. Papadakis. The binomial transform and its application to the analysis of skip lists. In ESA 95, pages 1–10. Springer-Verlag, 1995.Google Scholar
  4. [PVW83]
    W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2–3 trees. In J. Díaz, editor, Proc. 10th ICALP, LNCS 154, pages 597–609. Springer-Verlag, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • R. Baeza-Yates
    • 1
  • J. Gabarró
    • 2
  • X. Messeguer
    • 2
  1. 1.Departamento de Ciencias de la ComputacónUniversidad de ChileChile
  2. 2.Dep. LSIUniversitat Politècnica de CatalunyaBarcelonaSpain

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