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Algorithmica

, Volume 31, Issue 4, pp 501–512 | Cite as

Relaxed Balance Using Standard Rotations

  • Larsen
  • Soisalon-Soininen
  • Widmayer
Article

Abstract

Abstract. In search trees with relaxed balance, rebalancing transformations need not be connected with updates, but may be delayed. For standard AVL tree rebalancing, we prove that even though the rebalancing operations are uncoupled from updates, their total number is bounded by O(M log (M+N)) , where M is the number of updates to an AVL tree of initial size N . Hence, relaxed balancing of AVL trees comes at no extra cost asymptotically. Furthermore, our scheme differs from most other relaxed balancing schemes in an important aspect: No rebalancing transformation can be done in the wrong direction, i.e., no performed rotation can make the tree less balanced. Moreover, each performed rotation indeed corresponds to a real imbalance situation in the tree. Finally, and perhaps most importantly, our structure is capable of forgetting registered imbalance if later updates happen to improve the situation. Our results are of theoretical interest and have possible sequential and parallel applications.

Key words. Data structures, Search trees, AVL trees, Relaxed balance, Amortized analysis. 

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Copyright information

© Springer-Verlag 2001

Authors and Affiliations

  • Larsen
    • 1
  • Soisalon-Soininen
    • 2
  • Widmayer
    • 3
  1. 1.Department of Mathematics and Computer Science, University of Southern Denmark, Main campus: Odense University, Campusvej 55, \mbox{DK-5230} Odense M, Denmark. kslarsen@imada.sdu.dk.DK
  2. 2.Department of Computer Science and Engineering, Helsinki University of Technology, P.O. Box 540, FIN-02015 HUT, Finland. ess@cs.hut.fi.FI
  3. 3.Institut für Theoretische Informatik, ETH Zentrum, CH-8092 Zürich, Switzerland. widmayer@inf.ethz.ch.CH

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