Search Trees with Relaxed Balance and Near-Optimal Height

  • Rolf Fagerberg
  • Rune E. Jensen
  • Kim S. Larsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2125)


We introduce the relaxed k-tree, a search tree with relaxed balance and a height bound, when in balance, of (1 + ε)log2n + 1, for any g > 0. The rebalancing work is amortized O(1/ε) per update. This is the first binary search tree with relaxed balance having a height bound better than c · log2 n for a fixed constant c. In all previous proposals, the constant is at least 1/log2 φ τ; 1.44, where φ is the golden ratio.

As a consequence, we can also define a standard (non-relaxed) k-tree with amortized constant rebalancing per update, which is an improvement over the original definition.

Search engines based on main-memory databases with strongly fluctuating workloads are possible applications for this line of work.


Neighboring Node Search Tree Binary Search Tree Black Node External Node 
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  1. 1.
    G. M. Adel’son-Vel’skii and E. M. Landis. An Algorithm for the Organisation of Information. Doklady Akadamii Nauk SSSR, 146:263–266, 1962. In Russian. English translation in Soviet Math. Doklady, 3:1259–1263, 1962.MathSciNetGoogle Scholar
  2. 2.
    Joan Boyar, Rolf Fagerberg, and Kim S. Larsen. Amortization Results for Chromatic Search Trees, with an Application to Priority Queues. Journal of Computer and System Sciences, 55(3):504–521, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Joan F. Boyar and Kim S. Larsen. Efficient Rebalancing of Chromatic Search Trees. Journal of Computer and System Sciences, 49(3):667–682, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Leo J. Guibas and Robert Sedgewick. A Dichromatic Framework for Balanced Trees. In Proceedings of the 19th Annual IEEE Symposium on the Foundations of Computer Science, pages 8–21, 1978.Google Scholar
  5. 5.
    Kim S. Larsen. Amortized Constant Relaxed Rebalancing using Standard Rotations. Acta Informatica, 35(10):859–874, 1998.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Kim S. Larsen. AVL Trees with Relaxed Balance. Journal of Computer and System Sciences, 61(3):508–522, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. A. Maurer, Th. Ottmann, and H.-W. Six. Implementing Dictionaries using Binary Trees of Very Small Height. Information Processing Letters, 5(1):11–14, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Otto Nurmi and Eljas Soisalon-Soininen. Uncoupling Updating and Rebalancing in Chromatic Binary Search Trees. In Proceedings of the Tenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 192–198, 1991.Google Scholar
  9. 9.
    Otto Nurmi and Eljas Soisalon-Soininen. Chromatic Binary Search Trees—A Structure for Concurrent Rebalancing. Acta Informatica, 33(6):547–557, 1996.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Otto Nurmi, Eljas Soisalon-Soininen, and Derick Wood. Relaxed AVL Trees, Main-Memory Databases and Concurrency. International Journal of Computer Mathematics, 62:23–44, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Neil Sarnak and Robert E. Tarjan. Planar Point Location Using Persistent Search Trees. Communications of the ACM, 29:669–679, 1986.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Rolf Fagerberg
    • 1
  • Rune E. Jensen
    • 2
  • Kim S. Larsen
    • 3
  1. 1.BRICS, Department of Computer ScienceUniversity of AarhusÅrhus CDenmark
  2. 2.ALOC Bonnier A/SOdense CDenmark
  3. 3.Department of Mathematics and Computer ScienceUniversity of Southern Denmark Main Campus: Odense UniversityOdense MDenmark

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