Skip to main content
SpringerLink
Log in

O(log log n)‐competitive Binary Search Tree

2004; Demaine, Harmon, Iacono, Patrascu

  • Reference work entry
Encyclopedia of Algorithms

  • 239 Accesses

Keywords and Synonyms

Tango        

Problem Definition

Here is a precise definition of BST algorithms and their costs. This model is implied by most BST papers, and developed in detail by Wilber [22]. A static set of n keys is stored in the nodes of a binary tree. The keys are from a totally ordered universe, and they are stored in symmetric order. Each node has a pointer to its left child, to its right child, and to its parent. Also, each node may keep \( { o(\log n) } \) bits of additional information but no additional pointers.

A BST algorithm is required to process a sequence of m accesses (without insertions or deletions), \( S = s_1, s_2, s_3, s_4 \dots s_m \). The ith access starts from the root and follows pointers until s i is reached. The algorithm can update the fields in any node or rotate any edges that it touches along the way. The cost of the algorithm to execute an access sequence is defined to be the number of nodes touched plus the number of rotations.

Let Abe any BST...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 399.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Recommended Reading

  1. Blum, A., Chawla, S., Kalai, A.: Static optimality and dynamic search-optimality in lists and trees. Algorithmica 36, 249–260 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cole, R.: On the dynamic finger conjecture for splay trees II: The proof. SIAM J. Comput. 30(1), 44–85 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cole, R., Mishra, B., Schmidt, J., Siegel, A.: On the dynamic finger conjecture for splay trees I: Splay sorting \( { \log n } \)-block sequences. SIAM J. Comput. 30(1), 1–43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crane, C.A.: Linear lists and priority queues as balanced binary trees. Technical Report STAN-CS-72-259, Computer Science Dept., Stanford University (1972)

    Google Scholar 

  5. Culik II, K., Wood, D.: A note on some tree similarity measures. Inf. Process. Lett. 15(1), 39–42 (1982)

    Article  MATH  Google Scholar 

  6. Demaine, E.D., Harmon, D., Iacono, J., Patrascu, M.: Dynamic optimality —almost. SIAM J. Comput. 37(1), 240–251 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Derryberry, J., Sleator, D.D., Wang, C.C.: A lower bound framework for binary search trees with rotations. Technical Report CMU-CS-05-187, Carnegie Mellon University (2005)

    Google Scholar 

  8. Elmasry, A.: On the sequential access theorem and deque conjecture for splay trees. Theor. Comput. Sci. 314(3), 459–466 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Georgakopoulos, G.F.: How to splay for \( { \log\log n } \)‐competitiveness. In: Proc. 4th Int'l Workshop on Experimental and Efficient Algorithms (WEA), pp. 570–579 (2005)

    Google Scholar 

  10. Iacono, J.: Key-independent optimality. Algorithmica 42(1), 3–10 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Knuth, D.E.: Optimum binary search trees. Acta Informatica 1, 14–25 (1971)

    Article  MATH  Google Scholar 

  12. Luccio, F., Pagli, L.: On the upper bound on the rotation distance of binary trees. Inf. Process. Lett. 31(2), 57–60 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mäkinen, E.: On the rotation distance of binary trees. Inf. Process. Lett. 26(5), 271–272 (1988)

    Article  Google Scholar 

  14. Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sleator, D.D., Tarjan, R.E., Thurston, W.P.: Rotation distance, triangulations, and hyperbolic geometry. In: Proceedings 18th ACM Symposium on Theory of Computing (STOC), Berkeley, 1986, pp. 122–135

    Google Scholar 

  16. Sundar, R.: Twists, turns, cascades, deque conjecture, and scanning theorem. In: Proceedings 30th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 555–559 (1989)

    Google Scholar 

  17. Sundar, R.: On the deque conjecture for the splay algorithm. Combinatorica 12(1), 95–124 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tarjan, R.: Sequential access in play trees takes linear time. Combinatorica 5(4), 367–378 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tarjan, R.E.: Data structures and network algorithms, CBMS-NSF Reg. Conf. Ser. Appl. Math., vol. 44. SIAM, Philadelphia, PA (1983)

    Book  Google Scholar 

  20. Wang, C.C.: Multi-splay trees. Ph.D. Thesis, Carnegie Mellon University (2006)

    Google Scholar 

  21. Wang, C.C., Derryberry, J., Sleator, D.D.: \( { {O}(\log\log n) } \)‐competitive dynamic binary search trees. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Miami, 2006, pp. 374–383

    Google Scholar 

  22. Wilber, R.: Lower bounds for accessing binary search trees with rotations. SIAM J. Comput. 18(1), 56–67 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA

    Chengwen Chris Wang

Authors
  1. Chengwen Chris Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Wang, C. (2008). O(log log n)‐competitive Binary Search Tree. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_263

Download citation

Publish with us

Policies and ethics

Search

Navigation