Skip to main content
Springer Nature Link
Log in

Minimax Trees in Linear Time with Applications

  • Conference paper
Combinatorial Algorithms (IWOCA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5874))

Included in the following conference series:

Abstract

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves’ depths, it minimizes the maximum of any leaf’s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, \(\mathcal{O}{n \log n}\)-time algorithm for building them. Drmota and Szpankowski (2002) gave another \(\mathcal{O}{n \log n}\)-time algorithm, which takes linear time when the weights are already sorted by their fractional parts. In this paper we give the first linear-time algorithm for building minimax trees for unsorted real weights.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ahlswede, R., Wegener, I.: Search Problems. Wiley, Chichester (1987)

    MATH  Google Scholar 

  2. Aigner, M.: Combinatorial Search. Wiley, Chichester (1988)

    MATH  Google Scholar 

  3. Baer, M.B.: Tight bounds on minimum maximum pointwise redundancy. In: Proceedings of the International Symposium on Information Theory, pp. 1944–1948 (2008)

    Google Scholar 

  4. Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. Journal of Computer and System Sciences 7(4), 448–461 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  6. Coppersmith, D., Klawe, M.M., Pippenger, N.: Alphabetic minimax trees of degree at most t. SIAM Journal on Computing 15(1), 189–192 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill (2001)

    Google Scholar 

  8. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, Chichester (2006)

    MATH  Google Scholar 

  9. De Prisco, R., Persiano, G.: Characteristic inequalities for binary trees. Information Processing Letters 53(4), 201–207 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Drmota, M., Szpankowski, W.: Generalized Shannon code minimizes the maximal redundancy. In: Proceedings of the 5th Latin American Symposium on Theoretical Informatics, pp. 306–318 (2002)

    Google Scholar 

  11. Drmota, M., Szpankowski, W.: Precise minimax redundancy and regret. IEEE Transactions on Information Theory 50(11), 2686–2707 (2004)

    Article  MathSciNet  Google Scholar 

  12. Evans, W.S., Kirkpatrick, D.G.: Restructuring ordered binary trees. Journal of Algorithms 50(2), 168–193 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Faller, N.: An adaptive system for data compression. In: Record of the 7th Asilomar Conference on Circuits, Systems and Computers, pp. 593–597 (1973)

    Google Scholar 

  14. Franceschini, G., Muthukrishnan, S., Pǎtraşcu, M.: Radix sorting with no extra space. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 194–205. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  15. Gagie, T.: A new algorithm for building alphabetic minimax trees. Fundamenta Informaticae (to appear)

    Google Scholar 

  16. Gagie, T.: Dynamic Shannon coding. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 359–370. Springer, Heidelberg (2004)

    Google Scholar 

  17. Gagie, T.: Dynamic Shannon coding. Information Processing Letters 102(2-3), 113–117 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gallager, R.G.: Variations on a theme by Huffman. IEEE Transactions on Information Theory 24(6), 668–674 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Golin, M.J., Kenyon, C., Young, N.E.: Huffman coding with unequal letter costs. In: Proceedings of the 34th Symposium on Theory of Computing, pp. 785–791 (2002)

    Google Scholar 

  20. Golin, M.J., Li, J.: More efficient algorithms and analyses for unequal letter cost prefix-free coding. IEEE Transactions on Information Theory 52(8), 3412–3424 (2008)

    Article  MathSciNet  Google Scholar 

  21. Golumbic, M.C.: Combinatorial merging. IEEE Transactions on Computers 25(11), 1164–1167 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hoover, H.J., Klawe, M.M., Pippenger, N.: Bounding fan-out in logical networks. Journal of the ACM 31(1), 13–18 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hu, T.C., Kleitman, D.J., Tamaki, J.: Binary trees optimum under various criteria. SIAM Journal on Applied Mathematics 37(2), 246–256 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  24. Huffman, D.A.: A method for the construction of minimum-redundancy codes. Proceedings of the IRE 40, 1089–1101 (1952)

    Article  Google Scholar 

  25. Karpinski, M., Nekrich, Y.: A fast algorithm for adaptive prefix coding. Algorithmica 55(1), 29–41 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Katona, G.O.H., Nemetz, T.O.H.: Huffman codes and self-information. IEEE Transactions on Information Theory 22(3), 337–340 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kirkpatrick, D.G., Klawe, M.M.: Alphabetic minimax trees. SIAM Journal on Computing 14(3), 514–526 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kirkpatrick, D.G., Przytycka, T.M.: An optimal parallel minimax tree algorithm. In: Proceedings of the 2nd Symposium on Parallel and Distributed Processing, pp. 293–300 (1990)

    Google Scholar 

  29. Klawe, M.M., Mumey, B.: Upper and lower bounds on constructing alphabetic binary trees. SIAM Journal on Discrete Mathematics 8(4), 638–651 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Knuth, D.E.: Dynamic Huffman coding. Journal of Algorithms 6(2), 163–180 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kraft, L.G.: A device for quantizing, grouping, and coding amplitude-modulated pulses. MSc thesis, Massachusetts Institute of Technology (1949)

    Google Scholar 

  32. Krause, R.M.: Channels which transmit letters of unequal duration. Information and Control 5(1), 13–24 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nakatsu, N.: Bounds on the redundancy of binary alphabetical codes. IEEE Transactions on Information Theory 37(4), 1225–1229 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  34. Parker Jr., D.S.: Combinatorial merging and Huffman’s algorithm. IEEE Transactions on Computers 28(5), 365–367 (1979)

    Article  MATH  Google Scholar 

  35. Rezaei, F., Charalambous, C.D.: Robust coding for uncertain sources: a minimax approach. In: Proceedings of the International Symposium on Information Theory, pp. 1539–1543 (2005)

    Google Scholar 

  36. Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 379–423, 623–645 (1948)

    Google Scholar 

  37. van Leeuwen, J.: On the construction of Huffman trees. In: Proceedings of the 3rd International Colloquium on Automata, Languages and Programming, pp. 382–410 (1976)

    Google Scholar 

  38. Yeung, R.W.: Alphabetic codes revisited. IEEE Transactions on Information Theory 37(3), 564–572 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Wroclaw, Poland

    Paweł Gawrychowski

  2. Research Group in Genome Informatics, Bielefeld University, Germany

    Travis Gagie

Authors
  1. Paweł Gawrychowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Travis Gagie
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala  & Jan Kratochvíl  & 

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, NSW 2308, Callaghan, Australia

    Mirka Miller

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gawrychowski, P., Gagie, T. (2009). Minimax Trees in Linear Time with Applications. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10217-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10216-5

  • Online ISBN: 978-3-642-10217-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics