Skip to main content
SpringerLink
Log in
Book cover

International Computing and Combinatorics Conference

COCOON 2010: Computing and Combinatorics pp 338–347Cite as

The Longest Almost-Increasing Subsequence

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6196))

Abstract

Given a sequence of n elements, we introduce the notion of an almost-increasing subsequence in two contexts. The first notion is the longest subsequence that can be converted to an increasing subsequence by possibly adding a value, that is at most a fixed constant c, to each of the elements. We show how to optimally construct such subsequence in O(n logk) time, where k is the length of the output subsequence. As an exercise, we show how to produce in O(n 2 logk) time a special type of subsequences, that we call subsequences obeying the triangle inequality, by using as a subroutine our algorithm for the above case. The second notion is the longest subsequence where every element is at least the value of a monotonically non-decreasing function in terms of the r preceding elements (or even with respect to every r elements among those preceding it). We show how to construct such subsequence in O(n r logk) time.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adelson-Velskii, G., Landis, E.: On an information organization algorithm. Doklady Akademia Nauk SSSR 146, 263–266 (1962)

    MathSciNet  Google Scholar 

  2. Albert, M., Aldred, R., Atkinson, M., Ditmarsch, H., Handley, B., Handley, C., Opatrny, J.: Longest subsequences in permutations. Australian J. Combinatorics 28, 225–238 (2003)

    MATH  Google Scholar 

  3. Albert, M., Atkinson, M., Nussbaum, D., Sack, J., Santoro, N.: On the longest increasing subsequence of a circular list. Information Processing Letters 101, 55–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Albert, M., Golnski, A., Hamel, A., Lopez-Ortiz, A., Rao, S., Safari, M.: Longest increasing subsequences in sliding windows. Theoretical Computer Science 321, 405–414 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Apostolico, A., Atallah, M., Hambrusch, S.: New clique and independent set algorithms for circle graphs. Discrete Appl. Math. 36, 1–24 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baik, J., Deift, P., Johannsson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. American Math. Society 12, 1119–1178 (1999)

    Article  MATH  Google Scholar 

  7. Chang, M., Wang, F.: Efficient algorithms for the maximum weight clique and maximum weight independent set problems on permutation graphs. Information Processing Letters 76, 7–11 (2000)

    Article  MathSciNet  Google Scholar 

  8. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  9. Delcher, A., Kasif, S., Fleischmann, R., Paterson, J., White, O., Salzberg, S.: Alignment of whole genomes. Nucl. Acis. Res. 27, 2369–2376 (1999)

    Article  Google Scholar 

  10. Elmasry, A.: Deterministic jumplists. Nordic J. Comp. 12(1), 27–39 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Fredman, M.: On computing the length of longest increasing subsequence. Disc. Math. 11, 29–35 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hunt, J., Szymanski, T.: A fast algorithm for computing longest common subsequences. ACM Comm. 20, 350–353 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  13. Knuth, D.: Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34, 709–727 (1970)

    MATH  MathSciNet  Google Scholar 

  14. Ramanan, P.: Tight Ω(n logn) lower bound for finding a longest increasing subsequence. International J. Comp. Math. 65(3-4), 161–164 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Robinson, G.: On representations of the symmetric group. American J. Math. 60, 745–760 (1938)

    Article  Google Scholar 

  16. Schensted, C.: Longest increasing and decreasing subsequences. Canadian J. Math. 13, 179–191 (1961)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck Institut für Informatik, Saarbrücken, Germany

    Amr Elmasry

Authors
  1. Amr Elmasry
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Florida, CSE Building, Room 566, P.O. Box 116120, 32611-6120, Gainesville, Florida, USA

    My T. Thai

  2. Department of Computer and Information Science and Technology, University of Florida, P.O. Box, USA

    Sartaj Sahni

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Elmasry, A. (2010). The Longest Almost-Increasing Subsequence. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14031-0_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14030-3

  • Online ISBN: 978-3-642-14031-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation