Skip to main content
SpringerLink
Log in

The Incompressibility Method

  • Conference paper
  • First Online:
SOFSEM 2000: Theory and Practice of Informatics (SOFSEM 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1963))

Abstract

Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A. N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. This is impossible for classical probability theory. Kolmogorov complexity is known variously as ‘algorithmic information’, ‘algorithmic entropy’, ‘Kolmogorov-Chaitin complexity’, ‘descriptional complexity’, ‘shortest program length’, ‘algorithmic randomness’, and others. Using it, we developed a new mathematical proof technique, now known as the ‘incompressibility method’. The incompressibility method is a basic general technique such as the ‘pigeon hole’ argument, ‘the counting method’ or the ‘probabilistic method’. The new method has been quite successful and we present recent examples. The first example concerns a “static” problem in combinatorial geometry. From among (n 3) triangles with vertices chosen from among n points in the unit square, U, let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c/n 3 < µn < C/n 3 for all large enough n, where µn is the expectation of A. Moreover, c/n 3 <A < C/n 3 for almost all A, that is, almost all A are close to the expectation value so that we determine the area of the smallest triangle for an arrangement in “general position”. Our second example concerns a “dynamic” problem in average-case running time of algorithms. The question of a nontrivial general lower bound (or upper bound) on the average-case complexity of Shellsort has been open for about forty years. We obtain the first such lower bound.

The first and second authors were supported in part by NSERC and CITO grants, and UCR startup grants, the third author was supported in part by the European Union via the NeuroCOLT II Working Group and the QAIP Project. The second author is on leave from the University of Waterloo.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Barequet, A lower bound for Heilbronn’s triangle problem in d dimensions. In: Proc. 10th ACM-SIAM Symp. Discrete Algorithms, 1999, 76–81. 40, 41

    Google Scholar 

  2. A. Berthiaume, W. van Dam, S. Laplante, Quantum Kolmogorov complexity, Proc. 15th IEEE Computational Complexity Conference, 2000, 240–249. 37

    Google Scholar 

  3. J. Beck, Almost collinear triples among N points on the plane, in A Tribute to Paul Erdős, ed. A. Baker, B. Bollobas and A. Hajnal, Cambridge Univ. Press, 1990, pp. 39–57. 40

    Google Scholar 

  4. C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, Proc. 3rd Ann. Conf. Comput. and Combinatorics, T. Jiang and D. T. Lee (Eds), 1997, pp. 23–31. 40

    Google Scholar 

  5. H. Buhrman, T. Jiang, M. Li, and P. Vitányi, New applications of the incompressibility method, pp. 220–229 in the Proceedings of ICALP’99, LNCS 1644, Springer-Verlag, Berlin, 1999.

    Google Scholar 

  6. G. Cairns, M. McIntyre, and J. Strantzen, Geometric proofs of some recent results of Yang Lu, Math. Magazine, 66(1993), 263–265. 40

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Erdős, Problems and results in combinatorial geometry, In: Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, 440(1985), 1–11. 40

    Google Scholar 

  8. P. Erdős and G. Purdy, Extremal problems in combinatorial theory, In: Handbook of Combinatorics, R. L. Graham, M. Grötschel, L. Lovász, Eds., Elsevier/MIT Press, 1995, pp. 861–862. 40

    Google Scholar 

  9. M. Goldberg, Maximizing the smallest triangle made by N points in a square, Math. Magazine, 45(1972), 135–144. 40

    Article  MATH  Google Scholar 

  10. R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag 1994, pp. 242–244. 40

    Google Scholar 

  11. J. Incerpi and R. Sedgewick, Improved upper bounds on Shellsort, Journal of Computer and System Sciences, 31(1985), 210–224. 47

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Janson and D. E. Knuth, Shellsort with three increments, Random Struct. Alg., 10(1997), 125–142. 47, 51

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Jiang, M. Li, and P. Vitányi, New applications of the incompressibility method II, Theoretical Computer Science, 235:1(2000), 59–70.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Jiang, M. Li, and P. Vitányi, The average-case complexity of Shellsort, Preliminary version, pp. 453–462 in the Proceedings of ICALP’99, LNCS 1644, Springer-Verlag, Berlin, 1999. Also: J. Assoc. Comput. Mach., to appear. 37, 51

    Google Scholar 

  15. T. Jiang, M. Li, and P. M. B. Vitányi, The Expected Size of Heilbronn’s Triangles, Proc. 14th IEEE Computational Complexity Conference, 1999, 105–113. 37

    Google Scholar 

  16. D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching, Addison-Wesley, 1973(1st Edition), 1998 (2nd Edition). 47, 51

    Google Scholar 

  17. A. N. Kolmogorov, Three approaches to the quantitative definition of information. Problems Inform. Transmission, 1(1):1–7, 1965. 38

    MathSciNet  Google Scholar 

  18. J. Komlós, J. Pintz, and E. Szemerédi, On Heilbronn’s triangle problem, J. London Math. Soc., (2) 24(1981), 385–396. 40, 46

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Komlós, J. Pintz, and E. Szemerédi, A lower bound for Heilbronn’s problem, J. London Math. Soc., 25(1982), 13–24. 40

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Li and P. M. B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 2nd Edition, 1997. 37, 39, 41, 49

    MATH  Google Scholar 

  21. M. Nielsen, I. Huang, Quantum Computation and Quantum Information, Cambridge University Press, 2000. 37

    Google Scholar 

  22. D. Mackenzie, On a roll, New Scientist, November 6, 1999, 44–48. 37, 41

    Google Scholar 

  23. W. Blum, Geometrisch Eingekreist, Die Zeit, April 13, 2000 (#16), p. 40. 37

    Google Scholar 

  24. D. Mackenzie, Le hasard ne joue pas aux de’s, Courrier International, December 23, 1999–January 5, 2000 (#41), p. 477–478. 37

    Google Scholar 

  25. A. M. Odlyzko, J. Pintz, and K. B. Stolarsky, Partitions of planar sets into small triangles, Discrete Math., 57(1985), 89–97. 40

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Papernov and G. Stasevich, A method for information sorting in computer memories, Problems Inform. Transmission, 1:3(1965), 63–75. 47

    Google Scholar 

  27. C. G. Plaxton, B. Poonen and T. Suel, Improved lower bounds for Shellsort, Proc. 33rd IEEE Symp. Foundat. Comput. Sci., pp. 226–235, 1992. 47, 51

    Google Scholar 

  28. V. R. Pratt, Shellsort and Sorting Networks, Ph.D. Thesis, Stanford Univ., 1972. 47

    Google Scholar 

  29. K. F. Roth, On a problem of Heilbronn, J. London Math Society, 26(1951), 198–204. 40, 41

    Article  MATH  Google Scholar 

  30. K. F. Roth, On a problem of Heilbronn II, Proc. London Math Society, (3) 25(1972), 193–212. 40

    Article  MATH  Google Scholar 

  31. K. F. Roth, On a problem of Heilbronn III, Proc. London Math Society, (3) 25(1972), 543–549. 40

    Article  MATH  Google Scholar 

  32. K. F. Roth, Estimation of the area of the smallest triangle obtained by selecting three out of n points in a disc of unit area, Proc. Symp. Pure Mathematics 24, AMS, Providence, 1973, pp. 251–262. 40

    Google Scholar 

  33. K. F. Roth, Developments in Heilbronn’s triangle problem, Advances in Math. 22(1976), 364–385. 40

    Article  MATH  MathSciNet  Google Scholar 

  34. W. M. Schmidt, On a problem of Heilbronn, J. London Math. Soc., (2) 4(1972), 545–550. 40

    Article  MATH  Google Scholar 

  35. R. Sedgewick, Analysis of Shellsort and related algorithms, Proc. 4th Annual European Symposium on Algorithms, Lecture Notes in Computer Science, Vol. 1136, Springer-Verlag, Berlin, 1–11. 47

    Google Scholar 

  36. R. Sedgewick, Open problems in the analysis of sorting and searching algorithms, Presented at Workshop on Prob. Analysis of Algorithms, Princeton, 1997 (http://www.cs.princeton/). 47, 51

  37. D. L. Shell, A high-speed sorting procedure, Commun. ACM, 2:7(1959), 30–32. 47

    Article  Google Scholar 

  38. Tian Zheng Ping, On the problem of Heilbronn type, Northeast. Math. J., 10(1994), 215–216. 40

    MathSciNet  Google Scholar 

  39. L. Yang, J. Z. Zhang, and Z. B. Zeng, Heilbronn problem for five points, Int’l Centre Theoret. Physics preprint IC/91/252 (1991). 40

    Google Scholar 

  40. L. Yang, J. Z. Zhang, and Z. B. Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann. Math. Ser. A 13(1992) 503–515. 40

    MATH  MathSciNet  Google Scholar 

  41. L. Yang, J. Z. Zhang, and Z. B. Zeng, On the Heilbronn numbers of triangular regions, Acta Math. Sinica, 37(1994), 678–689. 40

    MATH  MathSciNet  Google Scholar 

  42. P. Vitányi, Three approaches to the quantitative definition of information in an individual pure quantum state, Proc. 15th IEEE Computational Complexity Conference, 2000, 263–270. 37

    Google Scholar 

  43. A. C. C. Yao, An analysis of (h, k, 1)-Shellsort, J. of Algorithms, 1(1980), 14–50. 47

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of California, 92521, Riverside, CA, USA

    Tao Jiang

  2. Computer Science Department, University of California, 93106, Santa Barbara, CA, USA

    Ming Li

  3. CWI and University of Amsterdam, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Paul Vitányi

Authors
  1. Tao Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul Vitányi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Cybernetics, Czech Technical University, Karlovo nám. 13, 121 35, Prague, Czech Republic

    Václav Hlaváč

  2. Information Technology Department, CLRC RAL, Chilton, Didcot, Oxfordshire, UK

    Keith G. Jeffery

  3. Insitute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, 182 07, Prague, Czech Republic

    Jiří Wiedermann

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jiang, T., Li, M., Vitányi, P. (2000). The Incompressibility Method. In: Hlaváč, V., Jeffery, K.G., Wiedermann, J. (eds) SOFSEM 2000: Theory and Practice of Informatics. SOFSEM 2000. Lecture Notes in Computer Science, vol 1963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44411-4_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-44411-4_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41348-6

  • Online ISBN: 978-3-540-44411-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation