Skip to main content
SpringerLink
Log in

Optimizing two-sequence functionals in competitive analysis

  • Special Issue Paper
  • Published:
Computer Science - Research and Development

Abstract

The efficiency of an on-line motion planning strategy often is measured by a constant competitive factor C. Competitivity means that the cost of a C-competitive on-line strategy with incomplete information is only C times worse than the optimal offline solution under full information. If a strategy is represented by an infinite sequence X=f 1,f 2,… of steps or values, the problem of finding a strategy with minimal C often results in minimizing functionals F k in X. For example \(F_{k}(f_{1},f_{2},\ldots):=\frac{\sum_{i=1}^{k+1}f_{i}}{f_{k}}\) represents a functional for the 2-ray search problem. There are two main paradigms for finding an optimal sequence f 1,f 2,… that minimizes F k for all k. Namely, optimality of the exponential function and equality approach.

If the strategy has to be defined by more than one interacting sequence both approaches may fail. In this paper we show that for such more sophisticated situations a combination of the paradigms is a good choice. As an example we consider an extension of the 2-ray search problem that can be formalized by two sequences.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alpern S, Gal S (2003) The theory of search games and rendezvous. Kluwer Academic, Norwell

    MATH  Google Scholar 

  2. Baeza-Yates R, Culberson J, Rawlins G (1993) Searching in the plane. Inf Comput 106:234–252

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck A, Newman DJ (1970) Yet more on the linear search problem. Isr J Math 8:419–429

    Article  MathSciNet  MATH  Google Scholar 

  4. Beck A, Warren P (1973) The return of the linear search problem. Isr J Math 14(2):169–183

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellman R (1963) An optimal search problem. SIAM Rev 274(5)

  6. Borodin A, El-Yaniv R (1998) Online computation and competitive analysis. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. Fiat A, Woeginger G (eds) (1998) On-line algorithms: the state of the art. Lecture notes comput sci, vol 1442. Springer, Berlin

    Google Scholar 

  8. Gal S (1974) Minimax solutions for linear search problems. SIAM J Appl Math 27:17–30

    Article  MathSciNet  MATH  Google Scholar 

  9. Gal S (1980) Search games. Mathematics in science and engineering, vol 149. Academic Press, New York

    MATH  Google Scholar 

  10. Gal S, Chazan D (1976) On the optimality of the exponential functions for some minimax problems. SIAM J Appl Math 30:324–348

    Article  MathSciNet  MATH  Google Scholar 

  11. Graham RL, Knuth DE, Patashnik O (1994) Concrete mathematics, 2nd edn. Addison-Wesley, Reading

    MATH  Google Scholar 

  12. Hammar M, Nilsson BJ, Schuierer S (1999) Parallel searching on m rays. In: Proc 16th sympos theoret aspects comput sci. Lecture notes comput sci, vol 1563. Springer, Berlin, pp 132–142

    Google Scholar 

  13. Hipke C, Icking C, Klein R, Langetepe E (1999) How to find a point on a line within a fixed distance. Discrete Appl Math 93:67–73

    Article  MathSciNet  MATH  Google Scholar 

  14. Icking C, Kamphans T, Klein R, Langetepe E (2002) On the competitive complexity of navigation tasks. In: Bunke H, Christensen HI, Hager GD, Klein R (eds) Sensor based intelligent robots. Lecture notes comput sci, vol 2238. Springer, Berlin, pp 245–258

    Chapter  Google Scholar 

  15. Kamphans T, Langetepe E (2005) On optimizing multi-sequence functionals for competitive analysis. In: Abstracts 21st European workshop comput geom, pp 111–114

    Google Scholar 

  16. Kamphans T, Langetepe E (2005) Optimal competitive online ray search with an error-prone robot. In: Proc 4th internat workshop efficient experim algorithms. Lecture notes comput sci, vol 3503. Springer, Berlin, pp 593–596

    Chapter  Google Scholar 

  17. Kao M-Y, Reif JH, Tate SR (1993) Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. In: Proc 4th ACM-SIAM sympos discrete algorithms, pp 441–447

    Google Scholar 

  18. Langetepe E (2000) Design and analysis of strategies for autonomous systems in motion planning. PhD thesis, Department of Computer Science, FernUniversität Hagen

  19. Langetepe E (2010) On the optimality of spiral search. In: SODA 2010: Proc 21st annu ACM-SIAM symp disc algor, pp 1–12

    Google Scholar 

  20. López-Ortiz A (1996) On-line target searching in bounded and unbounded domains. PhD thesis, Univ Waterloo, Waterloo, Canada

  21. López-Ortiz A, Schuierer S (2001) The ultimate strategy to search on m rays? Theor Comput Sci 261(2):267–295

    Article  MATH  Google Scholar 

  22. López-Ortiz A, Schuierer S (2004) Online parallel heuristics, processor scheduling, and robot searching under the competitive framework. Theor Comput Sci 310:527–537

    Article  MATH  Google Scholar 

  23. Rao NSV, Kareti S, Shi W, Iyengar SS (1993) Robot navigation in unknown terrains: introductory survey of non-heuristic algorithms. Technical Report ORNL/TM-12410, Oak Ridge National Laboratory

  24. Schuierer S (1998) Searching on m bounded rays optimally. Technical Report 112, Institut für Informatik, Universität Freiburg, Germany

  25. Schuierer S (2001) Lower bounds in on-line geometric searching. Comput Geom Theory Appl 18:37–53

    Article  MathSciNet  MATH  Google Scholar 

  26. Schuierer S (2003) A lower bound for randomized searching on m rays. In: Klein R, Six HW, Wegner L (eds) Computer science in perspective: essays dedicated to Thomas Ottoman. Lecture notes comput sci, vol 2598. Springer, Berlin, pp 264–277

    Google Scholar 

  27. Sleator DD, Tarjan RE (1985) Amortized efficiency of list update and paging rules. Commun ACM 28:202–208

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science I, University of Bonn, Friedrich-Ebert-Allee 144, 53113, Bonn, Germany

    Elmar Langetepe

Authors
  1. Elmar Langetepe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elmar Langetepe.

Additional information

A preliminary version of the paper was presented at [15].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Langetepe, E. Optimizing two-sequence functionals in competitive analysis. Comput Sci Res Dev 27, 207–216 (2012). https://doi.org/10.1007/s00450-011-0151-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00450-011-0151-7

Keywords

Search

Navigation