Skip to main content
SpringerLink
Log in

Testing the necklace condition for Shortest Tours and optimal factors in the plane

  • Algorithms And Complexity
  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1987)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 267))

Included in the following conference series:

Abstract

A tour τ of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in τ. It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour.

In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O(n 2logn) time. If a tour is given, then we can test in O(n 2) time whether or not this tour is a necklace tour. Both algorithms can be generalized to m-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.

Research of the first author reported in this paper was supported by Amoco Found. Fac. Dev. Comput. Sci. 1-6-44862. Research of the second author was supported by the Austrian Science Foundation (Fonds zur Förderung der wissenschaftlichen Forschung), Project S32/01.

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.

7. References

  1. Balas, E. Integer and fractional matchings. Ann. Discrete Math. 11 (1981), 1–13.

    Google Scholar 

  2. Bateman, P. and Erdös, P. Geometrical extrema suggested by a lemma of Besicovitch. Amer. Math. Monthly 58 (1951), 306–314.

    Google Scholar 

  3. Derigs, U. and Metz, A. On the use of optimal fractional matchings for solving the (integer) matching problem. Computing 36 (1986), 263–270.

    Google Scholar 

  4. Edelsbrunner, H., Rote, G. and Welzl, E. Testing the necklace condition for shortest tours and optimal factors in the plane. Rep. 86-83, Dept. Mathematics, Technical University and University of Graz, Austria, 1986.

    Google Scholar 

  5. Edmonds, J. and Karp, R. M. Theoretical improvements in algorithmic efficiency for network flow problems. J. Assoc. Comput. Mach. 19 (1972), 248–264.

    Google Scholar 

  6. Fredman, M. L. and Tarjan, R. E. Fibonacci heaps and their uses in improved network optimization algorithms. In "Proc. 25th Ann. IEEE Sympos. Found. Comput. Sci. 1984", 338–346.

    Google Scholar 

  7. Fulkerson, D. R., Hoffman, A. J. and McAndrew, M. M. Some properties of graphs with multiple edges. Canad. J. Math. 17 (1965), 166–177.

    Google Scholar 

  8. Gabow, H. N. An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In "Proc. 15th Ann. ACM Symp. Theory Comput. Sci. 1983", 448–456.

    Google Scholar 

  9. Garey, M. R. and Johnson, D. S. Computers and Intractability — A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.

    Google Scholar 

  10. Lawler, E. L. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, New York, 1976.

    Google Scholar 

  11. Lipton, R. J., Rose, D. J. and Tarjan, R. E. Generalized nested dissection. SIAM J. Numer. Anal. 16 (1979), 346–358.

    Google Scholar 

  12. Orlin, J. B. Genuinely polynomial simplex and non-simplex algorithms for the minimum cost flow problem. To appear in Operations Research.

    Google Scholar 

  13. Papadimitriou, C. H. The Euclidean TSP is NP-complete. Theoret. Comput. Sci. 4 (1977), 237–244.

    Google Scholar 

  14. Reifenberg, E. R. A problem on circles. Math. Gaz. 32 (1948), 290–292.

    Google Scholar 

  15. Rockafellar, R. T. Network Flows and Monotropic Optimization. Wiley-Interscience, New York, 1984.

    Google Scholar 

  16. Sanders, D. On extreme circuits. Ph.D. thesis, City College of New York, 1968.

    Google Scholar 

  17. Supnick, F. A class of combinatorial extrema. Ann. New York Acad. Sci. 175 (1970), 370–382.

    Google Scholar 

  18. Tomizawa, N. On some techniques useful for solution of transportation network problems. Networks 1 (1972), 179–194.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, 1304 W. Springfield Ave., 61801, Urbana, IL, USA

    Herbert Edelsbrunner

  2. Institute for Mathematics, Technical University of Graz, Kopernikusgasse 24, A-8010, Graz, Austria

    Günter Rote

  3. Institute for Information Processing, Technical University of Graz, Schiesstattgasse 4a, A-8010, Graz, Austria

    Emo Welzl

Authors
  1. Herbert Edelsbrunner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Günter Rote
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emo Welzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Thomas Ottmann

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Edelsbrunner, H., Rote, G., Welzl, E. (1987). Testing the necklace condition for Shortest Tours and optimal factors in the plane. In: Ottmann, T. (eds) Automata, Languages and Programming. ICALP 1987. Lecture Notes in Computer Science, vol 267. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18088-5_31

Download citation

  • DOI: https://doi.org/10.1007/3-540-18088-5_31

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18088-3

  • Online ISBN: 978-3-540-47747-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation