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The K-Gabriel graphs and their applications

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Algorithms (SIGAL 1990)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 450))

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Abstract

In this paper, we define and investigate the properties of k-Gabriel graphs and also propose an algorithm to construct the k-Gabriel graph of a points set in O(k 2 nlogn) time. The k-Gabriel graphs are also used to improve the running time of solving the Euclidean bottleneck biconnected edge subgraph problem from O(n 2) to 0(nlogn), and that of solving the Euclidean bottleneck matching problem from O(n 2) to O(n 1.5 log 0.5 n).

Tung-Hsin Su is with the Institute of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China.

Ruei-Chuan Chang is with the Institute of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan and the Institute of Information Science, Academia Sinica, Taipei, Taiwan, Republic of China.

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References

  1. Aho. A. V., Hopcroft, J. E. and Ullman, J. D.: The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.

    Google Scholar 

  2. Chang, M. S., Tang, C. Y. and Lee, R. C. T.: Solving the Euclidean bottleneck matching problem by k-relative neighborhood graph, to appear in Algorithmica.

    Google Scholar 

  3. Chang, M. S., Tang, C. Y. and Lee, R. C. T.: Solving Euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs, The First Workshop on Proximity Graphs, New Mexico State University, 1989.

    Google Scholar 

  4. Chazelle, B. and Edelsbrunner, H.: An improved algorithm for constructing kth-order Voronoi diagram, IEEE Transactions on Computers, Vol. C-36, No. 11, 1987, pp. 1349–1354.

    Google Scholar 

  5. Edelsbrunner, H.: Edge-skeleton in arrangements with applications, Algorithmica, Vol. 1, No. 1, 1986, pp. 93–109.

    Google Scholar 

  6. Gabow, H. N.: A scaling algorithm for weighted matching on general graphs, Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp. 90–100.

    Google Scholar 

  7. Gabow, H. N. and Tarjan, R. E.: Algorithms for two bottleneck optimization problems, Journal of Algorithms, Vol. 9, No. 3, 1988, pp. 411–417.

    Google Scholar 

  8. Gabriel, K. R. and Sokal, R. R.: A new statistical approach to geographical variation analysis, Systematic Zoology, Vol. 18, No. 2, 1969, pp. 259–287.

    Google Scholar 

  9. Lawer, E. L.: Combinatorial Optimization: Networks and Matoids, Holt, Rinehart and Winston, New York, 1976.

    Google Scholar 

  10. Lee, D. T.: On k-nearest neighbor Voronoi diagrams in the plane, IEEE Transactions on Computers, Vol. C-31, No. 6, 1982, pp. 478–487.

    Google Scholar 

  11. Manber, U. and Tompa, M.: Probabilistic, nondeterministic and alternating decision trees, Technical Report, No. 82-03-01, University Washington, 1982.

    Google Scholar 

  12. Matula, D. W. and Sokal, R. R.: Properties of Gabriel graphs relevant to geographic variation research and the clustering of points in the plane, Geographical Analysis, Vol. 12, No. 3, 1980, pp. 205–222.

    Google Scholar 

  13. Parker, R. G. and Rardin, R. L.: Guaranteed performance heuristics for the bottleneck traveling salesperson problem, Operations Research Letters, Vol. 2, No. 6, 1984, pp. 269–272.

    Google Scholar 

  14. Shamos, M. I. and Preparata, F. P.: Computational Geometry-an Introduction, Springer-Verlag, 1985.

    Google Scholar 

  15. Supowit, K. J.: The relative neighborhood graph with an application to minimum spanning trees, Journal of Association for Computing Machinery, Vol. 30, No. 3, 1983, pp. 428–448.

    Google Scholar 

  16. Tarjan, R. E.: Efficiency of a good but not linear set union algorithm, Journal of Association for Computing Machinery, Vol. 22, No. 2, 1975, pp. 215–225.

    Google Scholar 

  17. Toussaint, G. T.: The relative neighborhood graph of a finite planar set, Pattern Recognition, Vol. 12, No. 4, 1980, pp. 261–268.

    Article  Google Scholar 

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Authors
  1. Tung-Hsin Su
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  2. Ruei-Chuan Chang
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Editor information

Tetsuo Asano Toshihide Ibaraki Hiroshi Imai Takao Nishizeki

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© 1990 Springer-Verlag Berlin Heidelberg

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Su, TH., Chang, RC. (1990). The K-Gabriel graphs and their applications. In: Asano, T., Ibaraki, T., Imai, H., Nishizeki, T. (eds) Algorithms. SIGAL 1990. Lecture Notes in Computer Science, vol 450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52921-7_56

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  • DOI: https://doi.org/10.1007/3-540-52921-7_56

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52921-7

  • Online ISBN: 978-3-540-47177-6

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