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A tail estimate for Mulmuley's segment intersection algorithm

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Automata, Languages and Programming (ICALP 1992)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 623))

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Abstract

We analyze the distribution of the running time of Mulmuley's randomized algorithm for computing the intersections of n given segments in the plane. Its expectation has been known to be O(n log n+k), where k is the number of intersections. We show that for values of k not too close to n (k≥Cn log 15 n for a large enough constant C), the running time is sharply concentrated around the expected value; e.g., the probability that the expected value is exceeded more than twice is O(n − c), where c can be an arbitrarily large constant (its choice determines the value of C needed in the assumption). Our proof uses an isoperimetric inequality for permutations.

This research by the first author has been supported by International Computer Science Institute at Berkeley, CA, USA. Research by the second author was supported by NSF Presidential Young Investigator Award CCR-9058440.

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References

  1. H. Alt, L. J. Guibas, K. Mehlhorn, R. M. Karp, and A. Wigderson. A Method for Obtaining Randomized Algorithms with Small Tail Probabilities. Algorithmica, submitted.

    Google Scholar 

  2. B. Aronov, H. Edelsbrunner, L.J. Guibas, and M. Sharir. The number of edges of many faces in a line segment arrangement. Tech. Report 90-46, DIMACS Center, Piscataway, NJ, 1990. Combinatorica, to appear.

    Google Scholar 

  3. J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers, 28:643–647, 1979.

    Google Scholar 

  4. B. Bollobás. Martingales, isoperimetric inequalities and random graphs, pages 113–139. Math. Soc. J. Bolyai, 1987.

    Google Scholar 

  5. B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. In Proc. 29. IEEE Symposium on Theory of Computer Science, pages 590–600, 1988.

    Google Scholar 

  6. B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25(1), 1985.

    Google Scholar 

  7. B. Chazelle. Reporting and counting segment intersections. Journal of Computer and System Sciences, 32:156–182, 1986.

    Google Scholar 

  8. K. L. Clarkson and P. Shor. New applications of random sampling in computational geometry II. Discrete & Computational Geometry, 4, 1989.

    Google Scholar 

  9. H. Edelsbrunner. Algorithms in combinatorial geometry. Springer, 1987.

    Google Scholar 

  10. H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity and construction of many faces in arrangements of lines and segments. Discrete & Computational Geometry, 5:161–196, 1990.

    Google Scholar 

  11. H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of hyperplanes with applications. SIAM Journal on Computing, 15:341–363, 1986.

    Article  Google Scholar 

  12. T. Hagerup. An application of martingales to computational geometry. Lecture at Computational geometry workshop, Dagstuhl, FRG, October 1990.

    Google Scholar 

  13. T. Hagerup and C. Rüb. A Guided Tour of Chernoff Bounds. Inform. Proc. Letters 33 (1989/90), pp 305–308.

    Article  Google Scholar 

  14. K. Mehlhorn, M. Sharir, and E. Welzl. Tail estimates for the space complexity of randomized incremental algorithms. To appear in Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 1992.

    Google Scholar 

  15. V. D. Milman and G. Schechtman. Asymptotic theory of finite dimensional normed spaces. LNM 1200, Springer-Verlag, 1986.

    Google Scholar 

  16. K. Mulmuley. A fast planar partition algorithm: Part I. In Proc. 29. IEEE Symposium on Theory of Computer Science, pages 33–43, 1988.

    Google Scholar 

  17. R. Seidel. Backwards analysis of randomized geometric algorithms. International Computer Science Institute (ICSI) Berkeley, Report TR-92-014. Also to appear in New Trends in Discrete and Computational Geometry, J. Pach (ed.), Springer Verlag.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czechoslovakia

    Jiří Matoušek

  2. Computer Science Division, University of California, 94720, Berkeley, CA, USA

    Raimund Seidel

Authors
  1. Jiří Matoušek
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  2. Raimund Seidel
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W. Kuich

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© 1992 Springer-Verlag Berlin Heidelberg

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Matoušek, J., Seidel, R. (1992). A tail estimate for Mulmuley's segment intersection algorithm. In: Kuich, W. (eds) Automata, Languages and Programming. ICALP 1992. Lecture Notes in Computer Science, vol 623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55719-9_94

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  • DOI: https://doi.org/10.1007/3-540-55719-9_94

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55719-7

  • Online ISBN: 978-3-540-47278-0

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