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Planar realizations of nonlinear davenport-schinzel sequences by segments

Discrete & Computational Geometry volume 3, pages 15–47 (1988)Cite this article

Abstract

LetG={l 1,...,l n } be a collection ofn segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions ofx, letY G be their lower envelope (i.e., pointwise minimum).Y G is a piecewise linear function, whose graph consists of subsegments of the segmentsl i . Hart and Sharir [7] have shown thatY G consists of at mostO(nα(n)) segments (whereα(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a setG ofn segments for whichY G consists ofΩ(nα(n)) subsegments, proving that the Hart-Sharir bound is tight in the worst case.

Another interpretation of our result is in terms of Davenport-Schinzel sequences: the sequenceE G of indices of segments inG in the order in which they appear alongY G is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements ofE G are equal andE G contains no subsequence of the forma ...b ...a ...b ...a. Hart and Sharir have shown that the maximal length of such a sequence composed ofn symbols is Θ(nα(n)). Our result shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope ofn straight segments, thus settling one of the main open problems in this area.

References

  1. W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math. Ann 99 (1928), 118–133.

    MathSciNet  Article  MATH  Google Scholar 

  2. M. Atallah, Dynamic computational geometry,Proceedings of the 24th Symposium on Foundations of Computer Science, 92–99, 1983. (Also inComput. Math. Appl. 11 (1985), 1171–1181.)

  3. A. Baltsan and M. Sharir, On Shortest Paths Between Two Convex Polyhedra, Technical Report No. 180, Computer Science Department, New York University, 1985. (To appear inJ. Assoc. Comput. Mach.)

  4. R. Cole and M. Sharir, Visibility of a Polyhedral Surface from a Point, Technical Report No. 266, Computer Science Department, New York University, 1986. (To appear inJ. Symbolic Comput.)

  5. H. Davenport, A combinatorial problem connected with differential equations, II,Acta Arith. 17 (1971), 363–372.

    MathSciNet  Google Scholar 

  6. H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations,Amer. J. Math. 87 (1965), 684–694.

    MathSciNet  Article  MATH  Google Scholar 

  7. S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.

    MathSciNet  Article  Google Scholar 

  8. P. Komjath, private communication.

  9. D. Leven and M. Sharir, On the Number of Critical Free Contacts of a Convex Polygonal Object Moving in Two-Dimensional Polygonal Space, Technical Report No. 187, Computer Science Department, New York University, 1985. (Also inDiscrete Comput. Geom. 2 (1987), 255–270.)

  10. C. Ó'Dúnlaing, M. Sharir, and C. K. Yap, Generalized Voronoi diagrams for a ladder, II: Efficient construction of the diagram,Algorithmica 2 (1987), 27–59.

    MathSciNet  Article  MATH  Google Scholar 

  11. R. Pollack, M. Sharir, and S. Sifrony, Separating Two Simple Polygons by a Sequence of Translations, Technical Report No. 215, Computer Science Department, New York University, 1986. (To appear inDiscrete Comput. Geom.).

  12. D. P. Roselle and R. G. Stanton, Some properties of Davenport-Schinzel sequences,Acta Arith. 17 (1971), 355–362.

    MathSciNet  MATH  Google Scholar 

  13. M. Sharir, Almost Linear Upper Bounds on the Length of General Davenport-Schinzel Sequences,Combinatorica 7 (1987), 131–143.

    MathSciNet  Article  MATH  Google Scholar 

  14. M. Sharir, Improved Lower Bound on the Length of Davenport-Schinzel Sequences, Technical Report No. 204, Computer Science Department, New York University, 1986. (To appear inCombinatorica.)

  15. E. Szemeredi, On a problem of Davenport and Schinzel,Acta Arith. 25 (1974), 213–224.

    MathSciNet  MATH  Google Scholar 

  16. J. Pach and M. Sharir, The Upper Envelope of Piecewise Linear Functions and the Boundary of a Region Enclosed by Convex Plates: Combinatorial Analysis, Technical Report No. 279, Computer Science Department, New York University, 1987.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Ady Wiernik

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, NY, USA

    Micha Sharir

Authors
  1. Ady Wiernik
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  2. Micha Sharir
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Additional information

Work on this paper has been partially supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. This paper is part of the first author's M.Sc. thesis prepared at Tel Aviv University under the supervision of the second author. A preliminary version of this paper has appeared inProceedings of the 27th IEEE Symposium on Foundations of Computer Science, Toronto, 97–106, 1986.

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Wiernik, A., Sharir, M. Planar realizations of nonlinear davenport-schinzel sequences by segments. Discrete Comput Geom 3, 15–47 (1988). https://doi.org/10.1007/BF02187894

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  • DOI: https://doi.org/10.1007/BF02187894

Keywords

  • Induction Hypothesis
  • Piecewise Linear Function
  • Computer Science Department
  • Lower Envelope
  • Planar Realization