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Discrete & Computational Geometry

, Volume 3, Issue 1, pp 15–47 | Cite as

Planar realizations of nonlinear davenport-schinzel sequences by segments

  • Ady Wiernik
  • Micha Sharir
Article

Abstract

LetG={l1,...,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((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((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((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.

Keywords

Induction Hypothesis Piecewise Linear Function Computer Science Department Lower Envelope Planar Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Ady Wiernik
    • 1
  • Micha Sharir
    • 2
  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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