Planar realizations of nonlinear davenport-schinzel sequences by segments

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((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.

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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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Keywords

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