Planar realizations of nonlinear davenport-schinzel sequences by segments

Ady Wiernik*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let G={l1,..., ln} be a collection of n segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions of x, let YG be their lower envelope (i.e., pointwise minimum). YG is a piecewise linear function, whose graph consists of subsegments of the segments li. Hart and Sharir [7] have shown that YG consists of at most O(nα(n)) segments (where α(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a set G of n segments for which YG consists of Ω(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 sequence EG of indices of segments in G in the order in which they appear along YG is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements of EG are equal and EG contains no subsequence of the form a ... b ... a ... b ... a. Hart and Sharir have shown that the maximal length of such a sequence composed of n symbols is Θ(nα(n)). Our result shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope of n straight segments, thus settling one of the main open problems in this area.

Original languageEnglish
Pages (from-to)15-47
Number of pages33
JournalDiscrete and Computational Geometry
Volume3
Issue number1
DOIs
StatePublished - Dec 1988
Externally publishedYes

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