Planar realizations of nonlinear Davenport-Schinzel sequences by segments

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 lt. Hart and Sharir [HS] 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-Sehinzel 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 actually shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope of n straight segments, thus settling one the main open problems in this area.