Planar realizations of nonlinear davenport-schinzel sequences by segments

Let G={l₁,..., l_n} 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 Y_G be their lower envelope (i.e., pointwise minimum). Y_G is a piecewise linear function, whose graph consists of subsegments of the segments l_i. Hart and Sharir [7] have shown that Y_G 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 Y_G 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 E_G of indices of segments in G in the order in which they appear along Y_G is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements of E_G are equal and E_G 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.