Planar realizations of nonlinear davenport-schinzel sequences by segments

LetG={l1,...,ln} be a collection ofn segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions ofx, letYG be their lower envelope (i.e., pointwise minimum).YG is a piecewise linear function, whose graph consists of subsegments of the segmentsli. Hart and Sharir [7] have shown thatYG consists of at mostO(nα(n)) segments (whereα(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a setG ofn segments for whichYG consists ofΩ(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 sequenceEG of indices of segments inG in the order in which they appear alongYG is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements ofEG are equal andEG 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(nα(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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