TY - JOUR
T1 - Planar realizations of nonlinear davenport-schinzel sequences by segments
AU - Wiernik, Ady
AU - Sharir, Micha
PY - 1988/12
Y1 - 1988/12
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0002275985&partnerID=8YFLogxK
U2 - 10.1007/BF02187894
DO - 10.1007/BF02187894
M3 - מאמר
AN - SCOPUS:0002275985
VL - 3
SP - 15
EP - 47
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
SN - 0179-5376
IS - 1
ER -