## Abstract

Let G={l_{1},..., 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.

Original language | English |
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Pages (from-to) | 15-47 |

Number of pages | 33 |

Journal | Discrete and Computational Geometry |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1988 |

Externally published | Yes |