Planar point location using persistent search trees
Communications of the ACM
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Journal of Algorithms
Optimal point location in a monotone subdivision
SIAM Journal on Computing
A fast planar partition algorithm, I
Journal of Symbolic Computation
Computational Geometry: Theory and Applications
Journal of Algorithms
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Intersections with random geometric object
Computational Geometry: Theory and Applications
The design and implementation of panar maps in CGAL
Journal of Experimental Algorithmics (JEA)
Expected time analysis for Delaunay point location
Computational Geometry: Theory and Applications
On the stabbing number of a random Delaunay triangulation
Computational Geometry: Theory and Applications
Advanced programming techniques applied to Cgal's arrangement package
Computational Geometry: Theory and Applications
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
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We investigate the complexity of the outer face in arrangements of line segments of a fixed length in the plane, drawn uniformly at random within a square. We derive upper bounds on the expected complexity of the outer face, and establish a certain phase transition phenomenon during which the expected complexity of the outer face drops sharply as a function of the total number of segments. In particular we show that up till the phase transition the complexity of the outer face is almost linear in n, and that after the phase transition, the complexity of the outer face is roughly proportional to pn. Our study is motivated by the analysis of a practical point-location algorithm (so-called walk-along-a-line point-location algorithm) and indeed, it explains experimental observations of the behavior of the algorithm on arrangements of random segments.