Constructing Many Faces in Arrangements of Lines and Segments

  title={Constructing Many Faces in Arrangements of Lines and Segments},
  author={Haitao Wang},
  • Haitao Wang
  • Published 16 October 2021
  • Computer Science
  • ArXiv
We present new algorithms for computing many faces in arrangements of lines and segments. Given a set S of n lines (resp., segments) and a set P of m points in the plane, the problem is to compute the faces of the arrangements of S that contain at least one point of P . For the line case, we give a deterministic algorithm of O(mn log(n/ √ m)+(m+n) log n) time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of log n and improves the previously best… 

Figures from this paper


Computing faces in segment and simplex arrangements
This work gives the first work-optimal deterministic parallel algorithm for constructing a set of m = O(nd 1 logc n+k) cells of constant descriptive complexity that covers their arrangement, and describes a sequential algorithm for computing a single face in an arrangement of n line segments that improves on a previous O(n log n) time algorithm.
The complexity and construction of many faces in arrangements of lines and of segments
The proof takes an algorithmic approach, that is, an algorithm is described for the calculation of thesem faces and the upper bound for the total number of edges is derived from the analysis of the algorithm.
A Simple Algorithm for Computing the Zone of a Line in an Arrangement of Lines
  • Haitao Wang
  • Computer Science
    Symposium on Simplicity in Algorithms (SOSA)
  • 2022
This paper presents a simple algorithm that can compute Z(`) in O(n log n) time, once the sorted list of the intersections between ` and the lines of L is known.
The number of edges of many faces in a line segment arrangement
The number of edges bounding anym faces in an arrangement of line segments with a total oft intersecting pairs is O(m2/3t1/3+nα(t/n)+nmin{logm,logt/ n}), almost matching the lower bound of Ω(m 2/ 3t 1/3 + nα( t/n)) demonstrated.
Partitioning arrangements of lines II: Applications
  • P. Agarwal
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1990
An algorithm that preprocesses a set ofn points in the plane, into a data structure of sizeO(m) forn logn≤m≤n2, so that the number of points ofS lying inside a query triangle can be computed inO((n/√m) log3/2n) time.
Computing a face in an arrangement of line segments
This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of
Combinatorial complexity bounds for arrangements of curves and spheres
Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
A Fast Planar Partition Algorithm, I
  • K. Mulmuley
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 1990
Though the algorithm itself is simple, the global evolution of the underlying partition is non-trivial, which makes the analysis of the algorithm theoretically interesting in its own right.
Computing many faces in arrangements of lines and segments
We present randomized algorithms for computing many faces in an arrangement of lines or of segments in the plane, which are considerably simpler and slightly faster than the previously known ones.
Finding the Upper Envelope of n Line Segments in O(n log n) Time
The method can be used to compute the upper envelope of “segments” that intersect pairwise at most k times and computes theupper envelope in O(λk + 1(n)log n) time.