Computing a single face in an arrangement of line segments
|The unbounded single face (purple) of the given arrangement; in this instance the boundary consists of several connected components
|The unbounded single face of a random arrangement
While the complexity of a full arrangement of line segments is in O(n2), Sharir and Agarwal show that any single face in such arrangement has a maximum complexity of O(α(n)*n) where α(n) denotes the extremely slow-growing inverse Ackermann function which can be regarded as constant for any conceivable real-world input. Any single face can be constructed in time O(α(n)*n*log2n) using a deterministic divide and conquer algorithm including – in the words of Sharir and Agarwal – a “sophisticated sweep-line technique” in the merge step ("red blue merge").
- Jannis Warnat
Computing a single face in an arrangement of line segments with CGAL,
M.Sc. thesis, Rheinische Friedrich-Wilhelms-Universität Bonn,Institut für Informatik I, August, 2009.