Ham-Sandwich Cuts for Abstract Order Types

• Published in: Algorithmica Vol. 80; no. 1; pp. 234 - 257
• Main Authors: Felsner, Stefan, Pilz, Alexander
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2018; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Mathematics of Computing; Online securities trading; Theory of Computation; Ham-sandwich cut; Abstract order type; Pseudo-line arrangement
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-016-0246-4

The linear-time ham-sandwich cut algorithm of Lo, Matoušek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen vertical lines. We consider the setting where we are not given the coordinates of the point set, but only the orientation of each point triple (the order type) and give a deterministic linear-time algorithm for the mentioned sub-algorithm. This yields a linear-time ham-sandwich cut algorithm even in our restricted setting. We also show that our methods are applicable to abstract order types.