Ham-Sandwich Cuts for Abstract Order Types

TL;DR

This paper presents a deterministic linear-time algorithm for finding ham-sandwich cuts using only order type information, extending the approach to abstract order types.

Contribution

It introduces a linear-time algorithm for ham-sandwich cuts based solely on order type data, removing the need for explicit point coordinates.

Findings

The algorithm runs in linear time using only order type information.

It extends to abstract order types beyond realizable point sets.

Provides a deterministic approach to ham-sandwich cuts in restricted settings.

Abstract

The linear-time ham-sandwich cut algorithm of Lo, Matou\v{s}ek, 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.