An Efficient Algorithm for the 2-Central Path Problem

In this paper we consider the following [Formula: see text]-Central Path Problem (2CPP): Given a set of [Formula: see text] polygonal curves [Formula: see text] in the plane, find two curves [Formula: see text] and [Formula: see text], called [Formula: see text]-central paths, that best represent all curves in [Formula: see text]. Despite its theoretical interest and a wide range of practical applications, 2CPP has not been well studied. In this paper, we first establish criteria that [Formula: see text] and [Formula: see text] ought to meet in order for them to best represent [Formula: see text]. In particular, we require that there exists parametrizations [Formula: see text] and [Formula: see text] ([Formula: see text]) of [Formula: see text] and [Formula: see text] respectively such that the maximum distance from [Formula: see text] to curves in [Formula: see text] is minimized. Then an efficient algorithm is presented to solve 2CPP under certain realistic assumptions. Our algorithm constructs [Formula: see text] and [Formula: see text] in [Formula: see text] time, where [Formula: see text] is the total complexity of [Formula: see text] (i.e., the total number of vertices and edges), [Formula: see text] is the number of curves in [Formula: see text], and [Formula: see text] is the inverse Ackermann function. Our algorithm uses parametric search technique and is considerably faster than arrangement-related algorithms (i.e. [Formula: see text]) when [Formula: see text] as in most real applications.