Cartographic line simplification and polygon CSG formulæ in [formula omitted] time

Abstract

A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. By using a data structure that maintains convex hulls of polygonal lines under splits, both were known to have O( n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log ∗ n) algorithm for maintaining convex hulls under splits at extreme points. It opens the question of whether there are practical, linear-time solutions to these problems.