Cartographic line simplification and polygon CSG formulae in O(n log* n) time

A constructive solid geometry (CSG) conversion for a polygon takes a list o f vertices and produces a formula representing the polygon as an intersecti on and union of primitive halfspaces. The cartographers' favorite line simp lification algorithm recursively selects from a list of data points those t o 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-cas e. 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 poin ts. It opens the question of whether there are practical, linear-time solut ions to these problems. (C) 1998 Elsevier Science B.V. All rights reserved.