Finding the intersection of n half-spaces in time O(n log n)

Abstract

Given a set of n half-spaces in three dimensional space, we develop an algorithm for finding their common intersection in time O(n log n). The intersection, if nonempty, is presented as a convex polyhedron. The algorithm is summarized as follows: (i) the half-spaces are placed in two sets depending upon whether they contain or do not contain the origin; (ii) the half-spaces in each of these sets are dualized to points, and the convex hulls of the dualized sets are obtained in time O(n log n); (iii) since the half-space intersection is nonempty if and only if these two convex hulls are disjoint, a separating plane is found, also in time O(n log n); (iv) after applying a linear spatial transformation which maps the separating plane to infinity, the convex hull of the union of the two transformed convex hulls is the transformed intersection of the half-spaces. Since the letter can be found in time O(n), the overall running time of the procedure is O(n log n). A significant consequence of this result is that a three-variable linear, or convex, programming problem can be asymptotically solved faster than by the Simplex algorithm, in the worst case. © 1979.