A Combinatorial Proof of Universal Optimality for Computing a Planar Convex Hull

Abstract

For a planar point set $P$, its convex hull is the smallest convex polygon that encloses all points in $P$. The construction of the convex hull from an array $I_P$ containing $P$ is a fundamental problem in computational geometry. By sorting $I_P$ in lexicographical order, one can construct the convex hull of $P$ in $O(n \log n)$ time which is worst-case optimal. Standard worst-case analysis, however, has been criticized as overly coarse or pessimistic, and researchers search for more refined analyses. Universal analysis provides an even stronger guarantee. It fixes a point set $P$ and considers the maximum running time across all permutations $I_P$ of $P$. Afshani, Barbay, Chan [FOCS'07] prove that the convex hull construction algorithm by Kirkpatrick, McQueen, and Seidel is universally optimal. Their proof restricts the model of computation to any algebraic decision tree model where the test functions have at most constant degree and at most a constant number of arguments. They rely upon involved algebraic arguments to construct a lower bound for each point set $P$ that matches the universal running time of [SICOMP'86]. We provide a different proof of universal optimality. Instead of restricting the computational model, we further specify the output. We require as output (1) the convex hull, and (2) for each internal point of $P$ a witness for it being internal. Our argument is shorter, perhaps simpler, and applicable in more general models of computation.