Square Meshes Are Not Optimal for Convex Hull Computation

• Authors: Dharmavani Bhagavathi;Himabindu Gurla;Stephan Olariu;James L. Schwing;Jingyuan Zhang
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 1996

Recently it has been noticed that for semigroup computations and for selection rectangular meshes with multiple broadcasting yield faster algorithms than their square counterparts. The contribution of this paper is to provide yet another example of a fundamental problem for which this phenomenon occurs. Specifically, we show that the problem of computing the convex hull of a set of n sorted points in the plane can be solved in ${\rm O}(n^{{\textstyle{1 \over 8}}}\, {\rm log}^{{\textstyle{3 \over 4}}}\,n)$ time on a rectangular mesh with multiple broadcasting of size$$n^{{\textstyle{3 \over 8}}}\,{\rm log}^{{\textstyle{1 \over 4}}}\,n\times {\textstyle{{n^{{ \textstyle{5 \over 8}}}} \over {{\rm log}^{{\textstyle{1 \over 4}}}\,n}}}.$$The fastest previously-known algorithms on a square mesh of size $\sqrt n\times \sqrt n$ run in ${\rm O}(n^{{\textstyle{1 \over 6}}})$ time in case the n points are pixels in a binary image, and in ${\rm O}(n^{{\textstyle{1 \over 6}}}\,{\rm log}^{{\textstyle{2 \over 3}}}\,n).$ time for sorted points in the plane.