Reduction of Two-Dimensional Data for Speeding Up Convex Hull Computation

Abstract

An incremental approach for computation of convex hull for data points in two-dimensions is presented. The algorithm is not output-sensitive and costs a time that is linear in the size of data points at input. Graham's scan is applied only on a subset of the data points, represented at the extremal of the dataset. Points are classified for extremal, in proportion with the modular distance, about an imaginary point interior to the region bounded by convex hull of the dataset assumed for origin or center in polar coordinate. A subset of the data is arrived by terminating at until an event of no change in maximal points is observed per bin, for iteratively and exponentially decreasing intervals.Computation of convex hull of planar datasets is well studied in the literature of computational geometry [1,2,3,4,5,6]. A convex hull of a dataset is an ordering defined on the largest subset based on the geometry represented by the set. In two-dimensional dataset, the ordering leads to a closed polygon, where every line segment with any two vertices from the set, is contained entirely in the region bounded by the polygon. Algorithms for construction of convex hull, bears a correspondence with the algorithms for sorting.An algorithm for computation of convex hull is said to be output-sensitive, based on whether the complexity of computation depends on any structural attribute specific to the dataset at input other than its size. A consistent traversal (e.g. a turn in a counter-clockwise (CCW) sense) of the data points in a set, is often considered in computation of convex hull. An earliest output-sensitive algorithm by Jarvis et. al. [1], known as gift wrapping, computes convex hull in O(nh), for dataset of size n, and due to h points on the hull. The algorithm called Jarvis march, starts from a point that is lowest in one coordinate, and then selects the next point on the hull, with a consistent turn, and corresponds to selection sort.