Efficient computation of the convex hull on sets of points stored in a k-tree compact data structure

摘要

In this paper, we present two algorithms to obtain the convex hull of a set of points that are stored in the compact data structure called \(k^2\)-\(tree\). This problem consists in given a set of points P in the Euclidean space obtaining the smallest convex region (polygon) containing P. Traditional algorithms to compute the convex hull do not scale well for large databases, such as spatial databases, since the data does not reside in main memory. We use the \(k^2\)-\(tree\) compact data structure to represent, in main memory, efficiently a binary adjacency matrix representing points over a 2D space. This structure allows an efficient navigation in a compressed form. The experimentations performed over synthetical and real data show that our proposed algorithms are more efficient. In fact they perform over four order of magnitude compared with algorithms with time complexity of \(O(n \log n)\).