Probabilistic Convex Hull Queries over Uncertain Data

Abstract

The convex hull of a set of two-dimensional points, P, is the minimal convex polygon that contains all the points in P. Convex hull is important in many applications such as GIS, statistical analysis and data mining. Due to the ubiquity of data uncertainty such as location uncertainty in real-world applications, we study the concept of convex hull over uncertain data in 2D space. We propose the Probabilistic Convex Hull(PCH) query and demonstrate its applications, such as Flickr landscape photo extraction and activity region visualization, where location uncertainty is incurred by GPS devices or sensors. To tackle the problem of possible world explosion, we develop an O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) algorithm based on geometric properties, where N is the data size. We further improve this algorithm with spatial indices and effective pruning techniques, which prune the majority of data instances. To achieve better time complexity, we propose another O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log N) algorithm, by maintaining a probability oracle in the form of a circular array with nice properties. Finally, to support applications that require fast response, we develop a Gibbs-sampling-based approximation algorithm which efficiently finds the PCH with high accuracy. Extensive experiments are conducted to verify the efficiency of our algorithms for answering PCH queries.