Finding optimized bounding boxes of polytopes in d-dimensional space and their properties in k-dimensional projections

Using minimal bounding boxes to encapsulate or approximate a set of points in d-dimensional space is a non-trivial problem that has applications in a variety of fields including collision detection, object rendering, high dimensional databases and statistical analysis to name a few. While a significant amount of work has been done on the three dimensional variant of the problem (i.e. finding the minimum volume bounding box of a set of points in three dimensions), it is difficult to find a simple method to do the same for higher dimensions. Even in three dimensions existing methods suffer from either high time complexity or suboptimal results with a speed up in execution time. In this thesis we present a new approach to find the optimized minimum bounding boxes of a set of points defining convex polytopes in d-dimensional space. The solution also gives the optimal bounding box in three dimensions with a much simpler implementation while significantly speeding up the execution time for a large number of vertices. The basis of the proposed approach is a series of unique properties of the k-dimensional projections that are leveraged into an algorithm. This algorithm works by constructing the convex hulls of a given set of points and optimizing the projections of those hulls in two dimensional space using the new concept of Simultaneous Local Optimal. We show that the proposed algorithm provides significantly better performances than those of the current state of the art approach on the basis of time and accuracy. To illustrate the importance of the result in terms of a real world application, the optimized bounding box algorithm is used to develop a method for carrying out range queries in high dimensional databases. This method uses data transformation techniques in conjunction with a set of heuristics to provide significant performance improvement.