Novel Transformation Techniques Using Q-Heaps with Applications to Computational Geometry

Using the notions of Q-heaps and fusion trees developed by Fredman and Willard, we develop general transformation techniques to reduce a number of computational geometry problems to their special versions in partially ranked spaces. In particular, we develop a fast fractional cascading technique, which uses linear space and enables sublogarithmic iterative search on catalog trees in the case when the degree of each node is bounded by O(log n), for some constant 2 > 0, where n is the total size of all the lists stored in the tree. We apply the fast fractional cascading technique in combination with the other techniques to derive the first linear-space sublogarithmic time algorithms for the two fundamental geometric retrieval problems: orthogonal segment intersection and rectangular point enclosure.