We present linear-space sub-logarithmic algorithms for handling the 3-dimensional dominance reporting and the 2-dimensional dominance counting problems. Under the RAM model as described in [M. L. Fredman and D. E. Willard. "Surpassing the information theoretic bound with fusion trees", Journal of Computer and System Sciences, 47:424-436, 1993], our algorithms achieve O(log n/ log log n + f) query time for the 3-dimensional dominance reporting problem, where f is the output size, and O(log n/ log log n) query time for the 2-dimensional dominance counting problem. We extend these results to any constant dimension d ≥ 3, achieving O(n(log n/ log log n) d−3) space and O((log n/ log log n) d−2 + f) query time for the reporting case and O(n(log n/ log log n) d−2) space and O((log n/ log log n) d−1) query time for the counting case.

We propose a novel Persistent OcTree (POT) indexing structure for accelerating isosurface extraction and spatial filtering from volumetric data. This data structure efficiently handles a wide range of visualization problems such as the generation of view-dependent isosurfaces, ray tracing, and isocontour slicing for high dimensional data. POT can be viewed as a hybrid data structure between the interval tree and the Branch-On-Need Octree (BONO) in the sense that it achieves the asymptotic bound of the interval tree for identifying the active cells corresponding to an isosurface and is more efficient than BONO for handling spatial queries. We encode a compact octree for each isovalue. Each such octree contains only the corresponding active cells, in such a way that the combined structure has linear space. The inherent hierarchical structure associated with the active cells enables very fast filtering of the active cells based on spatial constraints. We demonstrate the effectiveness of our approach by performing view-dependent isosurfacing on a wide variety of volumetric data sets and 4D isocontour slicing on the time-varying Richtmyer-Meshkov instability dataset.

No abstract

We present in this paper fast algorithms for the 3-D dominance reporting and counting problems, and generalize the results to the d-dimensional case. Our 3-D dominance reporting algorithm achieves O( log n/ log log n+f) query time using O(n log ∊ n) space, where f is the number of points satisfying the query and ∊>0 is an arbitrarily small constant. For the 3-D dominance counting problem (which is equivalent to the 3-D range counting problem), our algorithm runs in O(( log n/ log log n)2) time using O(n log1+∊n/ log log n) space.

Abstract. 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 > 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.Key words. searching, computational geometry, geometric retrieval, fractional cascading, orthogonal segment intersection, rectangular point enclosure AMS subject classifications. 68P10, 68P05, 68Q251. Introduction. Q-heaps and fusion trees [10,11] are data structures that achieve sublogarithmic search time on one-dimensional data. In particular, a Q-heap supports constant time insertion, deletion and predecessor search on very "small" subsets of a larger set using linear space. In [30], Willard illustrated how upper bounds for several search problems can be improved using the Q-heap. In this paper, we further explore the Q-heap technique in the context of computational geometry by using it to develop several general techniques, which lead to faster algorithms for a number of geometric retrieval problems.A geometric retrieval problem is to preprocess a set S of n geometric objects so that, when given a query specifying a set of geometric constraints, the subset Q of S consisting of the objects that satisfy these constraints can be reported quickly. Examples of geometric retrieval problems include orthogonal segment intersection [26,3], rectangular point enclosure [19,27,3], and orthogonal range queries [2,29,20,3,4,21,25] and their special cases [19,30,5,18]. A typical data structure for handling such a problem often involves a primary constant-degree search tree T whose nodes are each equipped with secondary structures, which are built on a subset of S and are capable of handling special versions of the original query very quickly. There are two main ideas behind such a typical data structure. First, the objects in S are distributed among the nodes of T in such a way that, the number of nodes of T visited during a search process is bounded by the depth of T or the output size. Second, for each node v visited, the search query on the set S(v) of objects stored there can be performed very fast. The first idea is often realized by either ensuring that a non-root node v is visited only if it is on a specific path from the root of T to a leaf node (for example, in handling the segment intersection and rectangle point enclosure problems [3]), or the time spent at v can be compensated by the time spent at reporting Q ∩...

We consider the problem of dynamically indexing temporal observations about a collection of objects, each observation consisting of a key identifying the object, a list of attribute values and a timestamp indicating the time at which these values were recorded. We m a k e no assumptions about the rates at which these observations are collected, nor do we assume that the various objects have about the same number of observations. We d e v elop indexing structures that are almost linear in the total number of observations available at any g i v en time instant, and that support dynamic insertions in polylogarithmic time. Moreover, these structures allow the quick handling of queries to identify objects whose attribute values fall within a certain range at every time instance of a speci ed time interval. Provably good bounds are established.

No abstract

Given a set of n objects, each characterized by d attributes specified at m fixed time instances, we are interested in the problem of designing space efficient indexing structures such that a class of temporal range search queries can be handled efficiently. When m = 1, our problem reduces to the d-dimensional orthogonal search problem. We establish efficient data structures to handle several classes of the general problem. Our results include a linear size data structure that enables a query time of O(log n log m + f ) for one-sided queries when d = 1, where f is the number of objects satisfying the query. A similar result is shown for counting queries. We also show that the most general problem can be solved with a polylogarithmic query time using superlinear space data structures.