Abstract. We establish new upperbounds on the complexity ofmultidimensional 3earching. Our results include, in particular, linear-size data structures for range and rectangle counting in two dimensions with logarithmic query time. More generally, we give improved data structures for rectangle problems in any dimension, in a static as well as a dynamic setting. Several ofthe algorithms we give are simple to implement and might be the solutions of choice in practice. Central to this paper is the nonstandard approach followed to achieve these results. At its rootwe find a redefinition ofdata structures interms offunctional specifications.

In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worst-case efficient external memory dynamic data structures. We also briefly discuss some of the most popular external data structures used in practice.

We examine the problem of efficiently computing sum/count/avg aggregates over...

We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rectangles, spheres and loons; these lead to the graph of all dominance pairs, the Gabriel graph and the relative neighborhood graph, respectively. Other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph. In all cases, we obtain asymptotic lower bounds that do not depend uponf(and are hence useful in all applications involving these graphs, since we usually do not know f). For sparse graphs, exact asymptotic constants are obtained for E(N) that are valid for all densities.

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 survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas-convex hulls, intersections, searching, proximity, and combinatorial optimizations-are discussed. Seven algorithmic techniques incremental construction, plane-sweep, locus, divide-andconquer, geometric transformation, prune-and-search, and dynamization-are each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.

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We propose a new indexing scheme, called the CRB-tree, for efficiently answering range-aggregate queries. The range-aggregate problem is defined as follows: Given a set of weighted points in R , compute the aggregate of weights of points that lie inside a d-dimensional query rectangle. In this paper we focus on COUNT, SUM, AVG aggregates. First, we develop an indexing scheme for answering twodimensional range-COUNT queries that uses O(N=B) disk blocks and answers a query in O(log B N) I/Os, where N is the number of input points and B is the disk block size. This is the first optimal index structure for the 2D range-COUNT problem. The index can be extended to obtain a near-linearsize indexing structure for answering range-SUM queries using O(log B N) I/Os. We also obtain similar bounds for rectangle-intersection aggregate queries, in which the input is a set of weighted rectangles and a query asks to compute the aggregate of the weights of those input rectangles that overlap with the query rectangle. This result immediately improves a recent result on temporal-aggregate queries. Our indexing scheme can be dynamized and extended to higher dimensions. Finally, we demonstrate the practical efficiency of our index by comparing its performance against kdB-tree. For a dataset of around 100 million points, the CRB-tree query time is 8--10 times faster than the kdB-tree query time. Furthermore, unlike other indexing schemes, the query performance of CRB-tree is oblivious to the distribution of the input points and placement, shape and size of the query rectangle.

The study of simple families of geometric objects on the plane has always been of great interest in mathematics. Incidence relations among families of points, lines and circles on the Euclidean plane were studied intensely long ago. In fact, most of what is now considered as basic Euclidean Geometry was developed a few centuries before the advent of the Christian Era.

this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithms---one with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient than existing ones