Optimal External Memory Planar Point Enclosure

In this paper we study the external memory planar point enclosure problem: Given N axis-parallel rectangles in the plane, construct a data structure on disk (an index) such that all K rectangles containing a query point can be reported I/O-efficiently. This problem has important applications in e.g. spatial and temporal databases, and is dual to the important and well-studied orthogonal range searching problem. Surprisingly, despite the fact that the problem can be solved optimally in internal memory with linear space and O(log N+K) query time, we show that one cannot construct a linear sized external memory point enclosure data structure that can be used to answer a query in O(log  _B N+K/B) I/Os, where B is the disk block size. To obtain this bound, Ω(N/B ^1−ε ) disk blocks are needed for some constant ε>0. With linear space, the best obtainable query bound is O(log ₂ N+K/B) if a linear output term O(K/B) is desired. To show this we prove a general lower bound on the tradeoff between the size of the data structure and its query cost. We also develop a family of structures with matching space and query bounds. An extended abstract of this paper appeared in Proceedings of the 12th European Symposium on Algorithms (ESA’04), Bergen, Norway, September 2004, pp. 40–52. L. Arge’s research was supported in part by the National Science Foundation through RI grant EIA–9972879, CAREER grant CCR–9984099, ITR grant EIA–0112849, and U.S.-Germany Cooperative Research Program grant INT–0129182, as well as by the US Army Research Office through grant W911NF-04-01-0278, by an Ole Roemer Scholarship from the Danish National Science Research Council, a NABIIT grant from the Danish Strategic Research Council and by the Danish National Research Foundation. V. Samoladas’ research was supported in part by a grant co-funded by the European Social Fund and National Resources-EPEAEK II-PYTHAGORAS. K. Yi’s research was supported in part by the National Science Foundation through ITR grant EIA–0112849, U.S.-Germany Cooperative Research Program grant INT–0129182, and Hong Kong Direct Allocation Grant (DAG07/08).     

A bucketing algorithm for the orthogonal segment intersection search problem and its practical efficiency

The orthogonal segment intersection search problem is stated as follows: given a setS ofn orthogonal segments in the plane, report all the segments ofS that intersect a given orthogonal query segment. For this problem, we propose a simple and practical algorithm based on bucketing techniques. It constructs, inO(n) time preprocessing, a search structure of sizeO(n) so that all the segments ofS intersecting a query segment can be reported inO(k) time in the average case, wherek is the number of the reported segments. The proposed algorithm as well as existing algorithms is implemented in FORTRAN, and their practical efficiencies are investigated through computational experiments. It is shown that ourO(k) search time,O(n) space, andO(n) preprocessing time algorithm is in practice the most efficient among the algorithms tested.Supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan under Grant No. 62550270 (1987).     

Searching in metric spaces by spatial approximation

We propose a new data structure to search in metric spaces. A metric space is formed by a collection of objects and a distance function defined among them which satisfies the triangle inequality. The goal is, given a set of objects and a query, retrieve those objects close enough to the query. The complexity measure is the number of distances computed to achieve this goal. Our data structure, called sa-tree (“spatial approximation tree”), is based on approaching the searched objects spatially, that is, getting closer and closer to them, rather than the classic divide-and-conquer approach of other data structures. We analyze our method and show that the number of distance evaluations to search among n objects is sublinear. We show experimentally that the sa-tree is the best existing technique when the metric space is hard to search or the query has low selectivity. These are the most important unsolved cases in real applications. As a practical advantage, our data structure is one of the few that does not need to tune parameters, which makes it appealing for use by non-experts. Edited by R. Sacks-Davis Received: 17 April 2001 / Accepted: 24 January 2002 / Published online: 14 May 2002