We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an indexing structure that, for any given constant e> 0; uses OðN=BÞ disk blocks and answers a query in OððN=BÞ 1=2þe þ K=BÞ I/Os, where B is the block size. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in Oðlog 2 B NÞ I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a tradeoff between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between the query time stamp t and the current time. Finally, we develop an efficient indexing scheme to answer approximate

In the design of algorithms for large-scale applications it is essential to consider the problem of minimizing I/O communication. Geographical information systems (GIS) are good examples of such large-scale applications as they frequently handle huge amounts of spatial data. In this paper we develop efficient external-memory algorithms for a number of important problems involving line segments in the plane, including trapezoid decomposition, batched planar point location, triangulation, red–blue line segment intersection reporting, and general line segment intersection reporting. In GIS systems the first three problems are useful for rendering and modeling, and the latter two are frequently used for overlaying maps and extracting information from them.

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

In this paper we develop an optimal cache-oblivious priority queue data structure, supporting insertion, deletion, and deletemin operations in O ( 1 B logM/B N) amortized memory B transfers, where M and B are the memory and block transfer sizes of any two consecutive levels of a multilevel memory hierarchy. In a cache-oblivious data structure, M and B are not used in the description of the structure. The bounds match the bounds of several previously developed external-memory (cache-aware) priority queue data structures, which all rely crucially on knowledge about M and B. Priority queues are a critical component in many of the best known external-memory graph algorithms, and using our cache-oblivious priority queue we develop several cacheoblivious graph algorithms.

In recent years there has been an upsurge of interest in spatial databases. A major issue is how to manipulate efficiently massive amounts of spatial data stored on disk in multidimensional spatial indexes (data structures). Construction of spatial indexes (bulk loading) has been studied intensively in the database community. The continuous arrival of massive amounts of new data makes it important to update existing indexes (bulk updating) efficiently. In this paper we present a simple, yet efficient, technique for performing bulk update and query operations on multidimensional indexes. We present our technique in terms of the so-called R-tree and its variants, as they have emerged as practically efficient indexing methods for spatial data. Our method uses ideas from the buffer tree lazy buffering technique and fully utilizes the available internal memory and the page size of the operating system. We give a theoretical analysis of our technique, showing that it is efficient both in terms of I/O communication, disk storage, and internal computation time. We also present the results of an extensive set of experiments showing that in practice our approach performs better than the previously best known bulk update methods with respect to update time, and that it produces a better quality index in terms of query performance. One important novel feature of our technique is that in most cases it allows us to perform a batch of updates and queries simultaneously. To be able to do so is essential in environments where queries have to be answered even while the index is being updated and reorganized.

In this paper, we present lower bounds for permuting and sorting in the cache-oblivious model. We prove that (1) I/O optimal cache-oblivious comparison based sorting is not possible without a tall cache assumption, and (2) there does not exist an I/O optimalcache-oblivious algorithm for permuting, not even in the presence of a tall cache assumption.Our results for sorting show the existence of an inherent trade-off in the cache-oblivious model between the strength of the tall cache assumption and the overhead for the case M >> B, and show that Funnelsort and recursive binary mergesort are optimal algorithms in the sense that they attain this trade-off.

Abstract The cache oblivious model of computation is a two-level memory model with the assumption that the parameters of the model are unknown to the algorithms. A consequence of this assumption is that an algorithm efficient in the cache oblivious model is automatically efficient in a multi-level memory model. Arge et al. recently presented the first optimal cache oblivious priority queue, and demonstrated the importance of this result by providing the first cache oblivious algorithms for graph problems. Their structure uses cache oblivious sorting and selection as subroutines. In this paper, we devise an alternative optimal cache oblivious priority queue based only on binary merging. We also show that our structure can be made adaptive to different usage profiles. 1

We present the first provably I/O-efficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks to store a subdivision of size N , where B is the disk block size. Queries can be answered in ... I/Os in the worst-case, and insertions and deletions can be performed in ... and ... I/Os amortized, respectively. Previously, an I/O-efficient dynamic point location structure was only known for monotone subdivisions. Part of our data structure...

In recent years, many theoretically I/O-efficient algorithms and data structures have been developed. The TPIE project at Duke University was started to investigate the practical importance of these theoretical results. The goal of this ongoing project is to provide a portable, extensible, flexible, and easy to use C++ programming environment for efficiently implementing I/O-algorithms and data structures. The TPIE library has been developed in two phases. The first phase focused on supporting algorithms with a sequential I/O pattern, while the recently developed second phase has focused on supporting on-line I/O-efficient data structures, which exhibit a more random I/O pattern. This paper describes the design and implementation of the second phase of TPIE.

In this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for object-oriented and temporal databases and for constraint logic programming. Part of the structure uses a novel weight-balancing technique for efficient worst-case manipulation of balanced trees of independent interest. The external interval tree, as well at our new balancing technique, have recently been used to develop several efficient external data structures.