... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.

We examine I O-efficient data structures that provide indexing support for new data models. The database languages of these models include concepts from constraint programming (e.g., relational tuples are generated to conjunctions of constraints) and from object-oriented programming (e.g., objects are organized in class hierarchies). Let n be the size of the database, c the number of classes, B the page size on secondary storage, and t the size of the output of a query: (1) Indexing by one attribute in many constraint data models is equivalent to external dynamic interval management, which is a special case of external dynamic two-dimensional range searching. We present a semi-dynamic data structure for this problem that has worst-case space O(n B) pages, query I O time O(logB n+t B) and O(logB n+(logB n) 2 B) amortized insert I O time. Note that, for the static version of this problem, this is the first worst-case optimal solution. (2) Indexing by one attribute and by class name in an object-oriented model, where objects are organized

In Layer Four switching, the route and resources allocated to a packet are determined by the destination address as well as other header fields of the packet such as source address, TCP and UDP port numbers. Layer Four switching unifies firewall processing, RSVP style resource reservation filters, QoS Routing, and normal unicast and multicast forwarding into a single framework. In this framework, the forwarding database of a router consists of a potentially large number of filters on key header fields. A given packet header can match multiple filters, so each filter is given a cost, and the packet is forwarded using the least cost matching filter. In this paper, we describe two new algorithms for solving the least cost matching filter problem at high speeds. Our first algorithm is based on a grid-of-tries construction and works optimally for processing filters consisting of two prefix fields (such as destination-source filters) using linear space. Our second algorithm, cross-producting, provides fast lookup times for arbitrary filters but potentially requires large storage. We describe a combination scheme that combines the advantages of both schemes. The combination scheme can be optimized to handle pure destination prefix filters in 4 memory accesses, destination-source filters in 8 memory accesses worst case, and all other filters in 11 memory accesses in the typical case.

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.

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.

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In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

We consider the problem of dynamic range searching in tree structures that do not replicate data. We propose a new dynamic structure, called the O-tree, that achieves a query time complexity of O(n (d\Gamma1)=d ) on n d-dimensional points and an amortized insertion/deletion time complexity of O(log n). We show that this structure is optimal when data is not replicated. In addition to optimal query and insertion/deletion times, the O-tree also supports exact match queries in worst-case logarithmic time. 1 Introduction Given a set S of d-dimensional points, a range query q is specified by d 1-dimensional intervals [q s i ; q e i ], one for each dimension i, and retrieves all points p = (p 1 ; p 2 ; : : : p d ) in S such that h8i 2 f1; : : : ; dg : q s i p i q e i i. This type of searching in multidimensional space has important applications in geographic information systems, image databases, and computer graphics. Several structures such as the range trees [3], P-range trees [29...

We investigate the complexity of halfspace range searching: Given n points in d- space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worst-case query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n= log n) 1\Gamma d\Gamma1 d(d+1) =m 1=d : Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound of O \Gamma n=m 1=d \Delta upper bound established by Matousek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n 2\GammaO(1=d) . Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require\Omega \Gamma n 1=3 \Delta time (modulo a logarithmic factor).

ABSTRACT We develop cache-oblivious data structures for orthogonal range searching, the problem of finding all T points in a set of N points in Rd lying in a query hyper-rectangle. Cacheoblivious data structures are designed to be efficient in arbitrary memory hierarchies. We describe a dynamic linear-size data structure that answers d-dimensional queries in O((N/B)1-1/d + T/B) memory transfers, where B is the block size of any two levels of a multilevel memory hierarchy. A point can be inserted into or deleted from this data structure in O(log2B N) memory transfers. We also develop a static structure for the twodimensional case that answers queries in O(logB N + T /B) memory transfers using O(N log22 N) space. The analysis of the latter structure requires that B = 22 c for some nonnegative integer constant c. Categories and Subject Descriptors F.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems