... 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.

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.

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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A range query applies an aggregation operation over all selected cells of an OLAP data cube where the selection is specified by providing ranges of values for numeric dimensions. We present fast algorithms for range queries for two types of aggregation operations: SUM and MAX. These two operations cover techniques required for most popular aggregation operations, such as those supported by SQL. For range-sum queries, the essential idea is to precompute some auxiliary information (prefix sums) that is used to answer ad hoc queries at run-time. By maintaining auxiliary information which is of the same size as the data cube, all range queries for a given cube can be answered in constant time, irrespective of the size of the sub-cube circumscribed by a query. Alternatively, one can keep auxiliary information which is 1/b d of the size of the d-dimensional data cube. Response to a range query may now require access to some cells of the data cube in addition to the access to the auxiliary ...

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.

Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the well-known predecessor problem, where the tree is a path.

We prove \Omega\Gamma p log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the union-split-find problem, dynamic prefix problems and onedimensional range query problems. The proof techniques include a general technique using perfect hashing for reducing static data structure problems (with a restriction of the size of the structure) into partially dynamic data structure problems (with no such restriction), thus providing a way to transfer lower bounds. We use a generalization of a method due to Ajtai for proving the lower bounds on the static problems, but describe the proof in terms of communication complexity, revealing a striking similarity to the proof used by Karchmer and Wigderson for proving lower bounds on the monotone circuit depth of connectivity. 1 Introduction and summary of results In this paper we give lower bounds for the complexity of implementing several dynamic and sta...

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).