U. Glaeser

By the reduction in [52], a data structure for 3-sided 3-D queries also applies to 2-D homothetic range
search, in which the queries correspond to scaled and translated (but not rotated) transformations of an
arbitrary fixed polygon. An interesting special case is “fat” orthogonal 2-D range search, where the query
rectangles are required to have bounded aspect ratio. For example, every rectangle with bounded aspect
ratio can be covered by two overlapping squares. An interesting open problem is to develop linear-sized
optimal data structures for fat orthogonal 2-D range search. By the reduction, one possible approach
would be to develop optimal linear-sized data structures for 3-sided 3-D range search.
Agarwal et al. [6] consider halfspace range searching, in which a query is specified by a hyperplane
and a bit indicating one of its two sides, and the output of the query consists of all the points on that
side of the hyperplane. They give various data structures for halfspace range searching in two, three, and
higher dimensions, including one that works for simplex (polygon) queries in two dimensions, but with
a higher query I/O cost. They have subsequently improved the storage bounds for halfspace range queries
in two dimensions to obtain an optimal static data structure satisfying Criteria 1 and 2 of Section 32.12.
The number of I/Os needed to build the data structures for 3-D orthogonal range search and halfspace
range search is rather large (more than Ω(N)). Still, the structures shed useful light on the complexity
of range searching and may open the way to improved solutions. An open problem is to design efficient
construction and update algorithms and to improve upon the constant factors.
Callahan et al. [50] develop dynamic EM data structures for several online problems in d dimensions.
For any fixed � > 0, they can find an approximately nearest neighbor of a query point (within a 1 + � factor
of optimal) in O(log B N) I/Os; insertions and deletions can also be done in O(log B N) I/Os. They use a
related approach to maintain the closest pair of points; each update costs O(log B N) I/Os. Govindrajan
et al. [98] achieve the same bounds for closest pair by maintaining a well-separated pair decomposition.
For finding nearest neighbors and approximate nearest neighbors, two other approaches are partition trees
[5,6] and locality-sensitive hashing [95]. Numerous data structures and lower bounds have been developed
for range queries and related problems on spatial data. Refer to [10,91,153,199] for a broad survey.
Dynamic and Kinetic Data Structures
The preceding sections have outlined cases of data structures in which the data items change dynamically.
The bootstrapping paradigm discussed in the two previous subsections is a very useful approach for
converting static data structures that are efficient in internal memory into dynamic ones that are efficient
for external memory.
In another approach to dynamic data, Arge and Vahrenhold [26] obtain I/O bounds for dynamic point
location in general planar subdivisions similar to those of [4], but without use of level-balanced trees.
Their method uses a weight-balanced base structure at the outer level and a multislab structure for storing
segments similar to that of Arge and Vitter [28]. They use an externalization of Bentley’s logarithmic
method [44,161] to construct a data structure to answer vertical rayshooting queries in the multislab
structures. Arge et al. [8] apply the logarithmic method (in both the binary form and B-way variant) to
get EM versions of kd-trees, BBD trees, and BAR trees.
In some applications, the data items are moving and their spatial coordinates change in a regular manner.
Early work on temporal data generally concentrated on time-series data or multiversion data [171]. A
question of growing interest in this mobile age is how to store and index continuously moving items, such
as mobile telephones, cars, and airplanes (e.g., see [113,170,211]). Two main approaches are used for
storing moving items: The first technique is to use the same sort of data structure as for nonmoving data,
but to update it whenever items move sufficiently so far as to trigger important combinatorial events that
are relevant to the application at hand [38]. A different approach is to store each item’s location and speed
trajectory, so that no updating is needed as long as the item’s trajectory plan does not change. Such an
approach requires fewer updates, but the representation for each item generally has higher dimension, and
the search strategies are therefore less efficient.
Kollios et al. [126] developed a linear-space indexing scheme for moving points along a (1-D) line, based
upon the notion of partition trees. Their structure supports a variety of range search and approximate
© 2002 by CRC Press LLC