Algorithms and Data Structures for External Memory

112 Spatial Data Structures and Range Search
provide an elegant and optimal static data structure for three-sided
range search, which can be used in the EM priority search tree described
above to implement the child caches of size O(B 2 ). The static structure
is a persistent B-tree optimized for batched construction. When used
for O(B 2 ) points, it occupies O(B) blocks, can be built in O(B) I/Os,
and supports three-sided queries in O � ⌈Z ′ /B⌉ � I/Os per query, where
Z ′ is the number of points reported. The static structure is so simple
that it may be useful in practice on its own.
Both the three-sided structure developed by Arge et al. [50] and
the structure for two-sided diagonal queries discussed in Section 12.3
satisfy Criteria 1–3 of Chapter 12. So in a sense, the three-sided
query structure subsumes the diagonal two-sided structure, since
three-sided queries are more general. However, diagonal two-sided
structure may prove to be faster in practice, because in each of its
corner substructures, the data accessed during a query are always in
contiguous blocks, whereas the static substructures used in three-sided
search do not guarantee block contiguity.
On a historical note, earlier work on two-sided and three-sided
queries was done by Ramaswamy and Subramanian [289] using the
notion of path caching; their structure met Criterion 1 but had higher
storage overheads and amortized and/or nonoptimal update bounds.
Subramanian and Ramaswamy [323] subsequently developed the prange
tree data structure for three-sided queries, with optimal linear
disk space and nearly optimal query and amortized update bounds.
12.5 General Orthogonal 2-D Range Search
The dynamic data structure for three-sided range searching can be generalized
using the filtering technique of Chazelle [98] to handle general
four-sided queries with optimal I/O query bound O(log B N + z) and
optimal disk space usage O � n(logn)/log(log B N +1) � [50]. The update
bound becomes O � (log B N)(logn)/log(log B N +1) � , which may not be
optimal.
The outer level of the structure is a balanced (log B N + 1)-way 1-D
search tree with Θ(n) leaves, oriented, say, along the x-dimension. It
therefore has about (logn)/log(log B N + 1) levels. At each level of the