Algorithms and Data Structures for External Memory

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Fig. 12.1 Different types of 2-D orthogonal range queries: (a) Diagonal corner two-sided
2-D query (equivalent to a stabbing query, cf. Section 12.3); (b) Two-sided 2-D query;
(c) Three-sided 2-D query; (d) General four-sided 2-D query.
we discuss some of the linear-space data structures used
extensively in practice. None of them come close to satisfying
Criteria 1 and 3 for range search in the worst case, but
in typical-case scenarios they often perform well. We devote
Section 12.2 to R-trees and their variants, which are the
most popular general-purpose spatial structures developed
to date.
• Since the lower bound applies only to general 2-D rectangular
queries, are there any data structures that meet Criteria 1–3
for the important special cases of 2-D range searching pictured
in Figures 12.1(a), 12.1(b), and 12.1(c)? Fortunately
the answer is yes. We show in Sections 12.3 and 12.4 how to
use a “bootstrapping” paradigm to achieve optimal search
and update performance.
• Can we meet Criteria 1 and 2 for general four-sided
range searching if the disk space allowance is increased
to O � n(logn)/log(log B N +1) � blocks? Yes again! In Section
12.5, we show how to adapt the optimal structure for
three-sided searching in order to handle general four-sided
searching in optimal search cost. The update cost, however,
is not known to be optimal.
In Section 12.6, we discuss other scenarios of range search dealing
with three dimensions and nonorthogonal queries. We discuss the lower
bounds for 2-D range searching in Section 12.7.