U. Glaeser

of satellite images is over one terabyte in size [188]. A major challenge is to develop mechanisms for
processing the data, or else much of the data will be useless. ∗
For systems of this size to be efficient, fast EM algorithms and data structures are needed for basic
problems in computational geometry. Luckily, many problems on geometric objects can be reduced to
a small core of problems, such as computing intersections, convex hulls, or nearest neighbors. Useful
paradigms have been developed for solving these problems in external memory.
Theorem 32.8.1 The following batched problems involving N = nB input items, Q = qB queries, and
Z = zB output items can be solved using
I/Os with a single disk:
© 2002 by CRC Press LLC
O((n + q)log m n + z) (32.7)
1. Computing the pairwise intersections of N segments in the plane and their trapezoidal decomposition,
2. Finding all intersections between N nonintersecting red line segments and N nonintersecting blue line
segments in the plane,
3. Answering Q orthogonal 2-D range queries on N points in the plane (i.e., finding all the points within
the Q query rectangles),
4. Constructing the 2-D and 3-D convex hull of N points,
5. Voronoi diagram and Triangulation of N points in the plane,
6. Performing Q point location queries in a planar subdivision of size N,
7. Finding all nearest neighbors for a set of N points in the plane,
8. Finding the pairwise intersections of N orthogonal rectangles in the plane,
9. Computing the measure of the union of N orthogonal rectangles in the plane,
10. Computing the visibility of N segments in the plane from a point,
11. Performing Q ray-shooting queries in 2-D Constructive Solid Geometry (CSG) models of size N: The
parameters Q and Z are set to 0 if they are not relevant for the particular problem.
Goodrich et al. [97], Zhu [217], Arge et al. [27], Arge et al. [24], and Crauser et al. [64,65] develop
EM algorithms for those problems using the following EM paradigms for batched problems:
Distribution sweeping—a generalization of the distribution paradigm of Section 32.6 for “externalizing”
plane sweep algorithms.
Persistent B-trees—an offline method for constructing an optimal-space persistent version of the
B-tree data structure (see the subsection “B-trees and Variants”), yielding a factor of B improvement
over the generic persistence techniques of Driscoll et al. [74].
Batched filtering—a general method for performing simultaneous EM searches in data structures that
can be modeled as planar layered directed acyclic graphs; it is useful for 3-D convex hulls and
batched point location. Multisearch on parallel computers is considered in [73].
External fractional cascading—an EM analog to fractional cascading on a segment tree, in which the
degree of the segment tree is O(m α ) for some constant 0 < α ≤ 1. Batched queries can be performed
efficiently using batched filtering; online queries can be supported efficiently by adapting the
parallel algorithms of work of Tamassia and Vitter [187] to the I/O setting.
External marriage-before-conquest—an EM analog to the technique of Kirkpatrick and Seidel [123]
for performing output-sensitive convex hull constructions.
Batched incremental construction—a localized version of the randomized incremental construction
paradigm of Clarkson and Shor [56], in which the updates to a simple dynamic data structure
∗ For brevity, in the remainder of this chapter, only with the single-disk case D = 1 is presented. The single-disk
I/O bounds for the batched problems can often be cut by a factor of Θ(D) for the case D ≥ 1 by using the load
balancing techniques of Section 32.6. In practice, disk striping (cf., Section 32.5) may be sufficient. For online
problems, disk striping will convert optimal bounds for the case D = 1 into optimal bounds for D ≥ 1.