Algorithms and Data Structures for External Memory

8.1 Distribution Sweep 73
the current sweep line; vertical segments that are found to be no longer
active are deleted from the slabs.) The remaining two end portions of h
(which “stick out” past a slab boundary) are passed recursively to the
next level of recursion, along with the vertical segments. The downward
sweep then continues. After an initial one-time sorting (to order
the segments with respect to the y-dimension), the sweep at each of the
O(logm n) levels of recursion requires O(n) I/Os, yielding the desired
bound (8.1). Some timing experiments on distribution sweeping appear
in [104]. Arge et al. [48] develop a unified approach to distribution sweep
in higher dimensions.
A central operation in spatial databases is spatial join. A common
preprocessing step is to find the pairwise intersections of the bounding
boxes of the objects involved in the spatial join. The problem of
intersecting orthogonal rectangles can be solved by combining the previous
sweep line algorithm for orthogonal segments with one for range
searching. Arge et al. [48] take a more unified approach using distribution
sweep, which is extendible to higher dimensions: The active
objects that are stored in the data structure in this case are rectangles,
not vertical segments. The authors choose the branching factor to be
Θ( √ m). Each rectangle is associated with the largest contiguous range
of vertical slabs that it spans. Each of the possible Θ ��√ �� m
2 =Θ(m)
contiguous ranges of slabs is called a multislab. The reason why the
authors choose the branching factor to be Θ( √ m) rather than Θ(m)
is so that the number of multislabs is Θ(m), and thus there is room in
internal memory for a buffer for each multislab. The height of the tree
remains O(logm n).
The algorithm proceeds by sweeping a horizontal line from top to
bottom to process the N rectangles. When the sweep line first encounters
a rectangle R, we consider the multislab lists for all the multislabs
that R intersects. We report all the active rectangles in those multislab
lists, since they are guaranteed to intersect R. (Rectangles no longer
active are discarded from the lists.) We then extract the left and right
end portions of R that partially “stick out” past slab boundaries, and
we pass them down to process in the next lower level of recursion. We
insert the remaining portion of R, which spans complete slabs, into the
list for the appropriate multislab. The downward sweep then continues.