Algorithms and Data Structures for External Memory

13.2 Continuously Moving Items 121
O � (log B N) 2� I/Os amortized per insertion. By global rebuilding we can
do deletions in O(log B N) I/Os amortized. As in the internal memory
case, the amortized updates can typically be made worst-case.
Arge and Vahrenhold [56] obtain I/O bounds for dynamic point
location in general planar subdivisions similar to those of [6], 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 [58] described in Section
12.3. They use the logarithmic method to construct a data structure
to answer vertical rayshooting queries in the multislab structures.
The method relies upon a total ordering of the segments, but such an
ordering can be changed drastically by a single insertion. However, each
substructure in the logarithmic method is static (until it is combined
with another substructure), and thus a static total ordering can be used
for each substructure. The authors show by a type of fractional cascading
that the queries in the log B N substructures do not have to be done
independently, which saves a factor of log B N in the I/O cost, but at
the cost of making the data structures amortized instead of worst-case.
Agarwal et al. [11] 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.
13.2 Continuously Moving Items
Early work on temporal data generally concentrated on time-series or
multiversion data [298]. 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 [283, 297, 356]). There are two
main approaches to 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 far so as to trigger important combinatorial
events that are relevant to the application at hand [73]. For
example, an event relevant for range search might be triggered when
two items move to the same horizontal displacement (which happens
when the x-ordering of the two items is about to switch). A different
approach is to store each item’s location and speed trajectory, so that