Algorithms and Data Structures for External Memory

22 Fundamental I/O Operations and Bounds
Table 3.1 I/O bounds for the four fundamental operations. The PDM parameters are
defined in Section 2.1.
Operation I/O bound, D = 1 I/O bound, general D ≥ 1
Scan(N) Θ
Sort(N)
� �
� �
N
N
�
n
�
=Θ(n) Θ =Θ
B
DB D
�
N
Θ
B log �
N
M/B
B
=Θ(nlog m n)
�
N
Θ
DB log �
N
M/B
B
�
n
=Θ
D logm n
�
Search(N) Θ(log B N) Θ(log DB N)
Output(Z)
� �
Θ max 1, Z
��
B
=Θ � max{1,z} �
� �
Θ max 1, Z
��
DB
� �
=Θ max 1, z
��
D
The first two of these I/O bounds — Scan(N) and Sort(N) —
apply to batched problems. The last two I/O bounds — Search(N)
and Output(Z) — apply to online problems and are typically combined
together into the form Search(N) +Output(Z). As mentioned in
Section 2.1, some batched problems also involve queries, in which case
the I/O bound Output(Z) may be relevant to them as well. In some
pipelined contexts, the Z items in an answer to a query do not need to
be output to the disks but rather can be “piped” to another process, in
which case there is no I/O cost for output. Relational database queries
are often processed in such a pipeline fashion. For simplicity, in this
manuscript we explicitly consider the output cost for queries.
The I/O bound Scan(N) =O(n/D), which is clearly required to
read or write a file of N items, represents a linear number of I/Os in the
PDM model. An interesting feature of the PDM model is that almost
all nontrivial batched problems require a nonlinear number of I/Os,
even those that can be solved easily in linear CPU time in the (internal
memory) RAM model. Examples we discuss later include permuting,
transposing a matrix, list ranking, and several combinatorial graph
problems. Many of these problems are equivalent in I/O complexity to
permuting or sorting.