Algorithms and Data Structures for External Memory

4.2 Disk Striping and Parallelism with Multiple Disks 27
least (log N)/log B N = logB, which is significant in practice, and can
be as much as Z/z = B for large Z.
4.2 Disk Striping and Parallelism with Multiple Disks
It is conceptually much simpler to program for the single-disk case
(D = 1) than for the multiple-disk case (D ≥ 1). Disk striping [216,
296] is a practical paradigm that can ease the programming task with
multiple disks: When disk striping is used, I/Os are permitted only on
entire stripes, one stripe at a time. The ith stripe, for i ≥ 0, consists
of block i from each of the D disks. For example, in the data layout
in Figure 2.3, the DB data items 0–9 comprise stripe 0 and can be
accessed in a single I/O step. The net effect of striping is that the D
disks behave as a single logical disk, but with a larger logical block
size DB corresponding to the size of a stripe.
We can thus apply the paradigm of disk striping automatically to
convert an algorithm designed to use a single disk with block size DB
into an algorithm for use on D disks each with block size B: In the
single-disk algorithm, each I/O step transmits one block of size DB;
in the D-disk algorithm, each I/O step transmits one stripe, which
consists of D simultaneous block transfers each of size B. The number
of I/O steps in both algorithms is the same; in each I/O step, the DB
items transferred by the two algorithms are identical. Of course, in
terms of wall clock time, the I/O step in the multiple-disk algorithm
will be faster.
Disk striping can be used to get optimal multiple-disk algorithms
for three of the four fundamental operations of Chapter 3 — streaming,
online search, and answer reporting — but it is nonoptimal for sorting.
To see why, consider what happens if we use the technique of disk
striping in conjunction with an optimal sorting algorithm for one disk,
such as merge sort [220]. As given in Table 3.1, the optimal number
of I/Os to sort using one disk with block size B is
�
Θ(nlogm n)=Θ n logn
� � �
N log(N/B)
=Θ
. (4.1)
logm B log(M/B)
With disk striping, the number of I/O steps is the same as if we use
a block size of DB in the single-disk algorithm, which corresponds to