Algorithms and Data Structures for External Memory

54 External Sorting and Related Problems
example, the blocks for the subfiles can be allocated from the blocks
freed up from the file being partitioned; the disadvantage is that the
blocks in the individual subfiles are no longer consecutive on the disk.
The algorithms can be adapted to run on D disks with a speedup
of O(D) using the techniques described in Sections 5.1 and 5.2.
5.6 Permuting
Permuting is the special case of sorting in which the key values of the
N data items form a permutation of {1,2,...,N}.
Theorem 5.3 ([345]). The average-case and worst-case number
of I/Os required for permuting N data items using D disks is
� � ��
N
Θ min , Sort(N) .
D
(5.11)
The I/O bound (5.11) for permuting can be realized by one of
the optimal external sorting algorithms except in the extreme case for
which B logm = o(logn), when it is faster to move the data items one
by one in a nonblocked way. The one-by-one method is trivial if D =1,
but with multiple disks there may be bottlenecks on individual disks;
one solution for doing the permuting in O(N/D) I/Os is to apply the
randomized balancing strategies of [345].
An interesting theoretical question is to determine the I/O cost for
each individual permutation, as a function of some simple characterization
of the permutation, such as number of inversions. We examine
special classes of permutations having to do with matrices, such as
matrix transposition, in Chapter 7.
5.7 Fast Fourier Transform and Permutation Networks
Computing the Fast Fourier Transform (FFT) in external memory consists
of a series of I/Os that permit each computation implied by the
FFT directed graph (or butterfly) to be done while its arguments are
in internal memory. A permutation network computation consists of an
oblivious (fixed) pattern of I/Os that can realize any of the N! possible