Algorithms and Data Structures for External Memory

7.2 Matrix Transposition 67
When the matrices are stored using a sparse representation, transposition
is always as hard as sorting, unlike the B 2 ≤ M case for dense
matrix transposition (cf. Theorem 7.2).
Theorem 7.3 ([23]). For a matrix stored in sparse format and containing
Nz nonzero elements, the number of I/Os required to transpose
the matrix from row-major order to column-major order, and viceversa,
is Θ � Sort(Nz) � .
Sorting suffices to perform the transposition. The lower bound
follows by reduction from sorting: If the ith item to sort has key
value x �= 0, there is a nonzero element in matrix position (i,x).
Matrix transposition is a special case of a more general class of permutations
called bit-permute/complement (BPC) permutations, which
in turn is a subset of the class of bit-matrix-multiply/complement
(BMMC) permutations. BMMC permutations are defined by a logN ×
logN nonsingular 0–1 matrix A and a (logN)-length 0-1 vector c. An
item with binary address x is mapped by the permutation to the binary
address given by Ax ⊕ c, where ⊕ denotes bitwise exclusive-or. BPC
permutations are the special case of BMMC permutations in which A is
a permutation matrix, that is, each row and each column of A contain
a single 1. BPC permutations include matrix transposition, bit-reversal
permutations (which arise in the FFT), vector-reversal permutations,
hypercube permutations, and matrix reblocking. Cormen et al. [120]
characterize the optimal number of I/Os needed to perform any given
BMMC permutation solely as a function of the associated matrix A,
and they give an optimal algorithm for implementing it.
Theorem 7.4 ([120]). With D disks, the number of I/Os required to
perform the BMMC permutation defined by matrix A and vector c is
� �
n
Θ 1+
D
rank(γ)
��
, (7.2)
logm
where γ is the lower-left log n × logB submatrix of A.