Algorithms and Data Structures for External Memory

7
Matrix and Grid Computations
7.1 Matrix Operations
Dense matrices are generally represented in memory in row-major or
column-major order. For certain operations such as matrix addition,
both representations work well. However, for standard matrix multiplication
(using only semiring operations) and LU decomposition, a better
representation is to block the matrix into square √ B × √ B submatrices,
which gives the upper bound of the following theorem:
Theorem 7.1 ([199, 306, 345, 357]). The number of I/Os required
for standard matrix multiplication of two K × K matrices or to compute
the LU factorization of a K × K matrix is
�
K
Θ
3
min{K, √ �
.
M }DB
The lower bound follows from the related pebbling lower bound by
Savage and Vitter [306] for the case D = 1 and then dividing by D.
Hong and Kung [199] and Nodine et al. [272] give optimal EM algorithms
for iterative grid computations, and Leiserson et al. [233] reduce
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