II. Notes on Data Structuring * - Cornell University

154 c.A.R. HOARE
to ensure that the sizes and location of the sequences and sections be chosen
to correspond closely with the access characteristics of the storage medium.
10.1.4. Locally Dense Representation
A special case of a sparse array encountered in numerical computer appli-
cations is the sparse matrix. Quite frequently a sparse matrix can be split
into submatrices, only a few of which contain significant non-zero entries.
In this case, the matrix may be said to be locally dense, and should be
represented and processed in a manner which takes advantage of this fact.
One method of achieving this is to store with each significant submatrix
its position and size, and to represent the whole matrix as a table or sequence
off such submatrices, where each submatrix is stored contiguously in the
usual way, using multiplicative address calculation. However, the sub-
matrices will in general be of different sizes, and if the size varies during the
processing of the matrix, the problems will be quite severe. A possible way of
dealing with sparse matrices is to split them into submatrices of standard
size, say sixteen by sixteen, and set up a table of pointers to each of these
submatrices. A submatrix that is wholly zero is represented by a null pointer
and occupies no additional storage; otherwise, the submatrix is stored in the
usual way, using the following method of address calculation.
Each access to the array involves first "interleaving" the bit values of the
two subscripts, so that the least significant part of the result contains the least
significant part of both subscripts. The more significant part of the result is
then used to consult the table of addresses, to locate the desired submatrix,
and the less significant part to find the position off the required element
within the submatrix. This technique of interleaving subscripts may on
some machines be more efficient than general multiplication. If some of the
submatrices have to be held on backing store, this method of address calcu-
lation is particularly recommended, since it is equally efficient at processing
the matrix by rows as by columns; and the method can then be recommended
for all large arrays, whether sparse or not, particularly on a paged computer.
The inventor of this method is Professor E. W. Dijkstra.
10.1.5. Grid Representation
The phenomenon of cross-classification of files causes as many problems in a
computer as it does in real life. It is usually solved by standardising on one
of the classifications which is most convenient, and accepting the extra cost of
processing in accordance with the other classification, even if this involves
resorting the file. Thus the sparse mapping
sparse array (i:D1 ; j" DE) of R
is represented as:
sparse array D1 of (sparse array D2 of R)