Introduction
An
m
x
n
matrix (table) is said to be
sparse
if “many” of its elements are zero
(empty).
A matrix which is not sparse is
dense
.
The boundary between a
dense and a sparse matrix is not precisely defined.
Diagonal and tridiagonal
n
x
n
matrices are sparse since they have O(n) nonzero terms and O(n
2
) zero
terms.
Is an
n
x
n
triangular (either upper or lower) matrix sparse?
A triangular matrix
will have at least
n(n1)/2
zero terms and at most
n(n+1)/2
nonzero terms.
For
the representation schemes that we are about to examine to be competitive
over the standard twodimensional array representation, it will turn out that the
number of nonzero terms will need to be less than
n
2
/3
and in some cases less
than
n
2
/5
.
Thus, in the context of the representation schemes we are about to
see, a triangular matrix is considered to be dense rather than sparse.
Sparse matrices which are either diagonal or tridiagonal have sufficient
structure in their nonzero regions to allow fairly simple representation schemes
Sparse Matrices 
1
Advanced List Structures – Sparse Matrices (3)
Definition:
A matrix
M
is
diagonal
iff M(i, j) = 0 for i
≠
j.
Definition: A matrix
M
is
tridiagonal
iff M(i, j) = 0 for i – j > 1.
[Both of these special matrices are special cases of the more general
square band matrix in which the nonzero elements are on a band which
is centered about the main diagonal.]
Considering matrix
M
to be a 6 x 6 matrix, for purposes of an example, we have:
2
0
0
0
0
0
4
2
0
0
0
0
0
1
0
0
0
0
1
3
1
0
0
0
0
0
4
0
0
0
0
4
5
2
0
0
0
0
0
6
0
0
0
0
2
9
4
0
0
0
0
0
5
0
0
0
0
6
3
3
0
0
0
0
0
3
0
0
0
0
1
2
Diagonal Matrix
Tridiagonal Matrix
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View Full Documentto be developed whose space requirements equal the size of the nonzero
region.
We are not concerned with these simple cases, rather our focus will be
on sparse matrices with irregular or unstructured nonzero regions.
Irregular Sparse Matrices
An irregular sparse matrix has a nonzero region in which no discernable pattern
exists.
Without regularity in the nonzero region it is highly unlikely that a
standard representation, such as a twodimensional array, would provide an
efficient representation of the matrix.
On the other hand, if there is a high
degree of regularity or structure in the nonzero region, then an efficient
representation structure of the nonzero region can typically be developed using
standard linked lists that will require space equal in size to the nonzero region.
We will not examine these highly regular sparse matrices, our concern is finding
a suitable representation scheme for an irregular sparse matrix.
Consider the
following irregular 4 x 8 sparse matrix.
0
0
0
2
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 Summer '09
 Computer Science, Matrices, random access

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