A few more words about sparse matrices. In practice, sparse matrices are used for one of two reasons: To safe memory or to speed up computation. Hash based sparse matrices are neither the smallest possible matrix representation nor the fastest. They implement a reasonable trade-off between performance and memory: Very good average performance on get/set operations at quite small memory footprint. However, they are not particularly suited for special-purpose algorithms exploiting explicit knowledge about what regions are zero and non-zero. For example, sparse linear algebraic matrix multiplies, inversions, etc. better work on other sparse matrix representations like, for example, Harwell-Boeing. Harwell-Boeing also has smaller memory footprint. However, those alternative sparse matrix representations are really only usable for special purposes, because their get/set performance is typically very bad. In contrast, hash based sparse matrices are more generally applicable data structures.

Finally note, that some algorithms exploiting sparsity can be expressed in a generic manner, without needing to know or dictate a special internal storage format. For example, in many linear algebraic operations (like the matrix multiply) the dot product is in the inner-most loop of a cubic or quadratic loop, where one operand of the dot product "changes slowly". Detecting sparsity in the blocked "slow changing" operand and using a quick generic dot product algorithm summing only non-zero cells can drastically improve performance without needing to resort to special storage formats. Imagine a 500 x 500 `DenseDoubleMatrix2D` or `SparseDoubleMatrix2D` which is in fact populated with only one (or few) non-zero cells per row. The innermost loop of the cubic matrix multiply is reduced from 500 steps to 1 step, resulting in an algorithm that in benchmarks runs about 50 times quicker (up to 500 "virtual" Mflops on a now outdated processor Pentium 200Mhz, running NT, SunJDK1.2.2, java -classic, `DenseDoubleMatrix2D`. The theoretical speedup of 500 cannot be achieved). Because the performance overhead of sparsity detection is negligible (some 5%), this is the way the linear algebraic matrix-matrix and matrix-vector multiplications of this toolkit are implemented.

To summarize, generic algorithms can often detect and exploit sparsity with insignificant overhead, without needing to know or dictate a special matrix storage format.