  Communication lower bounds have long been established for matrix
multiplication algorithms. However, most methods of asymptotic analysis have
either ignored the constant factors or not obtained the tightest possible
values. Recent work has demonstrated that more careful analysis improves the
best known constants for some classical matrix multiplication lower bounds and
helps to identify more efficient algorithms that match the leading-order terms
in the lower bounds exactly and improve practical performance. The main result
of this work is the establishment of memory-independent communication lower
bounds with tight constants for parallel matrix multiplication. Our constants
improve on previous work in each of three cases that depend on the relative
sizes of the aspect ratios of the matrices.
