  This work studies the application of the discrete Holder-Brascamp-Lieb (HBL)
inequalities to the design of communication optimal algorithms. In particular,
it describes optimal tiling (blocking) strategies for nested loops that lack
data dependencies and exhibit linear memory access patterns. We attain known
lower bounds for communication costs by unraveling the relationship between the
HBL linear program, its dual, and tile selection. The methods used are
constructive and algorithmic. The case when all arrays have one index is
explored in depth, as a useful example in which a particularly efficient tiling
can be determined.
