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We investigate the problem of memory reuse in order to reduce the memory needed to store an array variable. We develop techniques that can lead to smaller memory requirements in the synthesis of dedicated processors or to more effective use by compiled code of software-controlled scratchpad memory. Memory reuse is well-understood for allocating registers to hold scalar variables. Its extension to arrays has been studied recently for multimedia applications, for loop parallelization, and for circuit synthesis from recurrence equations. In all such studies, the introduction of modulo operations to an otherwise affine mapping (of loop or array indices to memory locations) achieves the desired reuse. We develop here a new mathematical framework, based on critical lattices, that subsumes the previous approaches and provides new insight. We first consider the set of indices that conflict, those that cannot be mapped to the same memory cell. Next, we construct the set of differences of conflicting indices. We establish a correspondence between a valid modular mapping and a strictly-admissible integer lattice-one having no nonzero element in common with the set of conflicting index differences. The memory required by an optimal modular mapping is equal to the determinant of the corresponding lattice. The memory reuse problem is thus reduced to the (still interesting and nontrivial) problem of finding a strictly admissible integer lattice-of least determinant. We then propose and analyze several practical strategies for finding strictly admissible integer lattices, either optimal or optimal up to a multiplicative factor, and, hence, memory-saving modular mappings. We explain and analyze previous approaches in terms of our new framework.