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The amount of memory required by a parallel program may be spectacularly larger than the memory required by an equivalent sequential program, particularly for programs that use recursion extensively. Since most parallel programs are nondeterministic in behavior, even when computing a deterministic result, parallel memory requirements may vary from run to run, even with the same data. Hence, parallel memory requirements may be both large (relative to memory requirements of an equivalent sequential program) and unpredictable. Assume that each parallel program has an underlying sequential execution order that may be used as a basis for predicting parallel memory requirements. We propose a simple restriction that is sufficient to ensure that any program that will run in n units of memory sequentially can run in mn units of memory on m processors, using a scheduling algorithm that is always within a factor of two of being optimal with respect to time. Any program can be transformed into one that satisfies the restriction, but some potential parallelism may be lost in the transformation. Alternatively, it is possible to define a parallel programming language in which only programs satisfying the restriction can be written