Complete register allocation problems

The search for efficient algorithms for register allocation dates back to the time of the first Fortran compiler for the IBM 704. Since then, many variants of the problem have been considered; depending on two factors: (1) the particular model for registers, and (2) the definition of the term "computation of a program" e.g. Whether values may be computed more than once. We will show that several variants of the register allocation problem for straight line programs are polynomial complete. In particular we consider, (1) the case when each value is computed exactly once, and (2) the case when values may be recomputed as necessary. The completeness of the third problem considered is surprising. A straight line program starts with a set of initial values, and computes intermediate and final values. Suppose, for each value, the register that value must be computed into is preassigned. Then, (3) the problem of determining if there is a computation of the straight line program, that computes values into the assigned registers, is polynomial complete.