All Structured Programs Have Small Tree Width and Good Register Allocation

Abstract

The register allocation problem for an imperative program is often modeled as the coloring problem of the interference graph of the control-flow graph of the program. The interference graph of a flow graph G is the intersection graph of some connected subgraphs of G . These connected subgraphs represent the lives, or life times, of variables, so the coloring problem models that two variables with overlapping life times should be in different registers. For general programs with unrestricted gotos, the interference graph can be any graph, and hence we cannot in general color within a factor O ( n ε ) from optimality unless NP=P. It is shown that if a graph has tree width k , we can efficiently color any intersection graph of connected subgraphs within a factor (⌊ k /2⌋+1) from optimality. Moreover, it is shown that structured (≡goto-free) programs, including, for example, short circuit evaluations and multiple exits from loops, have tree width at most 6. Thus, for every structured program, we can do register allocation efficiently within a factor 4 from optimality, regardless of how many registers are needed. The bounded tree decomposition may be derived directly from the parsing of a structured program, and it implies that the many techniques for bounded tree width may now be applied in compiler optimization, solving problems in linear time that are NP-hard, or even P-space hard, for general graphs.