A new approximation algorithm for the register allocation problem

Abstract

In this paper we study the problem of register allocation in the presence of parallel conditional branches with a given branching depth d. We start from a scheduled flow graph and the goal is to find an assignment of the variables in the flow graph to a minimum number of registers. This problem can be solved by coloring the corresponding conflict graph G=(V, E). We describe a new approximation algorithm with constant worst case rate for flow graphs with constant branching depth. The algorithm works in two steps. In the first step, the lifetimes are enlarged such that the lifetimes form one unique interval across the different possible execution paths for each variable. We prove that the conflict graph ¯G with enlarged lifetimes satisfies ω(¯G)≤(2d+1)ω(G) where ω(G) is the cardinality of a maximum clique in G. In the second step, we propose an algorithm with approximation bound (d+1)ξ(¯G) for ¯G, using its specific structure.