Lambda-Lifting in Quadratic Time

Abstract

Lambda-lifting is a program transformation used in compilers and in partial evaluators and that operates in cubic time. In this article, we show how to reduce this complexity to quadratic time, and we present a flow-sensitive lambda-lifter that also works in quadratic time.

Lambda-lifting transforms a block-structured program into a set of recursive equations, one for each local function in the source program. Each equation carries extra parameters to account for the free variables of the corresponding local function and of all its callees. It is the search for these extra parameters that yields the cubic factor in the traditional formulation of lambda-lifting, which is due to Johnsson. This search is carried out by a transitive closure.

Instead, we partition the call graph of the source program into strongly connected components, based on the simple observation that all functions in each component need the same extra parameters and thus a transitive closure is not needed. We therefore simplify the search for extra parameters by treating each strongly connected component instead of each function as a unit, thereby reducing the time complexity of lambda-lifting from O(n^3 log n) to O(n^2 log n), where n is the size of the program.

Since a lambda-lifter can output programs of size O(n^2), we believe that our algorithm is close to optimal.