On Sparse Evaluation Representations

The Sparse Evaluation Graph has emerged over the past several years as an intermediate representation that captures the data ow information in a program compactly and helps perform data ow analysis e ciently. The contributions of this paper1 are three-fold: We present a linear time algorithm for constructing a variant of the Sparse Evaluation Graph for any data ow analysis problem. Our algorithm has two advantages over previous algorithms for constructing Sparse Evaluation Graphs. First, it is simpler to understand and implement. Second, our algorithm generates a more compact representation than the one generated by previous algorithms. (Our algorithm is also as e cient as the most e cient known algorithm for the problem.) We present a formal de nition of an equivalent ow graph, which attempts to capture the goals of sparse evaluation. We present a quadratic algorithm for constructing an equivalent ow graph consisting of the minimumnumber of vertices possible. We show that the problem of constructing an equivalent ow graph consisting of the minimum number of vertices and edges is NP-hard. We generalize the notion of an equivalent ow graph to that of a partially equivalent ow graph, an even more compact representation, utilizing the fact that the data ow solution is not required at every node of the controlow graph. We also present an e cient linear time algorithm for constructing a partially equivalent ow graph.