Loop identification is a necessary step in loop transformations for high-performance architectures. The Tarjan intervals are single-entry, strongly connected subgraphs, so they closely reflect the loop structure of a program [Tar74]. They have been used for loop identification. In this paper we give a simple algorithm for identifying both reducible and irreducible loops using DJ graphs. Our method can be considered as a generalization of Tarjan's interval-finding algorithm, since we can identify nested intervals (or loops) even in the presence of irreducibility. i Contents 1 Introduction 1 2 Background and Notation 1 3 Reducible and Irreducible Loops 3 4 Our Algorithm 6 5 Conclusion and Related Work 10 List of Figures 1 An example of a flowgraph and its DJ graph : : : : : : : : : : : : : : : : : : : : 2 2 Examples of reducible and irreducible flowgraphs : : : : : : : : : : : : : : : : : : 4 3 An irreducible flowgraph with two irreducible loops : : : : : : : : : : : : : : : : : 6 4 ...

In this paper we propose a new framework for elimination-based exhaustive and incremental data flow analysis using DJ graphs. In this paper we give an overview of our framework. The details can be found in our two long reports [SGL95a, SG95a]. These two reports are available from our WWW URL http://www-acaps.cs.mcgill.ca/~sreedhar/pubs.html OR http://www-acaps.cs.mcgill.ca/doc/memos.html i Contents 1 Introduction 1 2 Exhaustive Data Flow Analysis 3 2.1 The Eager Elimination Method : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Correctness of Eager Elimination : : : : : : : : : : : : : : : : : : : : : : : 8 2.2 The Delayed Elimination Method : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.3 The Complexity of Our Exhaustive Elimination Methods : : : : : : : : : : : : : 11 3 Handling Irreducibility 13 4 Incremental Data Flow Analysis 13 4.1 Updating the Final Flow Equations: Non-Structural Changes : : : : : : : : : : : 14 4.2 Updating the Final Flow Equa...

In this paper we present a new approach for incremental data flow analysis based on elimination methods. Unlike previous elimination-based incremental data flow analysis, our approach can handle arbitrary non-structural and structural changes to program flowgraphs, including irreducibility. We use properties of dominance frontiers and iterated dominance frontiers for updating data flow solutions. These properties are applicable to both reducible and irreducible flowgraphs. Since we use properties of dominance frontiers (and iterated dominance frontiers) we also give a simple algorithm for updating the dominance frontier relation. The dominance frontier update algorithm is based on our previous work for updating dominator trees. i Contents 1 Introduction 1 2 Exhaustive Elimination Method: An Overview 2 3 Our Approach 5 3.1 Initial and Final Flow Equations : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3.2 Basic Steps : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...