Generalized dominators and post-dominators

The notion of dominators is generalized to include multiple-vertex dominators in addition to single-vertex dominators. A multiple-vertex dominator of a vertex is a group of vertices that collectively dominate the vertex. Existing algorithms compute immediate single-vertex dominators, and an algorithm for computing immediate multiple-vertex dominators is presented in this paper. The immediate dominator information is expressed in the form of a directed acyclic graph referred to as the dominator DAG or the DDAG. The generalized dominator set of any vertex dominators of the vertex, can be computed from the DDAG. The single-vertex dominator information restricts the propagation of loop invariant statements and array bound checks out of loops. Generalized dominator information avoids these restrictions. In addition, it can be used to identify natural loops and improve the existing optimization algorithm for code hoisting. The dual notion of generalized post-dominators can be used for computing control dependences and automatic generation of compact test suites for programs.