GLB-Closures in Directed Acyclic Graphs and Their Applications

A subset S of the vertices of a directed acyclic graph is called glb-closed, if it contains the greatest lower bounds of all pairs of vertices of S. The glb-closure of S is the smallest glb-closed subset containing S. An eecient output sensitive algorithm for computing glb-closures is presented and two applications in the eld of object-oriented programming languages are discussed. 1 The problem in general Directed acyclic graphs are widely used in diierent areas of computer science. In compiler construction they can be used to identify common subexpressions or to represent type lattices in programming languages supporting subtypes. In the latter case lattice operations for determining least upper bounds or greatest lower bounds are frequently needed. In this paper the notion of closure of a subset of the vertices of a directed acyclic graph is introduced. A subset S of the vertices is called closed if it contains the greatest lower bounds of all pairs of vertices in S, i.e. S is closed with respect to taking greatest lower bounds. In order to avoid confusion with other already existing usages of the notion closure in graph theory (e.g. transitive closure) in this paper the notion glb-closed is used. The glb-closure of a subset S is the intersection of all glb-closed subsets containing S, i.e. the smallest glb-closed subset containing S. Glb-closed subsets have some interesting properties. In this paper an eecient algorithm for computing the glb-closure of a set of vertices is presented. With some preprocessing the glb-closure of a subset S can be computed in time O(e + n) where e is the number of edges in G and n is the number of vertices. Let G = (V; E) be a directed acyclic graph, V denotes the set of vertices and E the set of edges. The following notation is used throughout the paper. Let v 1 ; v 2 2 V then v 1 v 2 if v 1 = v 2 or if there exists a path in G from v 1 to v 2. Furthermore v 1 v 2 if v 1 v 2 and v 1 6 = v 2. In the following the lattice operations GLB and LUB are deened for directed graphs. The deenitions diier from that for lattices. GLB and LUB are sets of vertices ?