FlexDAGs: Navigation of Directed Acyclic Graphs by Depth Contracting/Expanding, Breadth Concealing/Exposing, and Incarnation Revolving

The paper introduces FlexDAGs as an interactive-diagram presentation of Directed Acyclic Graphs (DAGs). It also describes the design of a FlexDAG navigator for such DAG presentations. Each sharing of a subDAG by N parents is disentangled via N copies (incarnations) of its atomic root, the Ith occurrence of which being labeled by a number pair “I/N” (“I of N”, where 1 ≤ I ≤ N). While the Ith incarnation is thus, tree-like, below only one parent, its actual fan-in, N, can be readily seen. In FlexDAG normal form, each shared subDAG is itself visible only as a single copy (expansion), namely at position “1/N” of its (textually) “1 to N”-ordered place-holder roots. The FlexDAG navigator provides for Cartesian-planar on-demand expansion and contraction of arbitrary focused subDAG place-holders. It allows to ‘revolve’ from a focused node labeled “I/N”, with N > 1, to all of its other incarnations labeled “J/N”, with J = I+1, ..., N, 1, ..., I-1 or J = I-1,..., 1, N, ..., I+1, bringing them into focus. Moreover, it extends certain tree operations for DAG navigation. Besides generalizing the usual tree-depth contracting to DAGs, FlexDAGs permit a generalized tree-breadth concealing, thus providing two complementary ways of focusing the relevant regions of large DAGs. FlexDAGs are exemplified with DAGs that are used for multiple inheritance in ontologicial modeling (taxonomies) and OOP (class systems). 1 Motivation and Basic Concepts Directed Acyclic Graphs (DAGs), as a graph-theoretical representation of partial orders, have been widely used in many fields of science and engineering. DAG modeling permits good compromises between expressiveness and perspicuity. For example, generalizing the usual tree structure of file directories or folders to at least a DAG structure, Web links allow any given Web page to be reused from multiple other pages. (While Web links also permit an arbitrary, cyclic graph structure, a complete loss of hierarchical discipline may endanger global user orientation.) This ‘page’ sharing aspect of the Web was anticipated, e.g., by Unix-file (soft) links and by some module-, context-, or theory-composition languages. Also for the reuse of complex data, as in Lisp-list, Java-vector, or Prolog-structure sharing, DAGs constitute an appropriate graph model. Another important DAG application is Bayesian Networks. Finally, as focused in the following, for OOP classes, frame systems, lattice-based models, and the taxonomic parts of description logics and ontologies, DAGs can represent heterarchies, which allow multiple inheritance; but by virtue of being ‘overlayed trees’ DAGs do not lose the hierarchical structuring power of trees. Thus, to extend the well-known parallelogram DAG, Figure 1 shows that square inherits from both rhombus (a = b) and rectangle (a ⊥ b), and itself inherits to even-square and odd-square. While layout generation for the latter kind of tree-like fan-out branching is easy, layout generation for the former kind of DAG-like fan-in branching has long been a challenge. As a first step towards our approach, the two overlayed or shared subtrees rooted in ∗ On leave from DFKI GmbH, Postfach 2080, D-67608 Kaiserslautern, Germany.