Theory of placement by numDAG related with single-sequence, SP, BSG, and O-tree

After BSG and sequence-pair (SP) of the placement codes, a decade has passed with various proposals of new codes and ideas for applications, especially in VLSI physical design. Different in appearance, they are actually proposing generators of ABLR-relations, i.e. above, below, left-of, or right-of relations between two rectangles on a plane to be non-overlapping, together with methodologies how dimensions and peripheral constraints are integrated. This paper proposes another yet simplest generator numDAG which is simply a directed acyclic graph with vertices labelled with distinct numbers. Assuming each vertex representing a rectangle on a plane, an edge (i, j) is featured to imply the property "i is left-of j" or "i is above j" according to the edge being incremental or decremental, respectively, with respect to the numbers of end vertices. To demonstrate that the numDAG is the ABLR-relation generating system hierarchically above existing systems, we relate it with single-sequence, H-and V-constraint graph pair, SP, BSG, and O-tree. It is a future problem to find any practical merit by this idea but the contribution is believed in providing a base to the unified theory of constraint-driven placement. Several new ideas and problems are included