Constrained floorplans in 2D and 3D

A rectilinear dual of a plane graph refers to a partition of a rectangular area into nonoverlapping rectilinear polygonal modules, where each module corresponds to a vertex such that two modules have a side-contact iff their corresponding vertices are adjacent. An interesting observation is that most algorithms yielding low polygonal complexity rectilinear duals require the use of ⊤-shape modules or their extensions. To justify the intuition that ⊤-shape modules are more powerful than other 8-sided modules, we show the optimum polygonal complexity of ⊤-free rectilinear duals to be exactly 12. Our construction uses only monotone staircase modules. We continue this line of research by studying polygonal complexity and area-universality of rectilinear duals using modules of constrained shapes.Motivated by the design challenges in 3D IC technology, we also study the generalization of rectilinear duals in three dimensions by allowing each module to be an orthogonal polyhedron. Such a generalization to three dimensions opens the door for non-planar graphs to be accommodated in floorplan design. In particular, we prove that all chordal graphs admit 3D floorplans, which parallels the well-known result that all maximal plane graphs admit rectilinear duals.