Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct \(O(n^2)\)-area rectangular layouts for general contact graphs and \(O(n\log n)\)-area rectangular layouts for trees. (For trees, this is an \(O(\log n)\)-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require \(\Omega(n^2)\) (rsp., \(\Omega(n\log n)\)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.