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 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(n2)-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 (respectively, trees) that require Ω(n2) (respectively, Ω(n log n))area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts, using the related concept of rectangular duals. A corollary to our results relates the class of graphs that admit rectangular layouts to rectangle-of-influence drawings.