Reconstructing Generalized Staircase Polygons with Uniform Step Length

Abstract: Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an $O(n^2m)$-time reconstruction algorithm for orthogonally convex polygons, where $n$ and $m$ are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time $O(n^2m)$ under reasonable alignment restrictions.