Reconstructing Generalized Staircase Polygons with Uniform Step Length

TL;DR

This paper presents efficient algorithms for reconstructing specific classes of generalized staircase polygons with uniform step lengths from their visibility graphs, advancing understanding of polygon reconstruction complexity.

Contribution

It introduces new polynomial-time algorithms for reconstructing orthogonally convex polygons and staircase histograms from visibility graphs, including fixed-parameter tractability results.

Findings

Reconstruction of orthogonally convex polygons in O(n^2m) time.

Reconstruction of staircase histograms is fixed-parameter tractable.

Polynomial-time reconstruction under certain alignment restrictions.

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.