Recursively-Regular Subdivisions and Applications

We generalize regular subdivisions (polyhedral complexes resulting from the projection of the lower faces of a polyhedron) introducing the class of recursively-regular subdivisions. Informally speaking, a recursively-regular subdivision is a subdivision that can be obtained by splitting some faces of a regular subdivision by other regular subdivisions (and continue recursively). We also define the \emph{finest regular coarsening} and the \emph{regularity tree} of a polyhedral complex. We prove that recursively-regular subdivisions are not necessarily connected by flips and that they are acyclic with respect to the in-front relation. We show that the finest regular coarsening of a subdivision can be efficiently computed, and that whether a subdivision is recursively regular can be efficiently decided. As an application, we also extend a theorem known since 1981 on illuminating space by cones and present connections of recursive regularity to tensegrity theory and graph-embedding problems.Comment: 39 pages, 14 figure

Floodlight Illumination of Infinite Wedges

The floodlight illumination problem asks whether there exists a one-to-one placement of n floodlights illuminating infinite wedges of angles α1,..., αn at n sites p1,..., pn in a plane such that a given infinite wedge W of angle θ located at point q is completely illuminated by the floodlights. We prove that this problem is NP-hard, closing an open problem from 2001 [6]. In fact, we show that the problem is NP-complete even when αi = α for all 1 ≤ i ≤ n (the uniform case) and θ = Pn i=1 αi (the tight case). On the positive side, we describe sufficient conditions on the sites of floodlights for which there are efficient algorithms to find an illumination. We discuss various approximate solutions and show that computing any finite approximation is NP-hard while ε-angle approximations can be obtained efficiently.

Floodlight Illumination of Infinite Wedges

The floodlight illumination problem asks whether there exists a one-to-one placement of n floodlights illuminating infinite wedges of angles α1,..., αn at n sites p1,..., pn in a plane such that a given infinite wedge W of angle θ located at point q is completely illuminated by the floodlights. We prove that this problem is NPhard, closing an open problem posed by Demaine and O’Rourke (CCCG 2001). In fact, we show that the problem is NP-complete even when αi = α for all 1 ≤ i ≤ n (the uniform case) and θ = � n i=1 αi (the tight case). Key words: illumination, art gallery problem, floodlights, NP-completeness