We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

We consider beacons, an analog of geographical greedy routing, motivated by sensor network applications. A beacon b is a point object that can be activated to create a 'magnetic pull' towards itself everywhere in a polygonal domain P. We explore the properties of beacons and their effect on points in polygons, as well as demonstrate polynomial-time algorithms to compute a variety of structures defined by the action of beacons on P. We establish a polynomial-time algorithm for routing from a point s to a point t using a discrete set of candidate beacons, as well as a 2-approximation and a PTAS for routing between beacons placed without restriction in P.

/pdf/beacon-based-algorithms-for-geometric-routing-15ncoam1l7.pdf

Beacon-Based algorithms for geometric routing

Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s ⊆ P as its trajectory. The camera can see a point p ∈ P if there exists a point q ∈ s such that pq is a line segment normal to s that is completely contained in P. In the minimum-cardinality sliding cameras problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S) while in the minimum-length sliding cameras problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel.

/pdf/guarding-orthogonal-art-galleries-using-sliding-cameras-3c1mo84mvk.pdf

Guarding Orthogonal Art Galleries Using Sliding Cameras: Algorithmic and Hardness Results

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that $\lfloor\frac{m+1}{3} \rfloor$ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When $m \leq\frac{3n}{4} - 4$, the sufficiency condition yields a strictly lower guard number than $\lfloor\frac{n}{4}\rfloor$, given by the art gallery theorem for orthogonal polygons.

/pdf/the-art-gallery-theorem-for-polyominoes-4947y6db0z.pdf

The Art Gallery Theorem for Polyominoes

Let P be an orthogonal polygon with n vertices. A sliding camera travels back and forth along an orthogonal line segment s ⊆ P corresponding to its trajectory. The camera sees a point p ∈ P if there is a point q ∈ s such that p q ‾ is a line segment normal to s that is completely contained in P. In the Minimum-Cardinality Sliding Cameras (MCSC) problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S), while in the Minimum-Length Sliding Cameras (MLSC) problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we answer questions posed by Katz and Morgenstern (2011) by presenting the following results: (i) the MLSC problem is polynomially tractable even for orthogonal polygons with holes, (ii) the MCSC problem is NP -complete when P is allowed to have holes, and (iii) an O ( n 3 log ⁡ n ) -time 2-approximation algorithm for the MCSC problem on [NE]-star-shaped orthogonal polygons with n vertices (similarly, [NW]-, [SE]-, or [SW]-star-shaped orthogonal polygons).

Guarding orthogonal art galleries with sliding cameras

We consider combinatorial problems motivated by sensor networks for beacon-based point-to-point routing and covering. A beacon b is a point that can be activated to effect a ‘magnetic pull’ toward itself everywhere in a polygonal domain P . The routing problem asks how many beacons are required to route between any pair of points in a polygonal domain P . In simple polygons with n vertices we show that ⌊ n 2 ⌋ − 1 beacons are sometimes necessary and always sufficient. In polygons with h holes, we establish that ⌊ n 2 ⌋ −h−1 beacons are sometimes necessary while ⌊ n 2 ⌋ + h− 1 beacons are always sufficient. Loose bounds for simple orthogonal polygons are also shown. We consider art gallery problems where beacons function as guards. Loose bounds are given for covering simple polygons, polygons with holes and simple orthogonal polygons.

/pdf/combinatorics-of-beacon-routing-and-coverage-57c98zmm3d.pdf

Combinatorics of beacon routing and coverage

/pdf/combinatorics-of-beacon-based-routing-and-coverage-2a5ryhdk9i.pdf

Combinatorics of beacon-based routing and coverage

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P; in particular, we show that floor((m+1)/3) point guards are always sufficient and sometimes necessary to cover an m-polyomino. When m d 3n/4 - 4, the point guard sufficiency condition yields a strictly lower guard number than floor(n/4), given by the art gallery theorem for orthogonal polygons. When pixels behave themselves like guards (pixel guards), we prove that floor(3m/11) + 1 guards are sufficient and sometimes necessary to cover an m-polyomino. We also study the algorithmic complexity of computing optimal guard sets for polyominoes. We prove that determining the guard number of a given m-polyomino is NP-hard. We provide polynomial-time algorithms to solve exactly some special cases in which the polyomino is "thin".

Justin Iwerks

Papers

Beacon-Based algorithms for geometric routing

The Art Gallery Theorem for Polyominoes

Combinatorics of beacon routing and coverage

Combinatorics of beacon-based routing and coverage

Guarding polyominoes