Combinatorics of Beacon Routing and Coverage

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.