Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

A Nearly Optimal Algorithm for covering the interior of an Art Gallery

The problem of locating visual sensors can be often modeled as 2D Art Gallery problems. In particular, tasks such as surveillance require observing the interior of a polygonal environment (interior covering, IC), while for inspection or image based rendering observing the boundary (edge covering, EC) is sufficient. Both problems are NP-hard, and no technique is known for transforming one problem into the other. Recently, an incremental algorithm for EC has been proposed, and its near-optimality has been demonstrated experimentally. In this paper we show that, with some modification, the algorithm is nearly optimal also for IC. The algorithm has been implemented and tested over several hundreds of random polygons with and without holes. The cardinality of the solutions provided is very near to, or coincident with, a polygon-specific lower bound, and then suboptimal or optimal. In addition, our algorithm has been compared, for all the test polygons, with recent heuristic sensor location algorithms. In all cases, the cardinality of the set of guards provided by our algorithm was less than or equal to that of the set computed by the other algorithms. An enhanced version of the algorithm, also taking into account range and incidence constraints, has also been implemente

A Tight Bound for Edge Guards in Monotone Polygons

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