3 research outputs found
A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon
The problem of vertex guarding a simple polygon was first studied by Subir K.
Ghosh (1987), who presented a polynomial-time -approximation
algorithm for placing as few guards as possible at vertices of a simple -gon
, such that every point in is visible to at least one of the guards.
Ghosh also conjectured that this problem admits a polynomial-time algorithm
with constant approximation ratio. Due to the centrality of guarding problems
in the field of computational geometry, much effort has been invested
throughout the years in trying to resolve this conjecture. Despite some
progress (surveyed below), the conjecture remains unresolved to date. In this
paper, we confirm the conjecture for the important case of weakly visible
polygons, by presenting a -approximation algorithm for
guarding such a polygon using vertex guards. A simple polygon is weakly
visible if it has an edge , such that every point in is visible from
some point on . We also present a -approximation algorithm
for guarding a weakly visible polygon , where guards may be placed anywhere
on 's boundary (except in the interior of the edge ). Finally, we present
a -approximation algorithm for vertex guarding a polygon that is weakly
visible from a chord, given a subset of 's vertices that guards 's
boundary whose size is bounded by times the size of a minimum such subset.
Our algorithms are based on an in-depth analysis of the geometric properties
of the regions that remain unguarded after placing guards at the vertices to
guard the polygon's boundary. It is plausible that our results will enable
Bhattacharya et al. to complete their grand attempt to prove the original
conjecture, as their approach is based on partitioning the underlying simple
polygon into a hierarchy of weakly visible polygons
Finding all weakly-visible chords of a polygon in linear time, Manuscript
Abstract. A chord of a simple polygon P is weaidy-uiaibleffevery point on P is visible from some point on the chord. We give an optimal lineex-time algorithm which computea a/l weakly-visible chords of a es polygon P with n vertices.
Nordic Journal of Computing 1(1994), 433–457. FINDING ALL WEAKLY-VISIBLE CHORDS OF A POLYGON IN LINEAR TIME
Abstract. A chord of a simple polygon P is weakly-visible if every point on P is visible from some point on the chord. We give an optimal linear-time algorithm which computes all weakly-visible chords of a simple polygon P with n vertices. Previous algorithms for the problem run in O(n log n) time, and only compute a single weakly-visible chord, if one exists. CR Classification: F.2.2 Key words: computational geometry, visibility, polygons, chord