Let P be a polygon with r > 0 reflex vertices and possibly with holes and islands. A subsuming polygon of P is a polygon P such that P ⊆ P , each connected component R of P is a subset of a distinct connected component R of P , and the reflex corners of R coincide with those of R . A subsuming chain of P is a minimal path on the boundary of P whose two end edges coincide with two edges of P . Aichholzer et al. proved that every polygon P has a subsuming polygon with O(r) vertices, and posed an open problem to determine the computational complexity of computing subsuming polygons with the minimum number of convex vertices.We prove that the problem of computing an optimal subsuming polygon is NP-complete, but the complexity remains open for simple polygons (i.e., polygons without holes). Our NPhardness result holds even when the subsuming chains are restricted to have constant length and lie on the arrangement of lines determined by the edges of the input polygon. We show that this restriction makes the problem polynomial-time solvable for simple polygons.
Let P be a simple polygon with n vertices, and let q ∈ P be a point in P . Let k ∈ {0, . . . , n − 1}. A point p ∈ P is k-visible from q if and only if the line segment pq crosses the boundary of P at most k times. The k-visibility region of q in P is the set of all points that are k-visible from q. We study the problem of computing the k-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of O(n) words, each with Ω(log n) bits. The algorithm can read and write O(s) additional words of workspace, where s ∈ N is a parameter of the model. The output is written to a write-only stream.Given a simple polygon P with n vertices and a point q ∈ P , we present an algorithm that reports the k-visibility region of q in P in O(cn/s + c log s + min{ k/s n, n log log s n}) expected time using O(s) words of workspace. Here, c ∈ {1, . . . , n} is the number of critical vertices of P for q where the k-visibility region of q may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.
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