Near-Linear Algorithms for Visibility Graphs over a 1.5-Dimensional Terrain

We present several near-linear algorithms for problems involving visibility over a 1.5-dimensional terrain. Concretely, we have a 1.5-dimensional terrain T, i.e., a bounded x-monotone polygonal path in the plane, with n vertices, and a set P of m points that lie on or above T. The visibility graph VG(P,T) is the graph with P as its vertex set and {(p,q) | p and q are visible to each other} as its edge set. We present algorithms that perform BFS and DFS on VG(P,T), which run in O(nlog n + mlog³(m+n)) time. We also consider three optimization problems, in which P is a set of points on T, and we erect a vertical tower of height h at each p ∈ P. In the first problem, called the reverse shortest path problem, we are given two points s, t ∈ P, and an integer k, and wish to find the smallest height h^* for which VG(P(h^*),T) contains a path from s to t of at most k edges, where P(h^*) is the set of the tips of the towers of height h^* erected at the points of P. In the second problem we wish to find the smallest height h^* for which VG(P(h^*),T) contains a cycle, and in the third problem we wish to find the smallest height h^* for which VG(P(h^*),T) is nonempty; we refer to that problem as "Seeing the most without being seen". We present algorithms for the first two problems that run in O^*((m+n)^{6/5}) time, where the O^*(⋅) notation hides subpolynomial factors. The third problem can be solved by a faster algorithm, which runs in O((n+m)log³ (m+n)) time.