Approximate Shortest Paths in Polygons with Violations

Abstract

We study the problem of computing shortest k-violation path problem on polygons. Let P be a simple polygon in $\mathbb {R}^2$ with n vertices and let s, t be a pair of points in P. Let int(P) represent the interior of P. In other words, $int(P) = P \setminus \varDelta (P)$, where $\varDelta (P)$ is the boundary of P. Let $\tilde{P}=\mathbb {R}^2 \setminus int(P)$ represent the exterior of P. For an integer $k \ge 0$, the problem of k-violation shortest path in P is the problem of computing the shortest path from s to t in P, such that at most k path segments are allowed to be in $\tilde{P}$. The path segments are not allowed to bend in $\tilde{P}$. For any k, we present a $(1+\epsilon )$ factor approximation algorithm for the problem, that runs in $O(n^2 \sigma ^2 k\log n^2 \sigma ^2 + n^2 \sigma ^2 k)$ time. Here $\sigma =O( \log _\delta (\frac{|L|}{r}))$ and $\delta$, L, r are geometric parameters.