Let @math be a point in a polygonal domain @math of @math holes and @math vertices. We consider a quickest visibility query problem. Given a query point @math in @math , the goal is to find a shortest path in @math to move from @math to see @math as quickly as possible. Previously, (SoCG 2015) built a data structure of size @math that can answer each query in @math time, where @math is the inverse Ackermann function and @math is the size of the visibility polygon of @math in @math (and @math can be @math in the worst case). In this paper, we present a new data structure of size @math that can answer each query in @math time. Our result improves the previous work when @math is relatively small. In particular, if @math is a constant, then our result even matches the best result for the simple polygon case (i.e., @math ), which is optimal. As a by-product, we also have a new algorithm for a shortest-path-to-segment query problem. Given a query line segment @math in @math , the query seeks a shortest path from @math to all points of @math . Previously, gave a data structure of size @math that can answer each query in @math time, and another data structure of size @math with @math query time. We present a data structure of size @math with query time @math , which also favors small values of @math and is optimal when @math .
@cite_28: We address the point-to-face approximate shortest path problem in R: Given a set of polyhedral obstacles with a total of n vertices, a source point s, an obstacle face f, and a real positive parameter epsilon, compute a path from s to f that avoids the interior of the obstacles and has length at most (1+epsilon) times the length of the shortest obstacle avoiding path from s to f. We present three approximation algorithms that take by extending three well-known "point-to-point" shortest path algorithms.
Similar in spirit to the point-to-segment'' shortest path problem, Cheung and Daescu @cite_28 considered a point-to-face'' shortest path problem in 3D and approximation algorithms were given for the problem.