SHORTEST PATH TO A SEGMENT AND QUICKEST VISIBILITY QUERIES

Esther Arkin, Alon Efrat, Christian Knauer, Joseph Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, Topi Talvitie
2016 JoCG   unpublished
We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of
more » ... h may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain. 1 Introduction Finding shortest paths is a classical problem in computational geometry, and efficient algorithms are known for computing the paths both in simple polygons and polygonal domains with holes; see [36, 37] for surveys. In the query version of the problem one is given a fixed source point s in the domain, and the goal is to preprocess the domain so that the length of a shortest path from s to a query point q can be reported efficiently. The problem is solved by building the shortest path map (SPM) from s-the decomposition of the free space into cells such that for all points q within a cell the shortest s-q path is combinatorially the same, i.e., traverses the same sequence of vertices of the domain. Figure 1 shows an example of SPM. The query in the shortest path problem can be stated as Shortest path query: Given a query point q lying in the free space, how should one move, starting from s, in order to reach q as soon as possible? Queries like this arise in surveillance and security, search and rescue, aid and delivery, and various other applications of the shortest path problem. In this paper we introduce and study a related problem that has a very similar query: