Computing L1 Shortest Paths among Polygonal Obstacles in the Plane

Given a point $s$ and a set of $h$ pairwise disjoint polygonal obstacles of totally $n$ vertices in the plane, we present a new algorithm for building an $L_1$ shortest path map of size O(n) in $O(T)$ time and O(n) space such that for any query point $t$, the length of the $L_1$ shortest obstacle-avoiding path from $s$ to $t$ can be reported in $O(\log n)$ time and the actual shortest path can be found in additional time proportional to the number of edges of the path, where $T$ is the time for triangulating the free space. It is currently known that $T=O(n+h\log^{1+\epsilon}h)$ for an arbitrarily small constant $\epsilon>0$. If the triangulation can be done optimally (i.e., $T=O(n+h\log h)$), then our algorithm is optimal. Previously, the best algorithm computes such an $L_1$ shortest path map in $O(n\log n)$ time and O(n) space. Our techniques can be extended to obtain improved results for other related problems, e.g., computing the $L_1$ geodesic Voronoi diagram for a set of point sites in a polygonal domain, finding shortest paths with fixed orientations, finding approximate Euclidean shortest paths, etc.Comment: 48 pages; 19 figures; partial results appeared in ESA 201

Computing Shortest Paths among Curved Obstacles in the Plane

A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this paper, we consider the problem version on curved obstacles, commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons). Each curved edge is assumed to be of O(1) complexity. Given in the plane two points s and t and a set of $h$ pairwise disjoint splinegons with a total of $n$ vertices, we compute a shortest s-to-t path avoiding the splinegons, in $O(n+h\log^{1+\epsilon}h+k)$ time, where k is a parameter sensitive to the structures of the input splinegons and is upper-bounded by $O(h^2)$. In particular, when all splinegons are convex, $k$ is proportional to the number of common tangents in the free space (called "free common tangents") among the splinegons. We develop techniques for solving the problem on the general (non-convex) splinegon domain, which also improve several previous results. In particular, our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among $h$ pairwise disjoint convex splinegons with a total of $n$ vertices. Our algorithm runs in $O(n+h\log h+k)$ time and $O(n)$ space, where $k$ is the number of all free common tangents. Even for the special case where all splinegons are convex polygons, the previously best algorithm for this visibility problem takes $O(n+h^2\log n)$ time.Comment: 45 pages, 21 figures; to appear in TALG; an extended-abstract appeared in SoCG 201