Geodesic Spanners for Points in R3\mathbb{R}^3 amid Axis-parallel Boxes

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ amid a bounded number of obstacles. When obstacles are axis-parallel boxes, we prove that $P$ admits an $8\sqrt{3}$-spanner with $O(n\log^3 n)$ edges with respect to the geodesic distance

Walking Around Fat Obstacles

We prove that if an object O is convex and fat then, for any two points a and b on its boundary, there exists a path along the boundary, from a to b, whose length is bounded by the length of the line segment ab times some constant fi. This constant is a function of the fatness-constant and the dimension d. We prove bounds for fi and show how to efficiently find paths on the boundary of O whose lengths are within these bounds. As an application of this result, we present a method for computing short paths among convex, fat obstacles in R d by applying de Berg's method for producing a linear-size subdivision of the space. Given a start site and a destination site in the free space, a standard obstacle-avoiding "straight-line" path that is at most some multiplicative constant factor longer than the length of the segment between the sites can be computed efficiently. 1 Introduction An object O is ff-fat if for any hyperball B that does not entirely contain O and whose center ..