Coresets of obstacles in approximating Euclidean shortest path amid convex obstacles

Given a set \cal{P} of non-intersecting convex obstacles, we find a sketch \Omega of \cal{P} that helps in finding an obstacle avoiding approximate Euclidean shortest path between two given points amid \mathcal{P} efficiently. For both \mathbb{R}^2 and \mathbb{R}^3, we devise algorithms to compute a (1+\epsilon)-approximate shortest path when two points s and t are given a priori with the polygonal domain. Further, in \mathbb{R}^2, we devise an algorithm to preprocess obstacles to output a (2+\epsilon)-approximate distance between any given pair of query points. In \mathbb{R}^2, the data structures constructed at the end of the preprocessing are improving the space complexity of the known algorithms.