Chapter 14 Approximate Nearest Neighbor via Point - Location among Balls

x Today I know that everything watches, that nothing goes unseen, and that even wallpaper has a better memory than ours. It isn't God in His heaven that sees all. A kitchen chair, a coat-hanger a half-filled ash tray, or the wood replica of a woman name Niobe, can perfectly well serve as an unforgetting witness to every one of our acts. – The tin drum, Gunter Grass 14.1 Hierarchical Representation of Points In the following, it would be convenient to carry out our discussion in a more generalized settings than just low dimensional Euclidean space. In the following, we will slightly abuse notations, and use M to refer to the underlying set of points. Also, we assume that we are provided with a black box, such that given any two points x and y, we can compute the distances d M (x, y) in constant time. For example of a metric, IR 2 with the regular euclidean distance is a metric space. Another standard example is a graph G with non-negative weights ω(·) defined on the edges, and let d G (x, y) denote the shortest path (under the given weights) between any x, y ∈ V(G). It is easy to verify that d G (·, ·) is a metric. In fact, any finite metric (i.e., a metric defined over a finite set) can be represented by such a weighted graph.