Chapter 3 Well Separated Pairs Decomposition

x The fact remains that getting people right is not what living is all about anyway. It's getting them wrong that is living, getting them wrong and wrong and wrong and then, on careful reconsideration, getting them wrong again. That's how we known we're alive: we're wrong. Maybe the best thing would be to forget being right or wrong about people and just go along for the ride. But if you can do that-well, lucky you. In this chapter, we will investigate how to represent distances between points efficiently. Naturally , an explicit description of the distances between n points requires listing all the n 2 distances. Here we will show that there is a considerably more compact representation which is sufficient if all we care about are approximate distances. This representation would have many nice applications. Let P be a set of n points in IR d , and 1/4 > ε > 0 a parameter. One can represent all distances between points of P by explicitly listing the n 2 pairwise distances. Of course, the listing of the coordinates of each point gives us an alternative more compact representation (of size dn), but its not a very informative representation. We are interested in a representation that will capture the structure of the distances between the points. p q s Figure 3.1: As a concrete example, consider the three points on the right. We would like to have a representation that captures that p has similar distance to q and s, and furthermore, q and s are close together as far as p is concerned. As such, if we are interested in the closest pair among the three points, we will only check the distance between q and s, since they are the only pair (among the three) that might realize the closest pair. x This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit