Chapter 22

x Many things went on at Unseen University and, regrettably, teaching had to be one of them. The faculty had long ago confronted this fact and had perfected various devices for avoiding it. But this was perfectly all right because, to be fair, so had the students. In this chapter, we will introduce of coreset. Intuitively, a coreset is a small subset of the data that captures the key features that we care about. Surprisingly, in many cases such small coresets can be computed quickly and such facilitate fast approximation algorithms for various geometric optimization problems. Definition 22.1.1 Let P be a set of points in IR d. For a vector v ∈ IR d , such that v 0, let denote the directional width of P in the direction of v. The directional width has several interesting properties. (A) Its translation invariant. Indeed, for any vector s in IR d , we have that ω(v, s + P) = ω(v, P). (B) The directional width is a linear function as far as scaling; that is, for any constant c ≥ 0, we have that ω(v, cP) = c Pro jWidthXv, P, where cP denotes the scaling of P by c. (C) For any set P ⊆ IR d , we have that ω(v, P) = ω(v, CH(P)) , where CH(P) denotes the convex-hull of P. x This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit