5339 - Algorithm design under a geometric lens , Spring 2014 , CSE , OSU Homework 1

• For any x, y ∈ X, we have ρ(x, y) = 0 if and only if x = y. • For any x, y ∈ X, we have ρ(x, y) = ρ(y, x). • For any x, y, z ∈ X, we have ρ(x, y) ≤ ρ(x, z) + ρ(z, y). For any r ≥ 0, a r-partition of M is a partition of X, such that every cluster has diameter at most r. That is, for any cluster C in the partition, and for any x, y ∈ C, we have ρ(x, y) ≤ r. Let β > 0. A (β, r)-Lipschitz partition of (X, ρ) is a distribution D over r-partitions of M , such that for any x, y ∈ X: