Approximation Algorithms Fall Semester , 2003 Lecture 13 : October 20 , 2003

In today's lecture, we will present a randomized algorithm that embeds any arbitrary metric into a dominating tree metric such that each edge is distorted, in expectation, in the worst case by a factor of O(log n) and such that each edge is not contracted. Recall that a metric on graph G = (V, E) is a function d : E → R. One can define a metric M = (X, d) on an arbitrary set of vertices X with a distance function d such that the following properties hold: A tree metric is the shortest path metric of a weighted tree. In other words, d(i, j) is the length of the unique shortest path between node i and node j. A metric (V , d) is said to dominate metric (V, d) if ∀u, v ∈ V, d Let ∆ denote the diameter of the metric (V, d). WLOG, ∆ = 2 δ. Let S be a family of metrics over V , and D be a distribution of S. We say that (S, D) α-probabilistically approximates a metric (V, d) if every metric in S dominates d and for every pair of vertices (u, v) ∈ V , E d ∈(S,D) [d (u, v)] ≤ α · d(u, v). Formally, we're interested in O(log n)-probabilistically approximating an arbitrary metric (V, d) by a distribution over tree metrics. For a parameter r, an r-cut decomposition of (V, d) is a partitioning of V into clusters, each centered around a node and having radius at most r. Thus each cluster will have diameter at most 2r.