Approximating metrics by tree metrics Kunal Talwar Microsoft

Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with Jittat Fakcharoenphol Satish Rao Kasetsart University Thailand UC Berkeley

Metric 0 • 10 15 5 0 25 15 0 20 0 a 5 d 10 b 15 25 20 c 15 Princeton 2011

Bat. B Network design • T 1 Princeton 2011 Optical fiber

Tree metrics • Shortest path metric on a weighted tree • Simple to reason about a • Easier to design algorithms which are simple and/or fast. 5 d Princeton 2011 10 b 15 c

Bat. B Network design • T 1 Princeton 2011 Optical fiber

Question Can any metric be approximated by a tree metric? Approximately optimal solution Approximately Easy solution Princeton 2011

The cycle 1 • Shortest path metric on a cycle. 1 1 1 Princeton 2011 1

The cycle 1 • 1 1 1 Princeton 2011 1

The cycle 1 • 1 1 1 1 2 Princeton 2011 3 4 3 1 1

The cycle 1 • 1 1 1 2 2 2 Princeton 2011 2 2 2

…but Dice help 1 [Karp 89] Cut an edge at random ! 1 1 1 u 1 v

…but Dice help 1 [Karp 89] Cut an edge at random ! 1 1 • Expected stretch of any fixed edge is at most 2. 1 1 u 1 v

Probabilistic Embedding 1 1 1 1 Distortion Princeton 2011 u 1 v

Question Can any metric be probabilistically approximated by a tree metric? Approximately optimal solution (in Expectation) Approximately Easy solution Princeton 2011

Why? • Several problems are easy (or easier) on trees: Network design, Group Steiner tree, k-server, Metric labeling, Minimum communication cost spanning tree, metrical task system, Vehicle routing, etc. Princeton 2011

History • Princeton 2011

Approximating by tree metrics High level outline: 1. Hierarchically decompose the points in the metric – 2. Geometrically decreasing diameters Convert clustering into tree

Distances Increase High level outline: 1. Hierarchically decompose the points in the metric – 2. Geometrically decreasing diameters Convert clustering into tree

Bounding Distortion •

Low Diameter Decomposition • Princeton 2011

Our techniques • Techniques used in approximating 0 -extension problem by [Calinscu. Karloff-Rabani-01] • Improved algorithm and analysis used in [Fakcharoenphol-Harrelson-Rao. T. -03] Princeton 2011