Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with Jittat Fakcharoenphol Kasetsart University Thailand Satish Rao UC Berkeley
Metric a d c b Princeton 2011
BatB Network design T1 Optical fiber Princeton 2011
Tree metrics Shortest path metric on a weighted tree Simple to reason about Easier to design algorithms which are simple and/or fast a d c b Princeton 2011
BatB Network design T1 Optical fiber Princeton 2011
BatB Network design T1 Optical fiber Princeton 2011
Question Can any metric be approximated by a tree metric? Approximately Easy solution Approximately optimal solution Princeton 2011
The cycle Shortest path metric on a cycle Princeton 2011
The cycle Princeton 2011
The cycle Princeton 2011
The cycle Princeton 2011
[Karp 89] Cut an edge at random ! …but Dice help u v
[Karp 89] Cut an edge at random ! Expected stretch of any fixed edge is at most u v
Probabilistic Embedding u v Distortion Princeton 2011
Question Can any metric be probabilistically approximated by a tree metric? Approximately Easy solution Approximately optimal solution (in Expectation) 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 –Geometrically decreasing diameters 2.Convert clustering into tree
Distances Increase High level outline: 1.Hierarchically decompose the points in the metric –Geometrically decreasing diameters 2.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
Decomposition algorithm Princeton 2011
Decomposition algorithm Princeton 2011
Decomposition algorithm Princeton 2011
Decomposition algorithm Princeton 2011
Decomposition algorithm Princeton 2011
Decomposition algorithm Princeton 2011
Bounding Distortion Princeton 2011
The blaming game Princeton 2011
Thus… Princeton 2011
Few terminals case Princeton 2011
Remarks Princeton 2011
More remarks Princeton 2011
BatB Network Design Princeton 2011
Summary Princeton 2011