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Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with Jittat Fakcharoenphol Kasetsart University Thailand Satish Rao UC Berkeley
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Metric 010155 02515 020 0 10 20 5 25 15 a d c b Princeton 2011
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BatB Network design T1 Optical fiber Princeton 2011
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Tree metrics Shortest path metric on a weighted tree Simple to reason about Easier to design algorithms which are simple and/or fast. 10 5 15 a d c b Princeton 2011
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BatB Network design T1 Optical fiber Princeton 2011
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BatB Network design T1 Optical fiber Princeton 2011
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Question Can any metric be approximated by a tree metric? Approximately Easy solution Approximately optimal solution Princeton 2011
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The cycle Shortest path metric on a cycle. 1 1 1 1 1 11 1 Princeton 2011
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The cycle 1 1 1 1 11 1 Princeton 2011
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The cycle 1 1 1 1 11 1 1 2 3 1 1 4 3 Princeton 2011
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The cycle 1 1 1 1 11 1 2 2 2 2 2 2 2 2 Princeton 2011
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[Karp 89] Cut an edge at random ! …but Dice help 1 1 1 1 1 11 1 u v
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[Karp 89] Cut an edge at random ! Expected stretch of any fixed edge is at most 2. 1 1 1 1 1 11 1 u v
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Probabilistic Embedding 1 1 1 1 1 11 1 u v Distortion Princeton 2011
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Question Can any metric be probabilistically approximated by a tree metric? Approximately Easy solution Approximately optimal solution (in Expectation) Princeton 2011
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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
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History Princeton 2011
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Approximating by tree metrics High level outline: 1.Hierarchically decompose the points in the metric –Geometrically decreasing diameters 2.Convert clustering into tree
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Distances Increase High level outline: 1.Hierarchically decompose the points in the metric –Geometrically decreasing diameters 2.Convert clustering into tree
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Bounding Distortion
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Low Diameter Decomposition Princeton 2011
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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
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Decomposition algorithm Princeton 2011
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Decomposition algorithm Princeton 2011
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Decomposition algorithm Princeton 2011
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Decomposition algorithm Princeton 2011
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Decomposition algorithm Princeton 2011
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Decomposition algorithm Princeton 2011
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Bounding Distortion Princeton 2011
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The blaming game Princeton 2011
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Thus… Princeton 2011
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Few terminals case Princeton 2011
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Remarks Princeton 2011
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More remarks Princeton 2011
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BatB Network Design Princeton 2011
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Summary Princeton 2011
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