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

2. Metric 010155 02515 020 0 10 20 5 25 15 a d c b Princeton 2011

3. BatB Network design T1 Optical fiber Princeton 2011

4. 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

5. BatB Network design T1 Optical fiber Princeton 2011

6. BatB Network design T1 Optical fiber Princeton 2011

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

8. The cycle Shortest path metric on a cycle. 1 1 1 1 1 11 1 Princeton 2011

9. The cycle 1 1 1 1 11 1 Princeton 2011

10. The cycle 1 1 1 1 11 1 1 2 3 1 1 4 3 Princeton 2011

11. The cycle 1 1 1 1 11 1 2 2 2 2 2 2 2 2 Princeton 2011

12. [Karp 89] Cut an edge at random ! …but Dice help 1 1 1 1 1 11 1 u v

13. [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

14. Probabilistic Embedding 1 1 1 1 1 11 1 u v Distortion Princeton 2011

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

16. 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

17. History Princeton 2011

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

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

20. Bounding Distortion

21. Low Diameter Decomposition Princeton 2011

22. 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

23. Decomposition algorithm Princeton 2011

24. Decomposition algorithm Princeton 2011

25. Decomposition algorithm Princeton 2011

26. Decomposition algorithm Princeton 2011

27. Decomposition algorithm Princeton 2011

28. Decomposition algorithm Princeton 2011

29. Bounding Distortion Princeton 2011

30. The blaming game Princeton 2011

35. Thus… Princeton 2011

36. Few terminals case Princeton 2011

37. Remarks Princeton 2011

38. More remarks Princeton 2011

39. BatB Network Design Princeton 2011

40. Summary Princeton 2011