1. Tight Bounds for Online Vector Bin Packing Ilan Cohen Joint work with : Yossi Azar,Bruce Shepherd, Seny Kamara

3. Jobs Scheduling Scheduler Server 1 Server N

4. Jobs Scheduling CPU Server MemoryGPU…

5. Cloud Computing

6. Vector Bin Packing

7. On Line Algorithms

8. Competitive Ratio

9. Related Results

10. Our Results

11. Vector Bin Packing

12. Our Results

13. VBP(B=1) Lower Bound Online- VBP (B=1) α bins. Online- Graph coloring α colors

14. On-line Graph Coloring aabbccdde e

16. VBP(B=1) Lower Bound Online- VBP(B=1) α bins Online- Graph coloring α colors

17. Reduction VBP to Graph Coloring

18. Ind: Vec:

19. Reduction VBP to Graph Coloring aabbccddee

20. Ind: Vec:

21. Reduction VBP to Graph Coloring aabbccdd

22. VBP(B=1) Lower Bound Online- VBP(B=1) α bins Online- Graph coloring α colors

23. VBP(B=2) Lower Bound Online VBP(B=2) α bins α classes of Triangle-free sub graphs Online- Graph coloring α·k colors k colors for each sub graph

24. Reduction VBP to Graph Coloring

25. Reduction VBP(B=2) to Graph Coloring

26. a b c d

28. VBP(B=2) Lower Bound Online VBP(B=2) α bins α classes of Triangle-free sub graphs Online- Graph coloring α·k colors

29. Reduction VBP(B=2) to Graph Coloring

30. On-line Graph Coloring

32. VBP(B=2) Lower Bound Online VBP(B=2) α bins α classes of Triangle-free sub graphs Online- Graph coloring α·k colors

33. abcd Coloring vertices of bin j

34. Correctness

35. Analysis

36. VBP Lower Bound Online VBP (B) α bins α classes of B+1 clique free graphs Online- Graph coloring α· colors α·k classes of B clique free graphs

37. B Clique Free to B-1 Clique Free

38. Reduction VBP to Graph Coloring

40. Scheme

41. Analysis

42. Upper Bound (B > 1)

43. Pack Into Virtual bins B cBlog(d) OPT virtual bins All Vector Stream:

44. B r bins B B B cBlog(d) Distribute bin vectors Virtual Bin i:

45. The virtual VBP algorithm.

47. Proof Sketch

48. B r bins B B B cBlog(d) Distribute bin vectors Virtual Bin i:

49. Distributing the vectors

51. De-randomize the algorithm

54. Conclusions

55. Open Questions

56. Thank you!

57. The {0,1}-VBP Upper Bound

59. cBlog(d) Distribute bin vectors B r bins Virtual Bin i: B B B

60. Our Results

61. Distribute bin vectors B r machines Bin i’s vectors cBlog(d)

62. Lower Bounds (B = 1)

63. cBlog(d) Distribute bin vectors B r machines Virtual Bin i: B B B

64. Unrelated Machine Model