1. Joint work with Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick Overhang Bounds Mike Paterson DIMAP & Dept of Computer Science University of Warwick

2. The classical solution Harmonic Stacks Using n blocks we can get an overhang of

3. Diamonds The 4-diamond is balanced

4. Diamonds The 5-diamond is …

5. Diamonds? … unbalanced!

6. What really happens?

7. What really happens!

8. Small optimal stacks Overhang = 1.16789 Blocks = 4 Overhang = 1.30455 Blocks = 5 Overhang = 1.4367 Blocks = 6 Overhang = 1.53005 Blocks = 7

9. Small optimal stacks Overhang = 2.14384 Blocks = 16 Overhang = 2.1909 Blocks = 17 Overhang = 2.23457 Blocks = 18 Overhang = 2.27713 Blocks = 19 Note “spinality”

10. Support and balancing blocks Principal block Support set Balancing set

11. Support and balancing blocks Principal block Support set Balancing set

12. Principal block Support set Stacks with downward external forces acting on them Loaded stacks Size = number of blocks + sum of external forces.

13. Principal block Support set Stacks in which the support set contains only one block at each level Spinal stacks Assumed to be optimal in: J.F. Hall, Fun with stacking Blocks, American Journal of Physics 73(12), 1107-1116, 2005.

14. Optimal spinal stacks … Optimality condition:

15. Spinal overhang Let S (n) be the maximal overhang achievable using a spinal stack with n blocks. Let S * (n) be the maximal overhang achievable using a loaded spinal stack on total weight n. Theorem: A factor of 2 improvement over harmonic stacks! Conjecture:

16. Optimal 100-block spinal stack Spine Shield Towers

17. Optimal weight 100 loaded spinal stack

18. Loaded spinal stack + shield

19. spinal stack + shield + towers

20. Are spinal stacks optimal? No! Support set is not spinal! Overhang = 2.32014 Blocks = 20 Tiny gap

21. Optimal 30-block stack Overhang = 2.70909 Blocks = 30

22. Optimal (?) weight 100 construction Overhang = 4.2390 Blocks = 49 Weight = 100

23. “Parabolic” constructions 6-stack Number of blocks in d-stack: Overhang: Balanced!

24. “Parabolic” constructions 6-slab 5-slab 4-slab

25. r-slab

26. (r - 1 ) - slab within an r - slab (r-1)-slab Nested inductions

28. “Smooth” parabola? Stacks with monotonic right contour can achieve only about ln n overhang [theorem above] No good!

29. “Vases” Weight = 1151.76 Blocks = 1043 Overhang = 10

30. “Oil lamps” Weight = 1112.84 Blocks = 921 Overhang = 10

31. What about an upper bound?    n    is a lower bound for overhang with n blocks Can we do better?

32. Equilibrium F 1 + F 2 + F 3 = F 4 + F 5 x 1 F 1 + x 2 F 2 + x 3 F 3 = x 4 F 4 + x 5 F 5 Force equation Moment equation F1F1 F5F5 F4F4 F3F3 F2F2

33. Forces between blocks Assumption: No friction. All forces are vertical. Equivalent sets of forces

40. Distributions

41. Moments and spread j-th moment Center of mass Spread NB important measure

42. Signed distributions

43. Moves A move is a signed distribution  with M 0 [  ] = M 1 [  ] = 0 whose support is contained in an interval of length 1 A move is applied by adding it to a distribution. A move can be applied only if the resulting signed distribution is a distribution.

44. Equilibrium F 1 + F 2 + F 3 = F 4 + F 5 x 1 F 1 + x 2 F 2 + x 3 F 3 = x 4 F 4 + x 5 F 5 Force equation Moment equation F1F1 F5F5 F4F4 F3F3 F2F2 Recall!

45. Moves A move is a signed distribution  with M 0 [  ] = M 1 [  ] = 0 whose support is contained in an interval of length 1 A move is applied by adding it to a distribution. A move can be applied only if the resulting signed distribution is a distribution.

46. Move sequences

47. Extreme moves Moves all the mass within the interval to the endpoints

48. Lossy moves If  is a move in [c- ½,c+ ½ ] then A lossy move removes one unit of mass from position c Alternatively, a lossy move freezes one unit of mass at position c

49. Overhang and mass movement If there is an n-block stack that achieves an overhang of d, then n–1 lossy moves

50. Main theorem

51. Four steps Shift half mass outside intervalShift half mass across interval Shift some mass across interval and no further Shift some mass across interval

52. Simplified setting “Integral” distributions Splitting moves

53. 0123 -3-2

54. Basic challenge Suppose that we start with a mass of 1 at the origin. How many splits are needed to get, say, half of the mass to distance d ? Reminiscent of a random walk on the line O(d 3 ) splits are “clearly” sufficient To prove:  (d 3 ) splits are required

55. Effect of a split Note that such split moves here have associated interval of length 2.

56. Spread vs. second moment argument

57. That’s a start! Can we extend the proof to the general case, with general distributions and moves? Can we get improved bounds for small values of p? Can moves beyond position d help? But … We did not yet use the lossy nature of moves.

58. Spread vs. second moment argument

60. Spread vs. second moment inequalities If  1 is obtained from  0 by an extreme move, then Plackett (1947):

61. Spread vs. second moment argument (for extreme moves)

62. Splitting “Basic” splitting move A single mass is split into arbitrarily many parts, maintaining the total and center of mass if  1 is obtained from  0 by a sequence of splitting moves Def:

63. Splitting and extreme moves If V is a sequence of moves, we let V* be the corresponding sequence of extreme moves Lemma: Corollary:

64. Spread vs. second moment argument (for general moves) extreme

65. Notation

66. An extended bound

67. An almost tight bound

68. An almost tight bound - Proof

69. An asymptotically tight bound lossy moves

70. An asymptotically tight bound - Proof lossy

71. Our paper was in SODA’08 this week An early version is at http://arXiv.org/pdf/0707.0093

72. Some open questions ● What shape gives optimal overhang? ● We only consider frictionless 2D constructions here. This implies no horizontal forces, so, even if blocks are tilted, our results still hold. What happens in the frictionless 3D case? ● With friction, everything changes!

73. With friction ● With enough friction we can get overhang greater than 1 with only 2 blocks! ● With enough friction, all diamonds are balanced, so we get Ω(n 1/2 ) overhang. ● Probably we can get Ω(n 1/2 ) overhang with arbitrarily small friction. ● With enough friction, there are possibilities to get exponents greater than 1/2. ● In 3D, I think that when the coefficient of friction is greater than 1 we can get Ω(n) overhang.

74. The end Applications!

75. The end