1. Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick
Maximum Overhang
Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick

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

3. “Parabolic” constructions
6-stack
Number of blocks:
Overhang:
Balanced!

4. Optimal 30-block stack
Blocks = 30
Overhang =

5. Optimal(?) weight 100 construction
Blocks = 49
Overhang =

6. “Vases”
Weight =
Blocks = 1043
Overhang = 10

7. “Oil lamps”
Weight =
Blocks = 921
Overhang = 10

8. Equilibrium Force equation Moment equation F1 F2 F3 F4 F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5

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

11. Distributions

12. Signed distributions

13. Moments and spread
j-th moment
Center of mass
Spread

14. Elementary moves (informally)
μ1 is obtained from μ0 by an elementary move if they differ only in an interval of length 1 and have the same total mass and first moment

15. A move is applied by adding it to a distribution.
Moves
A move is a signed distribution  with M0[]= M1[]=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.

16. Move sequences

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

18. 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.

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

20. Main theorem

21. Four steps

22. “Integral” distributions
Simplified setting
“Integral” distributions
Splitting moves

23. 1 2 3 -3 -2 -1

24. 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(d3) splits are clearly sufficient
(d3) splits are required

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

26. Spread vs. second moment argument

27. That’s a start!
But …
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?
We did not yet use the lossy nature of moves.

28. Spread vs. second moment argument

29. Spread vs. second moment argument

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

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

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

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

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

35. Notation

36. An extended bound

37. An almost tight bound

38. An almost tight bound - Proof

39. An asymptotically tight bound
freezing moves

40. An asymptotically tight bound - Proof
freezing

41. Open problems Simpler proof ? The right constant ???
Several unresolved issues in 3D