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

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

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

Optimal 30-block stack Blocks = 30 Overhang = 2.70909

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

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

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

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

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

Distributions

Signed distributions

Moments and spread j-th moment Center of mass Spread

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

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.

Move sequences

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

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.

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

Main theorem

Four steps

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

1 2 3 -3 -2 -1

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

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.

Spread vs. second moment argument

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.

Spread vs. second moment argument

Spread vs. second moment argument

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

Spread vs. second moment argument (for extreme moves)

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

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

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

Notation

An extended bound

An almost tight bound

An almost tight bound - Proof

An asymptotically tight bound freezing moves

An asymptotically tight bound - Proof freezing

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