Joint work with Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick Overhang Bounds Mike...
Transcript of Joint work with Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick Overhang Bounds Mike...
Joint work with Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick
Overhang Bounds
Mike Paterson
DIMAP & Dept of Computer Science
University of Warwick
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The classical solution
Harmonic Stacks
Using n blocks we can get an overhang of
Diamonds
The 4-diamond is balanced
Diamonds
The 5-diamond is …
Diamonds?
… unbalanced!
What really happens?
What really happens!
Small optimal stacks
Overhang = 1.16789Blocks = 4
Overhang = 1.30455Blocks = 5
Overhang = 1.4367Blocks = 6
Overhang = 1.53005Blocks = 7
Small optimal stacks
Overhang = 2.14384Blocks = 16
Overhang = 2.1909Blocks = 17
Overhang = 2.23457Blocks = 18
Overhang = 2.27713Blocks = 19
Note “spinality”
Support and balancing blocks
Principalblock
Support set
Balancing
set
Support and balancing blocks
Principalblock
Support set
Balancing
set
Principalblock
Support set
Stacks with downward external
forces acting on them
Loaded stacks
Size =
number of blocks
+ sum of external
forces.
Principalblock
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.
Optimal spinal stacks
…
Optimality condition:
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:
Optimal 100-block spinal stack
Spine
Shield
Towers
Optimal weight 100 loaded spinal stack
Loaded spinal stack + shield
spinal stack + shield + towers
Are spinal stacks optimal?
No!
Support set is not spinal!
Overhang = 2.32014Blocks = 20
Tiny gap
Optimal 30-block stack
Overhang = 2.70909Blocks = 30
Optimal (?) weight 100 construction
Overhang = 4.2390Blocks = 49
Weight = 100
“Parabolic” constructions
6-stack
Number of blocks in d-stack: Overhang:Balanced!
“Parabolic” constructions
6-slab
5-slab
4-slab
r-slab
r-slab
(r -1) - slab within an r - slab
(r-1)-slab
Nested inductions
“Smooth” parabola?
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Stacks with monotonic right contour can achieve only about ln n overhang [theorem above]
No good!
“Vases”
Weight = 1151.76
Blocks = 1043
Overhang = 10
“Oil lamps”
Weight = 1112.84
Blocks = 921
Overhang = 10
What about an upper bound?
Ωnis a lower bound
for overhang with n blocks
Can we do better?
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
Forces between blocks
Assumption: No friction.All forces are vertical.
Equivalent sets of forces
Distributions
Moments and spread
j-th moment
Center of mass
Spread
NB important measure
Signed distributions
MovesA 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.
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
Recall!
MovesA 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
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 movementIf there is an n-block stack that achieves an overhang of d, then
n–1 lossy moves
Main theorem
Four stepsShift half mass outside interval Shift half mass across interval
Shift some mass across intervaland no further
Shift some mass across interval
Simplified setting
“Integral” distributions
Splitting moves
0 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
To prove: Ω (d3) splits are required
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!
Can we extend the proof to the general case, with general distributions and moves?
Can we get improved boundsfor small values of p?
Can moves beyond position d help?
But …
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
If 1 is obtained from 0 by an extreme move, then
Plackett (1947):
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
if 1 is obtained from 0 by a sequence of splitting moves
Def:
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
lossy moves
An asymptotically tight bound - Proof
lossy
Our paper was in SODA’08 this week
An early version is at http://arXiv.org/pdf/0707.0093
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!
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(n1/2) overhang.
Probably we can get Ω(n1/2) 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.
The end
Applications!
The end