Characterizing Generic Global Rigidity Steven J. Gortler (with A. Healy and D. Thurston)

Characterizing Generic Global Rigidity

Steven J. Gortler

(with A. Healy and D. Thurston)

Global Rigidity

• Given a graph: “G”,

• Given a “framework”: p, in Rd

– For some fixed d

R2

Global Rigidity

• p is GR in Rd : there is no second framework in Rd with the same edge lengths

R2 globally rigid

Global Rigidity


R2

Global Rigidity


• Else: p is GF in Rd

R2

Global Rigidity (notes)

• Edge crossing is allowed

• Euclidean: rot, trans, reflect is not considered different

R2

Global Rigidity (notes)

• Locally flexible => globally flexible

• Easier to characterize

R2

Motivation: Distance Geometry

• Input: some pairwise distances

• Output: geometric framework/Eucl– Chemistry, sensor networks


• Input: some pairwise distances

• Output: geometric framework/Eucl– GR: Well posed-ness of problem– GR: Divide and conquer


• Molecule problem

• MDS with partial information

• Rank constrained distance matrix completion

Generic

• Given a graph G, and framework p, the GR problem is NP-Hard [Saxe ‘79]

Generic


• The reductions all involve special “coincidences” in the framework

R1

Generic


• Problem seems simpler if we assume no coincidences

R1 globally rigid

Generic


• Problem seems simpler if we assume no coincidences

R1 globally rigid globally flexible

Generic

• Perhaps the problem is easier if we assume that the input framework is “generic”– Think: randomly perturbed


Generic

• Perhaps the problem is easier if we assume that the input framework is “generic”

• In 1D, a generic framework is GR iff the graph is 2-connected


Generic

• Perhaps the problem is easier if we assume that the input framework is “generic”

• In 1D, a generic framework is GR iff the graph is 2-connected– So GGR in R1 is a property of the graph alone


History of GGR

CC: (Connelly condition) Sufficient for all d [’89, H‘95, ‘05]

HC: (Hendrickson condition) Necessary for all d [’88, ’92]

History of GGR



HC = CC (nec & suff) for d=2 [JJ’05]

History of GGR



HC not sufficient for d >= 3 [C ’91]


K5,5

History of GGR



HC not sufficient for d >= 3 [C ’91]


CC is necessary for all d [this work]

Main result

• [Connelly] If CC is satisfied by a generic framework in Rd, it is globally rigid in Rd .

• [Thm] If CC is not satisfied by a generic framework in Rd then it is globally flexible in Rd

Main result



• Note: CC can be tested with an efficient randomized algorithm.

Main result



• Note: CC test gives same answer for all generic frameworks of G in Rd

Main result

• [Connelly] If CC is satisfied by a generic framework, it is globally rigid.


• [Cor] A graph is either GR for all generic fmwks in Rd, or is not GR for any generic fmwk in Rd.– So GGR in Rd is a property of the graph alone

On to the condition….

• A real number wuv on each edge euv

Σv wuv [p(v) – p(u)] = 0

Stress Vector satisfied by a framework

R2

f

a

c

b

e

dh

g

p(u)

u


Σv wuv [p(v) – p(u)] = 0


R2p(u)e

f

g

c

u


Σv wuv [p(v) – p(u)] = 0


u

R2

f

b

h

p(u)


• Equivalent to (symmetrically) writing each vertex as an affine comb of its nbrs

[1/ Σv wuv] Σv wuv p(v) = p(u)

R2

u

fc

e

g

p(u)


Equivalent to equilibrium point of quadratic spring/strut energy (no pins)

E(p)= Σu Σv wuv |p(v) – p(u)|2

R2

Stress vectors: easy facts

• Stress vectors satisfied by a fixed p form a linear space W(p)

p


• Stress vectors satisfied by a fixed p form a linear space W(p)

• Any affine transform ‘T(p)’ will satisfy all stresses in W(p)

p T(p)


• For some graphs, there are even more fmwks than the affine transforms of p,

• that still satisfy all stresses in W(p)

p not an affine tform of p

Connelly’s Condition

• CC := The only fmwks that satisfy all of W(p) are the affine transforms of p

• This will somehow describe GGR!

Stresses and lengths

• What is the relationship between stress vectors– Affine invariant

• ….. and lengths– Euclidean invariant

The mapping “L”

• “L” is the mapping from d-dim fmwks to Re

• Describing each edge’s squared length

Re

R2

L

The set “M”

• “M” is the image of “L” – All possible measurements

Re

R2

L

The set “M”

• “M” is the image of “L” – All possible measurements– A semi-algebraic set

• A smooth manifold a.e.

Re

R2

L

Re

Lengths and stresses: the connection

• At a generic fmwk p,

span L* = the tangent space of M at L(p)

R2

L

Re



span L* = the tangent space of M at L(p)


W(p) spans the normal space of M at L(p)– [Maxwell]

R2

L

Re


• So a generic fmwk that satisfies all the same stresses as p must have the same normal space

R2

L

Re


• So a generic fmwk that satisfies all the same stresses as p must have the same tangent space

R2

L

Re


• So a generic fmwk that satisfies all the same stresses as p must have the same tangent space– M is a cone: λM = M– Same tangent space, not just parallel

R2

L

Re


• So a generic fmwk that satisfies all the same stresses as p must have the same tangent space

• This is the key connection

• We will return to this in the proof

R2

L

On to the proof…

Degree mod two thm

Degree mod two thm

2

0

Degree mod two thm

2

0

4

Degree mod two thm

• Typical version:

• Given smooth map from a compact manifold to a connected manifold of same dimension

• Every generic point in the range has the same number of pre-images mod 2– The creases are singular

• This number {0,1} is called the degree

Degree mod two thm

• General version:

• Given smooth map from a compact manifold to a connected manifold of same dimension

• Every generic point in the range has the same number of pre-images mod 2

proper

Our plan

2

0

4

Our plan

• Assume (!CC)• Start with the map L• Define a domain

– Equate Euclidean transforms in the domain

• Define range– Connected smooth manifold

• Will need to remove some singularities while maintaining connectivity of range and properness of map

• Show the map has degree 0

• Each point in image has multiple pre-images– Framework is globally flexible

Domain

• Start with stress satisfiers: A(p)– Frameworks that satisfy all of the stresses

that are satisfied by p– Affine tforms of p plus maybe more

Domain

• Start with stress satisfiers: A(p)

• Mod out Euclideans: A(p)/Eucl

Domain



• Result is smooth manifold + singularities

Domain



• Result is smooth manifold + singularities– Singularities: fmwks stabilized by a n.t. euclidean.

Domain



• Result is smooth manifold + singularities– Singularities: deficient affine span.

Lemma 1

• Lemma: If (!CC)– A(p) is “big”

• then the singularities of A(p)/Eucl are of co-dimension >= 2.– Proof: counting

Codimension and cutting

• The singular co-dim will carry over to the range

• Removal a co-dimension 2 set does not disconnect– Needed for degree thm

co-dim 2

Codimension and cutting

• The singular co-dim will carry over to the range

• Removal a co-dimension 2 set does not disconnect– Needed for degree thm

co-dim 2 co-dim 1

The range

• Let B(p) := L(A(p))– Achievable measurements of stress satisfiers– Some subset of M

Re

L

The range

• For the degree to be well defined– Need to include B(p) as a full dimensional

subset of a connected range manifold

Re

L

The range



• To show the degree is 0– Sufficient for range to include pts not in B(p)

Re

L

0

The range



• To show the degree is 0– Sufficient for range to include pts not in B(p)

Re

L

The range

• So we need to understand the shape of B(p)

Re

L

0

Gauss fiber

Re

Digression

…

…

• Gauss fiber: points with same (not just parallel) tangent as chosen point

Gauss fiber

• Gauss fiber thm: The Gauss fiber at a generic point of an irreducible algebraic variety is an affine space

Re1d fiber

Digression

…

…

Gauss fiber

• Gauss fiber thm: The Gauss fiber at a generic point of an irreducible algebraic variety is a affine space

Re

1d non-affine fiber

Digression

Gauss fiber


Re

1d non-affine fiber, exceptional

Digression

Gauss fiber


Re

0-d fiber

Digression

The range

• Recall “the connection” – Same stresses = same tangent in M

• B(p) is a gauss fiber in M

L

Re

The range

• Recall “the connection” – Same stresses = same tangent in M

• B(p) is a gauss fiber in M

• M is not an irreducible algebraic variety • But it is a full dimensional semi-algebraic subset of one

L

Re

Lemma 2

• Lemma: B(p) is a flat space– Full dimensional subset of an affine space

L

Re

Lemma 2


• So define the range to be this whole affine space

L

Re

Lemma 2


• M is contained in first octant, the affine space is not– The degree will be zero

L

Re

0

Lemma 2


• Note: domain and range have same dimension

L

Re

0

Last step

L

• Remove the images of the singularities of A(p)/Eucl from the range and their pre-images– Range remains connected if (!CC)– Domain and range are smooth manifolds

0

Re

Last step

L

• Remove the images of the singularities of A(p)/Eucl from the range and their pre-images

• Can now apply degree thm

0

Re

24

Main Theorem

L

0

24

Re

Main Theorem

L

0

24

•[Thm] If CC is not satisfied by a generic framework in Rd then it is globally flexible in Rd

Re

Review

• Defined a domain– A(p)/Eucl– Created some singularities

• Defined same dimensional connected smooth range – Affine space containing B(p) (due to flatness)

• Removed singularities– To get smooth domain manifold– Maintained range connectedness (due to high co-dim if !CC)

• Now we there is a well defined degree• Need to know degree is 0 (due to flatness)

Deep Breath…..

Algorithm

• Input: Graph, d

Algorithm

• Input: Graph, d• Pick random framework p in Rd

• Compute stress vector space W(p)– Linear algebra

• Pick a random stress vector w from W(p)• Compute m: dimensionality of fmwks that satisfy w

– Linear algebra

• If m = d(d+1) output “GR in Rd”

• If m > d(d+1) output “GGR in Rd”

Algorithm

• With high probability, p will behave like a generic fmwk

• With high probability, fmwks that satisfy w will satisfy all of W(p)

• Can be done with integer linear algebra• The exceptions satisfy a “low” degree

polynomial• No false positives• GGR in RP

One more breath…

Bonus result

• If a graph is generically globally flexible

• One can continuously flex in one higher dimension back down to second framework

Future work

• More algebraic: – Not just the L function on graphs– More general field– More general metric signature

• More combinatorial:– Deterministic efficient algorithm

Future work

• Given lengths, it is NP-hard to figure out the framework in Rd

• Semi-definite programming will typically find solution in Rv

• But sometimes it happens to give an answer in Rd

– These are frameworks for which higher dimensions don’t help

• Can these cases be nicely characterized?

Characterizing Generic Global Rigidity Steven J. Gortler (with A. Healy and D. Thurston)

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