powerpoint: a semidefinite programming approach to tensegrity theory by so, el–chammas, ye
TRANSCRIPT
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
1/41
Graph Realization Waterloo, MOPTA 06 1
A Semidefinite Programming Approach to Tensegrity Theory
and Graph Realization
Anthony So
Department of Computer Science
Manar ElChammas
Department of Electrical EngineeringYinyu Ye
Department of Management Science and Engineering and
by courtesy, Electrical Engineering
Stanford University
http://www.stanford.edu/yyye
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
2/41
Graph Realization Waterloo, MOPTA 06 2
Outlines
Graph Realization Problem
d-Realizable Graphs
SDP Formulation
Realization Algorithm
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
3/41
Graph Realization Waterloo, MOPTA 06 3
The Graph Realization Problem
Given a graph G = (V, E) and a set of nonnegative edge weights
{dij : (i, j) E}, the goal is to compute a realization of G in the Euclidean
space Rd for a given dimension d, i.e. to place the vertices of G in Rd such that
the Euclidean distance between every pair of adjacent vertices (i, j) in E equals
the prescribed weight dij .
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
4/41
Graph Realization Waterloo, MOPTA 06 4
Figure 1: 50-node Graph Realization in 2D
0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.50.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
5/41
Graph Realization Waterloo, MOPTA 06 5
Applications
Global Position System (GPS)
Sensor network localization
Molecular conformation
Data dimension reduction
Euclidean ball packing
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
6/41
Graph Realization Waterloo, MOPTA 06 6
Related Work
Schoenberg and Young/Householder studied the case where all pairwisedistances are given. Their theories formed the basis of various
multidimensional scaling algorithms.
Barvinok, Pataki, and Alfakih/Wolkowicz used SDP models to show that the
problem is solvable in polynomial time if the dimension of the realization is not
restricted. Moreover, they have given bounds on the dimension needed to
realize the given distances.
However, if we require the realization to be in Rd for some fixed d, then the
problem becomes NPcomplete (e.g., Saxe 1979, Aspnes, Goldenberg, and
Yang 2004).
Identify families of graph instances that admit polynomial time algorithms for
computing a realization in the required dimension (Biswas, So, Toh, and Ye
2004-2005, Jin/Saunders 2005; SODA05, ACM, IEEE, ...).
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
7/41
Graph Realization Waterloo, MOPTA 06 7
Todays Talk: dRealizable Graphs
A graph is drealizable if it can always be realized in Rd whenever it is realizable
(the edge weights are Euclidean metric) for every instance of the graph.
Connelly and Sloughter have recently given a complete characterization of
the class of drealizable graphs, where d = 1, 2, 3 It is trivial to find a realization of an 1realizable graph, since a graph is
1realizable iff it is a forest.
A polynomial time algorithm for realizing 2realizable graphs exists:
triangulation.
Finding a corresponding algorithm for 3realizable graphs is posed as an
open question.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
8/41
Graph Realization Waterloo, MOPTA 06 8
3realizable graph I
A graph is 3realizable iff it does not contain K5 or K2,2,2 as a minor (Connelly
and Sloughter 2004).
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
9/41
Graph Realization Waterloo, MOPTA 06 9
Figure 2: K-5
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
10/41
Graph Realization Waterloo, MOPTA 06 10
Figure 3: K-2-2-2
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
11/41
Graph Realization Waterloo, MOPTA 06 11
3realizable graph II
Using the forbidden minor characterization of partial 3trees, one can show that a
graph is 3realizable if it either
contains an V8 or an C5 C2 as a minor
or does not contain either graphs as a minor.
Indeed, if it is the latter, then G is a partial 3tree.
An k-tree is defined recursively as follows. The complete graph on k vertices is
an ktree. An ktree with n + 1 vertices (where n k) can be constructed from
an ktree with n vertices by adding a vertex adjacent to all vertices of one of its
kvertex complete subgraphs, and only to those vertices.
A partial ktree is a subgraph of an ktree.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
12/41
Graph Realization Waterloo, MOPTA 06 12
Figure 4: V-8
8 1
2
3
45
6
7
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
13/41
Graph Realization Waterloo, MOPTA 06 13
Figure 5: C-5C-2
1
3
5
7
102
4
6
8
9
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
14/41
Graph Realization Waterloo, MOPTA 06 14
Our Result
We resolve the above open question by giving a polynomial time algorithm for
(approximately) realizing 3realizable graphs.
The main bottleneck in the proof is to show that two graphs, V8 and C5 C2,
are 3realizable.
There exists a realization p of H {V8, C5 C2} such that the distance
between a certain pair of nonadjacent vertices (i, j) is maximized. Such a
realization induces a nonzero equilibrium stress on the graph H obtained from
H by adding the edge (i, j). Then use this equilibrium force to prove that H
must be in R3.
We show that the problem of computing the desired p can be formulated as an
SDP. More interesting is that the optimal dual multipliers of our SDP give rise to a
nonzero equilibrium stress.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
15/41
Graph Realization Waterloo, MOPTA 06 15
Tensegrity Theory: Realization and Labels
Let G = (V, E; d) be a weighted connected graph that contains neither loops
nor multiple edges, such that dij 0 for all (i, j) E.
A tensegrity G(p) is a graph G = (V, E) together with a realization
p = (pi) RD RD = R|V|D, and a configuration such that each
edge is labelled as a cable, strut, or bar and each vertex is labelled as pinned or
unpinned.
The label on each edge is intended to indicate its functionality: cables
(resp. struts) are allowed to decrease (resp. increase) in length (or stay the same
length), but not to increase (resp. decrease) in length; bars are forced to remain
the same length.
A pinned vertex is forced to remain where it is.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
16/41
Graph Realization Waterloo, MOPTA 06 16
Tensegrity Theory: Equilibrium Stress
An equilibrium stress for G(p) is an assignment of real numbers ij = ji to
each edge (i, j) E such that for each unpinned vertex i of G, we have
j:(i,j)E
ij(pi pj) = 0.
Furthermore, we say that the equilibrium stress = {ij} is proper if
ij = ji 0 (resp. 0) if (i, j) is a cable (resp. strut).
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
17/41
Graph Realization Waterloo, MOPTA 06 17
A Semidefinite Programming (SDP) for Realization
Consider a simple model with C (or S) is a set of cables (or strut):
max
(i,j)S xi xj2
(i,j)Cxi xj2
s.t. xi xj2 = d2ij , (i, j) Nx, i < j,
ak xj2 = d2kj , (k, j) Na.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
18/41
Graph Realization Waterloo, MOPTA 06 18
Matrix Representation
Let X = [x1 x2 ... xn] be the d n matrix that needs to be determined. Then
xixj2 = (eiej)
TXTX(eiej) and akxj2 = (ak; ej)
T[I X]T[I X](ak; ej),
where ej is the vector of all zero except 1 at thejth position.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
19/41
Graph Realization Waterloo, MOPTA 06 19
Matrix Representation
Let X = [x1 x2 ... xn] be the d n matrix that needs to be determined. Then
xixj2 = (eiej)
TXTX(eiej) and akxj2 = (ak; ej)
T[I X]T[I X](ak; ej),
where ej is the vector of all zero except 1 at thejth position.
max
(i,j)S(ei ej)TY(ei ej)
(i,j)C(ei ej)
TY(ei ej)
s.t. (ei ej)TY(ei ej) = d
2ij , i, j Nx, (i, j) Nx, i < j,
(ak; ej)T
I X
XT
Y
(ak; ej) = d2kj , k, j Na,
Y = XTX.
where Y denotes the Gram matrix XTX.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
20/41
Graph Realization Waterloo, MOPTA 06 20
Semidefinite Programming (SDP)
(SDP) inf C Z
s.t. Ai Z = bi, i = 1, 2,...,m, Z 0,
where C, Ai Mn
, the set of n-dimension symmetric matrices.
The dual problem to (SDP) can be written as:
(SDD) sup bTy
s.t.m
i yiAi + S = C, S 0,
where b = (b1; ...; bm) Rm, variables y Rm and S Mn.
An generalization of linear programming.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
21/41
Graph Realization Waterloo, MOPTA 06 21
SDP Relaxation for Graph Realization
Change
Y = XTX
to
Y XTX.
This matrix inequality is equivalent to
I X
XT
Y
0.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
22/41
Graph Realization Waterloo, MOPTA 06 22
SDP standard form
Z =
I X
XT Y
.
Find a symmetric matrix Z R(2+n)(2+n) such that
max
(i,j)S
(0; ei ej)(0; ei ej)T
(i,j)C
(0; ei ej)(0; ei ej)T
Z
s.t. Z1:d,1:d = I
(0; ei ej)(0; ei ej)T Z = d2ij , i, j Nx, i < j,
(ak; ej)(ak; ej)T Z = d2kj , k, j Na,
Z 0.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
23/41
Graph Realization Waterloo, MOPTA 06 23
The Dual of the SDP Relaxation
min I V +
i
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
24/41
Graph Realization Waterloo, MOPTA 06 24
Analysis of the SDP Formulation
Theorem 1. LetX = [x1, . . . , xn] be the positions of the unpinned vertices
obtained from the optimal primal matrixZ, and let{ij , wkj} be a set of optimal
dual multipliers. Suppose that we assign the stress ij (resp. wkj) to the bar
(i, j) Nx (resp. (k, j) Na), a stress of1 to all the cables, and a stress of
1 to all the struts. Then, the resulting assignment yields a nonzero properequilibrium stressfor the realization{(a1; 0), . . . , (am; 0), x1, . . . , xn}.
Proof: The primal is feasible and the dual is strictly feasible. Let Z (resp. U) be
the optimal primal (resp. dual) solution matrix. Then, the absence of a duality gap
implies complementarity:
ZU = 0.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
25/41
Graph Realization Waterloo, MOPTA 06 25
Algorithm Tasks
1. Realizing a partial 3tree;
2. finding a subdivision of V8 or C5 C2 in an 3realizable graph;
3. realizing an V8 and its subdivisions;
4. realizing an C5 C2 and its subdivisions.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
26/41
Graph Realization Waterloo, MOPTA 06 26
1: Realizing Partial 3Trees
Suppose that we are given a 3tree G with feasible edge lengths, and that G isconstructed by adding the vertices v1, v2, . . . , vn, in that order. Then, to find a
realization of G in R3 can be done in linear time. A partial 3tree can be
completed into a 3tree by solving an SDP.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
27/41
Graph Realization Waterloo, MOPTA 06 27
2: Finding a Subdivision of V8 or C5 C2
Let G be an 3realizable graph. We now show how the algorithm of Matouvsekand Thomas can be used to obtain a subgraph of G that is a subdivision of V8 or
C5 C2. We shall also use the term homeomorphic for subdivision a graph
H1 is homeomorphic to H2 if H1 is a subdivision of H2.
1. (Asano) For an 3connected graph H, a graph H has a subgraph
homeomorphic to H iff there is an 3connected component of H that has a
subgraph homeomorphic to H.
2. (Connelly and Sloughter) If an edge is added between a nonadjacent pair of
vertices of V8 (resp. C5 C2), then the resulting graph has K5 (resp. K5 or
K2,2,2) as a minor.
3. (Connelly and Sloughter) Let G be an 3realizable graph. Suppose that G
contains a subdivision of H, where H {V8, C5 C2}. Remove the
subdivision of H from G and consider the components of the resulting graph.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
28/41
Graph Realization Waterloo, MOPTA 06 28
Then, each component is connected in G to exactly one of the subdivided
edges of H.
Theorem 2. LetG be an3realizable graph containing a subgraph
homeomorphic toH {V8, C5 C2}. Then, one of thetriconnected
componentsofG is isomorphic toH.
Algorithm: First, decompose G into triconnected components. Then, we check
each of the triconnected components for the presence or absence of V8 orC5 C2. For this we can run the algorithm on each of those components and
see if the component reduces to a null graph or not. If the component does not
reduce to a null graph, then it is isomorphic to either V8 or C5 C2, and the
number of vertices in the component will determine which one it is. The desired
subdivision can then be extracted from G.Proposition 1. LetG be an3realizable graph withn vertices. Then, a
subdivision ofV8 orC5 C2 inG can be found inO(n) time.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
29/41
Graph Realization Waterloo, MOPTA 06 29
3: Realizing V8 and its Subdivisions
The graph V8 is 3realizable. We first augment V8 to V8 by adding a strut
between vertices 1 and 4 Then, we pin vertex 1 at the origin. In other words, we
would like to find a realization that maximizes the length of the strut.
max (0; e4)(0; e4)T Z
s.t. Z1:3,1:3 = I3
(0; ei ej)(0; ei ej)T Z = d2ij (i, j) E(V8)
1 = i < j
(0; ej)(0; ej)T Z = d21j (1, j) E(V8)
Z 0
(1)
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
30/41
Graph Realization Waterloo, MOPTA 06 30
Figure 6: V-8
8 1
2
3
45
6
7
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
31/41
Graph Realization Waterloo, MOPTA 06 31
Rank of SDP Solutions
We can find an SDP solution whose rank r satisfying
r(r + 1)2
m,
where m is the number of constraints.
This general result fells short what we hope for.
But an optimal stress associated with the chosen SDP objective will help ...
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
32/41
Graph Realization Waterloo, MOPTA 06 32
4: Realizing C5 C2 and its Subdivisions
The graph C5 C2 is 3realizable. We first augment C5 C2 to G by adding a
strut between vertices 1 and 6, and we pin vertex 1 at the origin.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
33/41
Graph Realization Waterloo, MOPTA 06 33
Putting Everything Together
Theorem 3. There is a polynomial time algorithm for (approximately) realizing
3realizable graphs.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
34/41
Graph Realization Waterloo, MOPTA 06 34
The Kissing Problem
Given a unit sphere at the center, the maximum number of unit spheres, in d
dimensions, can touch or kiss the center sphere?
K(1)=2, K(2)=6; general Solutions does not exist.
Delsarte Method uses linear programming to provide an upper bound on thenumber of spheres.
K(8) = 240, K(24) = 196650.
K(4) = 24: proved using Delsarte Method by Oleg Musin only 3 years ago.
For other dimensions, lower bounds have been provided by constructing alattice structure. There also exists a bound using the Riemann zeta function,
but is non-constructive.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
35/41
Graph Realization Waterloo, MOPTA 06 35
Our Approach
Given n unit spheres, can they all kiss the center sphere in d dimension space?
Treat it as a graph realization;
Use SDP to provide a lower bound;
Offer a completely constructive approach.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
36/41
Graph Realization Waterloo, MOPTA 06 36
The Kissing Problem as a Graph Realization
Can be formulated as a SDP feasibility problem; but SDP solution may notprovide proper rank.
(ei ej)TY(ei ej) 4, i = j,
eTi Y ei = 4, i
Y 0.
Construct a nonzero SDP objective function to reduce the rank of a solution.
min C Y,
s.t. (ei ej)TY(ei ej) 4, i = j,
eTi Y ei = 4, i
Y 0.
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
37/41
Graph Realization Waterloo, MOPTA 06 37
The objective construction
Use pull some struts and/or push some cables in order to force SDP solutioninto low rank.
For example, for 2 dimensions, 6 spheres can be connected as follows (thick
lines are bars, red lines are struts, green lines are cables).
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
38/41
Graph Realization Waterloo, MOPTA 06 38
Figure 7: 6 Spheres in 2-D
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
39/41
Graph Realization Waterloo, MOPTA 06 39
Realizing the 3-D and 12-Sphere Kissing Problem
This objective structure can be extended to dimension 3. For 12 spheres, SDP
method provides the following realization
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
40/41
Graph Realization Waterloo, MOPTA 06 40
Figure 8: 12 Spheres in 3-D
-
8/9/2019 powerpoint: A Semidefinite Programming Approach to Tensegrity Theory by So, ElChammas, Ye
41/41
Graph Realization Waterloo, MOPTA 06 41
Conclusion
We have studied a connection between SDP and tensegrity theories, as well
as the notion of drealizability of graphs.
We have shown that the problem of finding an equilibrium stress can be
formulated as an SDP. This gives a constructive proof of (a variant of) a result
in tensegrity theory that is previously established by nonconstructive means.
We then combine this result with other techniques to design an algorithm for
realizing 3realizable graphs, thus answering an open question posed before;
and realizing the 3-D Kissing graph.
We believe that our techniques can be applied to derive some other
interesting properties of tensegrity frameworks.