the parameterized complexity of geometric graph isomorphismvraman/exact/gaurav.pdf · complexity of...
TRANSCRIPT
![Page 1: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/1.jpg)
The Parameterized Complexity of GeometricGraph Isomorphism
Gaurav Rattan
The Institute of Mathematical Sciences, Chennai, India
Workshop on New Developments in Exact Algorithms andLower Bounds, IIT Delhi, Dec. 2014
![Page 2: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/2.jpg)
Outline
Geometric Graph IsomorphismGraph IsomorphismProblem Definition
Algorithms for Geom-GIImprovements using latticesFaster isomorphism testing algorithm
![Page 3: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/3.jpg)
Table of Contents
Geometric Graph IsomorphismGraph IsomorphismProblem Definition
Algorithms for Geom-GIImprovements using latticesFaster isomorphism testing algorithm
![Page 4: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/4.jpg)
Graph Isomorphism
GIInput: Graphs G and HQuestion: Is G isomorphic to H? I.e., is there an adjacency-preserving bijection between the vertex sets of G and H?
a b
c d
e f 1 2
3 4
5 6
![Page 5: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/5.jpg)
Complexity of GI
Best known algorithm for GI runs in time 2O(√n log n) [Babai, Luks
1983]. Polynomial time algorithms known for restricted graphclasses
I bounded genus graphs Miller 1980
I bounded degree graphs Luks 1982
I bounded eigenvalue multiplicity graphs Babai et al. 1982
I bounded treewidth graphs Bodlaender 1990
I graphs with excluded topological minors Grohe, Marx 2012
GI ∈ FPT, parameterized by
I eigenvalue multiplicity Evdikomov et al. 1997
I treewidth Lokshtanov et al. 2014
Approaches: Graph-theoretic, group-theoretic, combinatorial,geometrical . . .
![Page 6: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/6.jpg)
Table of Contents
Geometric Graph IsomorphismGraph IsomorphismProblem Definition
Algorithms for Geom-GIImprovements using latticesFaster isomorphism testing algorithm
![Page 7: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/7.jpg)
Geometric Graph Isomorphism
Geom-GIInput: Point sets A,B ⊆ Qk of size nParameter: k , dimension of the host vector spaceQuestion: Is there is a distance preserving bijection from A toB?
![Page 8: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/8.jpg)
Geometric Graph Isomorphism
x
y
z
ab
c
d12
3
4
T
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Geometric Graph Isomorphism
(Equivalent Formulation:) Does there exist a transformation T ofthe host space (T : Rk → Rk) such that TA = B?
I T preserves lengths: ‖Tx‖ = ‖x‖ and ‖Tx − Ty‖ = ‖x − y‖I T preserves dot product: (Tx)t(Ty) = x ty for all x , y ∈ Rk
Formally, we call such a T to be an orthogonal transformation.We call A and B to be geomterically-isomorphic via T .
![Page 10: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/10.jpg)
Fixed Parameter Tractability of Geom-GI
Dimension of the host space is an important parameter.
LemmaGeom-GI can be solved in polynomial time for boundeddimension k.
LemmaGI ≤p Geom-GIn.
The reduction maps graphs on n vertices to point-sets in ndimensional host space.
![Page 11: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/11.jpg)
Fixed Parameter Tractability of Geom-GI
Parameterized Algorithms?
I O∗(2O(k4)) time complexity, uses cellular algebras [Evdikomov,
Ponomarenko]
Theorem (Arvind, R.)
Given point-sets A,B ∈ Qk , there is a deterministic O∗(kO(k))time algorithm which decides whether A is isomorphic to B.
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P time algorithm for bounded dimension
For a k dimensional space, the transformation T can be uniquelydescribed by its action on k independent vectors.
Exhaustive Search Algorithm: On input sets A and B in Qk , size n,
1. Fix k linearly independent vectors {a1, . . . , ak} in A.
2. Branch on the possible images {b1, . . . , bk} inside B.
3. Let T be the unique transformation which sends {ai} to {bi}.4. Check if T sends A to B in a distance preserving manner.
The algorithm runs in O(nk) time.
![Page 13: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/13.jpg)
P time algorithm for bounded dimension
For a k dimensional space, the transformation T can be uniquelydescribed by its action on k independent vectors.
Exhaustive Search Algorithm: On input sets A and B in Qk , size n,
1. Fix k linearly independent vectors {a1, . . . , ak} in A.
2. Branch on the possible images {b1, . . . , bk} inside B.
3. Let T be the unique transformation which sends {ai} to {bi}.4. Check if T sends A to B in a distance preserving manner.
The algorithm runs in O(nk) time.
![Page 14: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/14.jpg)
Table of Contents
Geometric Graph IsomorphismGraph IsomorphismProblem Definition
Algorithms for Geom-GIImprovements using latticesFaster isomorphism testing algorithm
![Page 15: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/15.jpg)
Improving over exhaustive search
Build sets SA and SB with the following properties
I |SA| = |SB | = f (k) (small-sized).
I If A and B are isomorphic via an orthogonal T , then SA andSB are also isomorphic via T (isomorphism-invariant).
Improvement: Would imply a (f (k))k branching, instead of nk
branching.
![Page 16: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/16.jpg)
Lattices
b1
b2
b1 + b2
2b1 + b2
Figure: Integer lattice generated by vectors b1 and b2.
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Lattices
DefinitionThe lattice generated by a set A ∈ Rk , denoted by LA, is the setof all integer linear combinations of the set A.
LA = {α1a1 + · · ·+ αnan |αi ∈ Z}
ClaimLet SA be the set of shortest vectors in LA. Then, SA isisomorphism-invariant.
Proof.Suppose A and B are isomorphic via T , i.e. TA = B. Then,T (LA) = LB . Since T preserves lengths, T (SA) = SB .
![Page 18: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/18.jpg)
Lattices
DefinitionThe lattice generated by a set A ∈ Rk , denoted by LA, is the setof all integer linear combinations of the set A.
LA = {α1a1 + · · ·+ αnan |αi ∈ Z}
ClaimLet SA be the set of shortest vectors in LA. Then, SA isisomorphism-invariant.
Proof.Suppose A and B are isomorphic via T , i.e. TA = B. Then,T (LA) = LB . Since T preserves lengths, T (SA) = SB .
![Page 19: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/19.jpg)
Shortest vectors in lattices
x
y
z
x
yr
r
> r
x , y ∈ L ⇒ x − y ∈ L
r2
r2
3r2
f (k) ≤O((32 r)k
)O(( r2)k
) = 3k
![Page 20: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/20.jpg)
Shortest vectors in lattices
x
y
z
x
yr
r
> r
x , y ∈ L ⇒ x − y ∈ Lr2
r2
3r2
f (k) ≤O((32 r)k
)O(( r2)k
) = 3k
![Page 21: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/21.jpg)
Improved Algorithm for Geom-GI
Given point sets A,B ⊂ Qk as input,
1. Compute sets SA,SB of shortest vectors in LA, LB using theSVP algorithm of [MV10].
2. If dim(Span(SA)) = k , follow the enumerative strategyI Fix basis in SA, branch in SB , . . .
3. If dim(Span(SA)) = k1 < k , recursively construct theisomorphism
I Construct all T1 : Span(SA)→ Span(SB) enumeratively.I Project sets A and B out of Span(A) and Span(B).I Construct all T2 : Span(SA)⊥ → Span(SB)⊥ recursively.I Check if T = T1 ⊕ T2 sends A to B.
Branching: B(k) = (3k1)k1 · B(k − k1), which yields a O∗(2O(k2))
branching.
![Page 22: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/22.jpg)
Improved Algorithm for Geom-GI
Given point sets A,B ⊂ Qk as input,
1. Compute sets SA,SB of shortest vectors in LA, LB using theSVP algorithm of [MV10].
2. If dim(Span(SA)) = k , follow the enumerative strategyI Fix basis in SA, branch in SB , . . .
3. If dim(Span(SA)) = k1 < k , recursively construct theisomorphism
I Construct all T1 : Span(SA)→ Span(SB) enumeratively.I Project sets A and B out of Span(A) and Span(B).I Construct all T2 : Span(SA)⊥ → Span(SB)⊥ recursively.I Check if T = T1 ⊕ T2 sends A to B.
Branching: B(k) = (3k1)k1 · B(k − k1), which yields a O∗(2O(k2))
branching.
![Page 23: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/23.jpg)
Improved Algorithm for Geom-GI
Given point sets A,B ⊂ Qk as input,
1. Compute sets SA,SB of shortest vectors in LA, LB using theSVP algorithm of [MV10].
2. If dim(Span(SA)) = k , follow the enumerative strategyI Fix basis in SA, branch in SB , . . .
3. If dim(Span(SA)) = k1 < k , recursively construct theisomorphism
I Construct all T1 : Span(SA)→ Span(SB) enumeratively.I Project sets A and B out of Span(A) and Span(B).I Construct all T2 : Span(SA)⊥ → Span(SB)⊥ recursively.I Check if T = T1 ⊕ T2 sends A to B.
Branching: B(k) = (3k1)k1 · B(k − k1), which yields a O∗(2O(k2))
branching.
![Page 24: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/24.jpg)
Improved Algorithm for Geom-GI
Given point sets A,B ⊂ Qk as input,
1. Compute sets SA,SB of shortest vectors in LA, LB using theSVP algorithm of [MV10].
2. If dim(Span(SA)) = k , follow the enumerative strategyI Fix basis in SA, branch in SB , . . .
3. If dim(Span(SA)) = k1 < k , recursively construct theisomorphism
I Construct all T1 : Span(SA)→ Span(SB) enumeratively.I Project sets A and B out of Span(A) and Span(B).I Construct all T2 : Span(SA)⊥ → Span(SB)⊥ recursively.I Check if T = T1 ⊕ T2 sends A to B.
Branching: B(k) = (3k1)k1 · B(k − k1), which yields a O∗(2O(k2))
branching.
![Page 25: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/25.jpg)
Table of Contents
Geometric Graph IsomorphismGraph IsomorphismProblem Definition
Algorithms for Geom-GIImprovements using latticesFaster isomorphism testing algorithm
![Page 26: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/26.jpg)
Exploiting the structure of the basis
We are searching for an isomorphism T which maps A to B
I length-preserving, i.e. ‖Tx‖ = ‖x‖I preserves dot product, i.e. (Tx)t(Ty) = x ty
Improvement: An isomorphism preserves the geometry of the basisset. (e.g. consider orthogonal bases . . . )
![Page 27: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/27.jpg)
Exploiting the structure of the basis
x
y
z
a
b
c
d e S
v
v td < v te < v tb
![Page 28: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/28.jpg)
Exploiting the structure of the basis
x
y
z
a
b
c
d e S
v
v td < v te < v tb
![Page 29: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/29.jpg)
Exploiting the structure of the basis
x
y
z
a
b
c
d e S
v
v td < v te < v tb
![Page 30: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/30.jpg)
Exploiting the structure of the basis
x
y
z
a
b
c
d e S
v
v td < v te < v tb
![Page 31: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/31.jpg)
Exploiting the structure of the basis
Let S be a set of vectors.
Definition (Haviv Regev)
A vector v defines a chain of length k in S if for some lin. ind.vectors a1, . . . , ak ∈ S
I a1 uniquely minimizes x tv over all x ∈ S
I a2 uniquely minimizes x tv over all x ∈ S\Span(a1)
I and in general, ai uniquely minimizes x tv over allx ∈ S\Span(a1, . . . , ai−1).
I.e. v ta1 < · · · < v tak
Remark: A random v defines a chain oflength 1 w.h.p. (Isolation Lemma [MVV]).
![Page 32: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/32.jpg)
Exploiting the structure of the basis
Let S be a set of vectors.
Definition (Haviv Regev)
A vector v defines a chain of length k in S if for some lin. ind.vectors a1, . . . , ak ∈ S
I a1 uniquely minimizes x tv over all x ∈ S
I a2 uniquely minimizes x tv over all x ∈ S\Span(a1)
I and in general, ai uniquely minimizes x tv over allx ∈ S\Span(a1, . . . , ai−1).
I.e. v ta1 < · · · < v tak Remark: A random v defines a chain oflength 1 w.h.p. (Isolation Lemma [MVV]).
![Page 33: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/33.jpg)
Chain isolation in lattices
Given a lattice L, call a vector dual if
I it has integral dot product with every vector in L.
The set of all dual vectors forms a lattice, the dual lattice L∗A.
There are short dual vectors in L∗A which define chains inside theset SA of shortest vectors.
![Page 34: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/34.jpg)
Chain isolation in lattices
Given a lattice L, call a vector dual if
I it has integral dot product with every vector in L.
The set of all dual vectors forms a lattice, the dual lattice L∗A.
There are short dual vectors in L∗A which define chains inside theset SA of shortest vectors.
![Page 35: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/35.jpg)
Chain isolation in lattices
Theorem (Haviv, Regev 14)
There exists a dual vector v ∈ L∗A such that
I (defines chain) v defines a chain of length k in SA and
I (small length) ‖v‖ ≤ kO(1) · λ(L∗)
Moreover, the set of all such isolating vectors has size at mostO∗(kO(k)), and can be computed in time O∗(kO(k)).
![Page 36: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/36.jpg)
A faster O∗(kO(k)) isomorphism algorithm
x
y
z
a
b
c
d e
SA
v∗
dual v∗ → chain d-e-b in SA
![Page 37: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/37.jpg)
A faster O∗(kO(k)) isomorphism algorithm
x
y
z
Ta
Tb
Tc
TdTe
Tv∗
SB
Suppose A ≡ B via T
Tv∗ → chain Td-Te-Tb in TSA
![Page 38: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/38.jpg)
Outline of O∗(kO(k)) isomorphism algorithm
Algorithm: On input sets A,B ∈ Qk ,
1. Compute the set of shortest vectors SA,SB in LA, LB .
2. Compute the set ΓA, ΓB of O∗(kO(k)) dual vectors whichinduce chains.
3. Pick a dual vector u ∈ ΓA. Let {a1, . . . , ak} be the chain-basisdefined by u inside SA.
4. For every dual vector v ∈ ΓB ,I Let {b1, . . . , bk} be the chain-basis defined by v inside SB .I Check if T : {ai} → {bi} is an orthogonal map which sends A
to B.
5. In case SA is not k-dimensional, recurse . . .
![Page 39: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/39.jpg)
Outline of O∗(kO(k)) isomorphism algorithm
Algorithm: On input sets A,B ∈ Qk ,
1. Compute the set of shortest vectors SA,SB in LA, LB .
2. Compute the set ΓA, ΓB of O∗(kO(k)) dual vectors whichinduce chains.
3. Pick a dual vector u ∈ ΓA. Let {a1, . . . , ak} be the chain-basisdefined by u inside SA.
4. For every dual vector v ∈ ΓB ,I Let {b1, . . . , bk} be the chain-basis defined by v inside SB .I Check if T : {ai} → {bi} is an orthogonal map which sends A
to B.
5. In case SA is not k-dimensional, recurse . . .
![Page 40: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/40.jpg)
Outline of O∗(kO(k)) isomorphism algorithm
Algorithm: On input sets A,B ∈ Qk ,
1. Compute the set of shortest vectors SA,SB in LA, LB .
2. Compute the set ΓA, ΓB of O∗(kO(k)) dual vectors whichinduce chains.
3. Pick a dual vector u ∈ ΓA. Let {a1, . . . , ak} be the chain-basisdefined by u inside SA.
4. For every dual vector v ∈ ΓB ,I Let {b1, . . . , bk} be the chain-basis defined by v inside SB .I Check if T : {ai} → {bi} is an orthogonal map which sends A
to B.
5. In case SA is not k-dimensional, recurse . . .
![Page 41: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/41.jpg)
Outline of O∗(kO(k)) canonization algorithm
Given a point set A ⊂ Qk , f : Qk → Qk is a canonizing function ifit has the following properties
I f (A) is isomorphic to A
I if A is isomorphic to B, then f (A) = f (B).
Computing such a f is least as hard as the isomorphism problem.
![Page 42: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/42.jpg)
Outline of O∗(kO(k)) canonization algorithm
Algorithm: On input set A ⊂ Qk ,
I Compute the set of shortest vectors SA in LA.
I Compute a set ΓA of O∗(kO(k)) special bases inside SA.I For each basis J = {u1, . . . , uk} ∈ ΓA,
I compute the Gram matrix Gi,j = uti uj .
I compute the coordinates C (ai ) of every point ai ∈ A in thebasis J.
I Output the lexicographically least description σ = (G ,K )obtained above.
If A ≡B T , then TJ generates same description for B as Jgenerates for A.
I Compute a canonical A∗ using σ as follows.I Find unique lower triangular matrix L such that LLT = G .I Use row vectors of L to get a basis J∗ = {v1, . . . , vk}.I Compute the point set A∗ s.t. a∗i is K (ai )-linear combination
of J∗.
![Page 43: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/43.jpg)
Outline of O∗(kO(k)) canonization algorithm
Algorithm: On input set A ⊂ Qk ,
I Compute the set of shortest vectors SA in LA.
I Compute a set ΓA of O∗(kO(k)) special bases inside SA.I For each basis J = {u1, . . . , uk} ∈ ΓA,
I compute the Gram matrix Gi,j = uti uj .
I compute the coordinates C (ai ) of every point ai ∈ A in thebasis J.
I Output the lexicographically least description σ = (G ,K )obtained above.
If A ≡B T , then TJ generates same description for B as Jgenerates for A.
I Compute a canonical A∗ using σ as follows.I Find unique lower triangular matrix L such that LLT = G .I Use row vectors of L to get a basis J∗ = {v1, . . . , vk}.I Compute the point set A∗ s.t. a∗i is K (ai )-linear combination
of J∗.
![Page 44: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/44.jpg)
Future Directions
I A O∗(2O(k)) algorithm for geometric graph isomorphism andcanonization in Euclidean metric?
I Faster canonization would give a 2O(n) algorithm forhypergraph canonization
I Geom-GI in other lp metrics?I Linear algebra breaks down for non-Euclidean metrices;
combinatorial algorithms work.I Two-dimensional case is polynomial time.I Reductions between Geom-GI for various metrics, similar to
embeddings.
![Page 45: The Parameterized Complexity of Geometric Graph Isomorphismvraman/exact/gaurav.pdf · Complexity of GI Best known algorithm for GI runs in time 2O( p nlogn) [Babai, Luks 1983]. Polynomial](https://reader033.vdocuments.net/reader033/viewer/2022053007/5f0b44907e708231d42fac89/html5/thumbnails/45.jpg)
Thank you!