a combinatorial construction of almost-ramanujan graphs using the zig-zag product avraham ben-aroya...
TRANSCRIPT
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A Combinatorial Construction of Almost-Ramanujan
Graphs Using the Zig-Zag product
Avraham Ben-Aroya Amnon Ta-Shma
Tel-Aviv University
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Expander graphs
- Sparse graphs with strong connectivity
- Fundamental objects in Math and CSApplications in: Communication networks Derandomization Error correcting codes PCPs Proof complexity …
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Properties of expanders
Many pseudorandom properties: No small cuts Every “not-too-large” set expands Random walks mix fast Spectral gap …
Challenge: Explicit constructions of “good”
expanders
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Spectral gap
0 0 … 1/D 0
0 1/D … 0 1/D
…
1/D 0 … 1/D 0
0 1/D … 0 0
Eigenvector basis v1,…,vn, eigenvalues 1=1 … n
Denote 2(G)=max{|2|,|n|}.
Smaller 2(G) better expansion
1 = 12
3
n
D-regular graph operator spectrum
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What is the optimal spectral gap?
[AlonBoppana] : Any D-regular graph satisfies
2(G) 2(D-1) /D – o(1) 2D-1/2
o(1)0 as the number of vertices grows
[Friedman] : Random D-regular graphs satisfy 2(G) 2(D-1) /D +
A graph is Ramanujan if 2(G) 2(D-1) /D
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Expander constructions
Construction Type 2(G)
[LubotzkyPhilipsSarnak86,
Margulis88,Morgenstern94]
Algebraic
Ramanujan
2(G) 2(D-1) /D
2D-1/2
[ReingoldVadhanWigderson00]
Combinatorial
O(D-1/3)
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Why do we look for combinatorial constructions?
A fundamental object in CS deserves a combinatorial proof [RVW00]
Central component in:• Good combinatorial expanders [CRVW02]• Undirected connectivity in L [Reingold05]• Cayley expanders [ALW01, MW02, RSW04]
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Expander constructions
Construction Type Explicitness
2(G)
[LPS86,Margulis88,
Morgenstern94]Algebraic Full
Ramanujan,
2D-1/2
[ReingoldVadhanWigderson00]
Combinatorial
Full O(D-1/3)
Full= The neighbors of a vertex are computable in time polylog(|V|)
![Page 9: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/9.jpg)
Expander constructions
Construction Type Explicitness
2(G)
[LPS86,Margulis88,
Morgenstern94]Algebraic Full
Ramanujan,
2D-1/2
[ReingoldVadhanWigderson00]
Combinatorial
Full O(D-1/3)
[BiluLinial04] Combinatorial
MildO(D-1/2log1.5D)=
D-1/2+o(1)
Mild= The graph is computable in time poly(|V|)
![Page 10: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/10.jpg)
Expander constructions
Construction Type Explicitness
2(G)
[LPS86,Margulis88,
Morgenstern94]Algebraic Full
Ramanujan,
2D-1/2
[ReingoldVadhanWigderson00]
Combinatorial
Full O(D-1/3)
[BiluLinial04] Combinatorial
MildO(D-1/2log1.5D)=
D-1/2+o(1)
Our result Combinatorial
Full D-1/2+o(1)
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Outline
The result
The technique: a new variant of the zig-zag product
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The construction scheme [ReingoldVadhanWigderson00]
Small expander Increase size
Improve 2Reduce degree
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Heart of [ReingoldVadhanWigderson00] the
zig-zag product
G
H
zig-zag G zigzag H
(N,DG,G)
(DG,DH,H)(N·DG, DH
2, ≈H+G)
vertices degree 2
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The replacement product
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The replacement product
Slightly incomplete…
.
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The replacement product
For simplicity assume G is DG edge-colorable
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The replacement product
Cloud
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Vertices: same as in replacement product Edges: (v,u)E there is a path of length 3
on the replacement product such that:• The first step is inside a cloud• The second step is inter-cloud• The third step is inside a cloud
The Zig-Zag product
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The Zig-Zag product
vuExample:
v and u are connected
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Formally: spectral analysis – not in this talk
Intuitively: analyze “entropy flow”Entropy- a measure of randomness
(H2()=-log Prx,y~[x=y])
If G is D-regular then H2(G())H2()+log(D)
When G has a good spectral gap this almost tight
Intuition vs. formality
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Intuitive analysis: start with some - a entropy deficient distribution on its vertices
Show that H2(G())>>H2() We’ll analyze only one illuminating
distribution• Formal analysis follows a similar
argument
Intuition vs. formality
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Why does the zig-zag work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
The graph afterreplacement
![Page 23: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/23.jpg)
Why does the zig-zag work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
A uniform distribution over a subset of clouds
H step
G step
H step
Wasted
![Page 24: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/24.jpg)
Why does the zig-zag work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
![Page 25: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/25.jpg)
Why does the zig-zag work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
![Page 26: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/26.jpg)
Why does the zig-zag work?Buffer point-of-view
Cloud-label Edge-label [N][DG]
A vertex in the new graph:
A H-step:
65 3
65 10
65 7
0.5
0.5
310
7
H
![Page 27: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/27.jpg)
Why does the zig-zag work?Buffer point-of-view
Cloud-label Edge-label [N][DG]
A vertex in the new graph:
A G-step:
65 3
65 9
G
9 3
3For simplicity
assume G is DG edge-colorable
![Page 28: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/28.jpg)
Why does the zig-zag work?Buffer point-of-view
Cloud-label Edge-label
The support of the distribution
[N][DG]
A uniform distribution over a subset of clouds:• The marginal on is uniform• Conditioned on every cloud in the support, is uniform
H step
G step
H step
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Why does the zig-zag work?Buffer point-of-view
H step
G step
H step
Wasted
Cloud-label Edge-label
H does nothing
Cloud=k Uniform
![Page 30: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/30.jpg)
Edge-label
Why does the zig-zag work?Buffer point-of-view
H step
G step
H step
Cloud-label Edge-label
- Applying G is just taking a step from according to
- Since G is an expander, after this step, should have more entropy
- G is a permutation, hence the entropy in is reduced
![Page 31: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/31.jpg)
Why does the zig-zag work?Buffer point-of-view
H step
G step
H step
Cloud-label Edge-label
H adds entropy!
Cloud=k
![Page 32: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/32.jpg)
Spectral gap of the zig-zag
We take two H-steps, hence the degree is DH
2. On some inputs, only one of the two
steps is useful This incurs a quadratic loss in 2.
![Page 33: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/33.jpg)
First Attempt: zig-zag with 3 cloud steps
v
u
![Page 34: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/34.jpg)
The problem
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
The graph afterreplacement
![Page 35: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/35.jpg)
The problem
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
A uniform distribution over a subset of clouds
H step
G step
H step
G step
H step
Wasted
![Page 36: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/36.jpg)
The problem
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
![Page 37: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/37.jpg)
The problem
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
cloud
![Page 38: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/38.jpg)
Ideally:
H step
G step
H step
G step
H step cloud
cloudcloud
cloud
cloud
cloud
cloud
cloud
cloud
![Page 39: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/39.jpg)
The problem
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
cloud
![Page 40: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/40.jpg)
The problem
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
Potentially: 1 out of 2 cloud steps is wasted
Wasted
![Page 41: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/41.jpg)
The problem:Buffer point-of-view
H step
G step
H step
G step
H step
A uniform distribution over a subset of clouds
Cloud-label Edge-label
Cloud=k Uniform
H does nothing
![Page 42: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/42.jpg)
Edge-label
The problem:Buffer point-of-view
Cloud-label Edge-label
H step
G step
H step
G step
H step
- Applying G is just taking a step from according to
- Since G is an expander, after this step, has more entropy
- G is a permutation, hence the entropy in is reduced
![Page 43: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/43.jpg)
H step
G step
H step
G step
H step
The problem:Buffer point-of-view
Cloud-label Edge-label
H adds entropy!
Cloud=k
![Page 44: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/44.jpg)
H step
G step
H step
G step
H step Edge-label
The problem:Buffer point-of-view
Cloud-label
Since the second component is not uniform, we don’t know
how the entropy flows in the G step (this is like taking a randomstep over the graph according to an unknown distribution).
Edge-label
??
![Page 45: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/45.jpg)
The problem:Buffer point-of-view
Cloud-label Edge-label
Cloud=k Uniform
We might be in this case, in which H does nothing
Potentially: 1 out of 2 cloud steps is wasted
H step
G step
H step
G step
H step
![Page 46: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/46.jpg)
Our solution
Once an H-step is wasted, all the following H-steps are not
We shall make sure that all the following G-steps are cloud-dispersing.• This is done by taking thicker clouds and
choosing H in a special way
![Page 47: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/47.jpg)
Replacement with thicker clouds
An expander over the cloud vertices
![Page 48: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/48.jpg)
Replacement with thicker clouds
Multiple paralleledges
![Page 49: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/49.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
The graph aftermodified replacement
DG10 vertices
![Page 50: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/50.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
A uniform distribution over a subset of clouds
H step
G step
H step
G step
H step
Wasted
![Page 51: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/51.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
DG9 vertices
DG10 vertices
![Page 52: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/52.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
Almost uniform over outgoing
edges
![Page 53: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/53.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H step
![Page 54: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/54.jpg)
Why does this work?
cloud
cloud
cloud
cloud
cloud
cloudcloud
cloud
cloud
H step
G step
H step
G step
H stepNot Wasted
![Page 55: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/55.jpg)
Why does this work?Buffer point-of-view
A vertex in the new graph:
A H-step:
651,… 65
2,…
654,…
0.5
0.5
3,1,…10,2,…
7,4,…
H
H-vertex-label
Cloud-label
Edge-label[N][DG]
10
310
7
![Page 56: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/56.jpg)
Why does this work?Buffer point-of-view
A vertex in the new graph:
A G-step:
651,…
91,…
H-vertex-label
Cloud-label
Edge-label[N][DG]
10
3 3
65 9
G3
For simplicity assume G is DG edge-colorable
![Page 57: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/57.jpg)
H-vertex-label
Why does this work? Buffer point-of-view
Cloud-label
Edge-label
The support of the distribution
[N][DG]10
A uniform distribution over a subset of clouds:• The marginal on is uniform• Conditioned on every cloud in the support, is uniform
H step
G step
H step
G step
H step
![Page 58: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/58.jpg)
Why does this work? Buffer point-of-view
H step
G step
H step
G step
H step
A uniform distribution over a subset of clouds
Cloud=k Uniform
H does nothing
H-vertex-label
Cloud-label
Edge-label
![Page 59: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/59.jpg)
H-vertex-label
Why does this work? Buffer point-of-view
H step
G step
H step
G step
H step
- Applying G is just taking a step from according to
- Since G is an expander, after this step, has more entropy
- G is a permutation, hence the entropy in is reduced
Cloud-label
Edge-labelEdge-label
![Page 60: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/60.jpg)
Why does this work? Buffer point-of-view
H step
G step
H step
G step
H step
Cloud=k
H-vertex-label
Cloud-label
Edge-label
We choose H such that it also “mixes” the second component
H adds entropy!
Now the Edge-label part is close to uniform!
![Page 61: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/61.jpg)
H step
G step
H step
G step
H step
H-vertex-label
Why does this work? Buffer point-of-view
Cloud-label
Edge-labelEdge-label
G moves entropy in the right direction again
Not Wasted
following H adds entropy
![Page 62: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/62.jpg)
The parameters We take 3 steps and at least 2 “work” degree= DH
3 eigenvalue H2
The general case: We take k steps and at least k-1
“work” degree= DHk eigenvalue H
k-1
![Page 63: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/63.jpg)
How to choose H
H is a graph with DG10 vertices that satisfies
two properties: “Mixing”: for every d[DG],
H(d,UDG9) (UDG
,¿) (i.e. almost uniform over the D outgoing
edges) Expansion
How to achieve both? Choose a random H and verify it has both properties
A simple permutationsatisfies this
![Page 64: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/64.jpg)
A problem The previous is only true when G is DG-
edge-colorable In general, G may change the edge-
label in an arbitrary way (may depend on the cloud)
The graphs H is of constant size The number of clouds =The size of the
large graph grows to infinity. It seems unlikely that there is a choice for
H that is good for all distributions.
65 3
G
9 ¿¿¿
![Page 65: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/65.jpg)
A solution
Take the large graph G to be “locally invertible”
G(v,i)=(v[i],(i))
This property of G is preserved throughout the construction
Now things are similar to the DG-edge-colorable case
![Page 66: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/66.jpg)
The final product
G – a “locally invertible” (N, DG, G) H1,…,Hk – a sequence of “mixing”
expanders (found by exhaustive search) each is (DG
10k, DH, H) Do modified replacement and take
the graph of “paths” of length k The resulting graph is
(NDG10k, DH
k, O(Hk-1 + G))
![Page 67: A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv](https://reader038.vdocument.in/reader038/viewer/2022110303/55165cd45503469d698b4ff2/html5/thumbnails/67.jpg)
Our result: a fully-explicit combinatorial construction of D-regular graphs with 2(G)=D-1/2+o(1) .
Can we push this further to O(D-1/2) ? Ramanujan+ ?
Thank you!
Conclusions