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![Page 1: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/1.jpg)
Order Optimal Information SpreadingUsing Algebraic Gossip
Chen Avin, Michael Borokhovich, Keren Censor-Hillel, Zvi Lotker
Communication Systems Engineering, BGU, Israel
Computer Science and Artificial Intelligence Laboratory, MIT, USA
PODC 2011
![Page 2: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/2.jpg)
The k -Dissemination Problem
1/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 = 5 x2 = 15
x3 = 10 x4 = 12
all nodes need all the k values
as quickly as possible
UniformAlgebraicGossip
Tree BasedAlgebraicGossip
![Page 3: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/3.jpg)
The k -Dissemination Problem
1/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 = 5 x2 = 15
x3 = 10 x4 = 12
all nodes need all the k values
as quickly as possible
UniformAlgebraicGossip
Tree BasedAlgebraicGossip
![Page 4: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/4.jpg)
Gossip Algorithm
2/12
what is algebraic gossip?
![Page 5: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/5.jpg)
Gossip Algorithm
2/12
Time is measuredin rounds
choose a partner
uniformlynon
uniformly
send a message
push pull exch.
message content
?
v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8v1 v2
v3 v4
v5
v6
v7
v8
![Page 6: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/6.jpg)
Gossip Algorithm
2/12
Time is measuredin rounds
choose a partner
uniformlynon
uniformly
send a message
push pull exch.
message content
?
v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8
![Page 7: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/7.jpg)
Gossip Algorithm
2/12
Time is measuredin rounds
choose a partner
uniformlynon
uniformly
send a message
push pull exch.
message content
?
v1 v2
v3 v4
v5
v6
v7
v8v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8
![Page 8: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/8.jpg)
Gossip Algorithm
2/12
Time is measuredin rounds
choose a partner
uniformlynon
uniformly
send a message
push pull exch.
message content
?
v1 v2
v3 v4
v5
v6
v7
v8v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8
![Page 9: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/9.jpg)
Algebraic Gossip
3/12
instead of sending randomly chosen values...
every message – a single value
v1 v2
v3 v4
v5
v6
v7
v8
x1
x4
x3
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
120
random coefficients ∈ Fq
1× 1x1 + 0x2 + 2x3 + 7x4 = 250
2× 2x1 + 0x2 + 0x3 + 7x4 = 45
5x1 + 0x2 + 2x3 + 21x4 = 340
rank k – node finishesonly helpful
messages are stored
![Page 10: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/10.jpg)
Algebraic Gossip
3/12
instead of sending randomly chosen values...
every message – a single value
v1 v2
v3 v4
v5
v6
v7
v8
x1
x4
x3
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
120
random coefficients ∈ Fq
1× 1x1 + 0x2 + 2x3 + 7x4 = 250
2× 2x1 + 0x2 + 0x3 + 7x4 = 45
5x1 + 0x2 + 2x3 + 21x4 = 340
rank k – node finishesonly helpful
messages are stored
![Page 11: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/11.jpg)
Algebraic Gossip
3/12
instead of sending randomly chosen values...
every message – a single value
v1 v2
v3 v4
v5
v6
v7
v8
x1
x4
x3
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
120
random coefficients ∈ Fq
1× 1x1 + 0x2 + 2x3 + 7x4 = 250
2× 2x1 + 0x2 + 0x3 + 7x4 = 45
5x1 + 0x2 + 2x3 + 21x4 = 340
rank k – node finishesonly helpful
messages are stored
![Page 12: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/12.jpg)
Algebraic Gossip
3/12
instead of sending randomly chosen values...
every message – a single value
v1 v2
v3 v4
v5
v6
v7
v8
x1
x4
x3
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
120
random coefficients ∈ Fq
1× 1x1 + 0x2 + 2x3 + 7x4 = 250
2× 2x1 + 0x2 + 0x3 + 7x4 = 45
5x1 + 0x2 + 2x3 + 21x4 = 340
rank k – node finishesonly helpful
messages are stored
![Page 13: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/13.jpg)
Algebraic Gossip
3/12
instead of sending randomly chosen values...
every message – a single value
v1 v2
v3 v4
v5
v6
v7
v8
x1
x4
x3
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes send random linear combinations
every message – linear equation
v1 v2
v3 v4
v5
v6
v7
v8
x1+ 2x3
+ 7x4= 250
3x1 + 4x4 =134
4x2+ x4
+ 8x3= 730
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
nodes store equations in a matrix form:
1 0 2 72 0 0 71 1 0 00 0 0 0
·
x1x2x3x4
=
25045780
120
random coefficients ∈ Fq
1× 1x1 + 0x2 + 2x3 + 7x4 = 250
2× 2x1 + 0x2 + 0x3 + 7x4 = 45
5x1 + 0x2 + 2x3 + 21x4 = 340
rank k – node finishesonly helpful
messages are stored
![Page 14: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/14.jpg)
So, Why Algebraic Gossip is Faster?
4/12
A B
Pr (helpful) = 1k
[x1, x2, x3, . . . xk ] [x1, x2=?, x3, . . . xk ]
Without Algebraic Gossip (Random Message Selection)
A has all values B is missing one valueA sends a random message
A B
Pr (helpful) = 1k
[x1, x2, x3, . . . xk ] [x1, x2=?, x3, . . . xk ]
Without Algebraic Gossip (Random Message Selection)
A has all values B is missing one valueA sends a random message
A B
Pr (helpful) = 1 − 1q
rank k
rank k − 1
With Algebraic Gossip (Random Linear Equations)
k indep. equations k − 1 indep. equationsA sends a random linear equation
qk−qk−1
qk = 1− 1q
![Page 15: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/15.jpg)
So, Why Algebraic Gossip is Faster?
4/12
A B
Pr (helpful) = 1k
[x1, x2, x3, . . . xk ] [x1, x2=?, x3, . . . xk ]
Without Algebraic Gossip (Random Message Selection)
A has all values B is missing one valueA sends a random message
A B
Pr (helpful) = 1k
[x1, x2, x3, . . . xk ] [x1, x2=?, x3, . . . xk ]
Without Algebraic Gossip (Random Message Selection)
A has all values B is missing one valueA sends a random message
A B
Pr (helpful) = 1 − 1q
rank k
rank k − 1
With Algebraic Gossip (Random Linear Equations)
k indep. equations k − 1 indep. equationsA sends a random linear equation
qk−qk−1
qk = 1− 1q
![Page 16: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/16.jpg)
Research Goal
5/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 = 5 x2 = 15
x3 = 10 x4 = 12k -Disseminationproblem
Analyze uniformalgebraic gossip
is itoptimal?
for whichgraphs?
Study non-uniformalgebraic gossip
we propose treebased algebraic gossip
![Page 17: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/17.jpg)
Research Goal
5/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 = 5 x2 = 15
x3 = 10 x4 = 12k -Disseminationproblem
Analyze uniformalgebraic gossip
is itoptimal?
for whichgraphs?
Study non-uniformalgebraic gossip
we propose treebased algebraic gossip
![Page 18: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/18.jpg)
Related Work
[Deb et al., 2006] – complete graphFor k ln3 n : Tight bound – Θ(k).
[Mosk-Aoyama and Shah, 2006] – any graph, k = nNot tight bound.
[Borokhovich et al., 2010] – k = nUpper bound – O(∆n) for any graph.Tight bound – Θ(n) for constant max degree graphs.
[Haeupler, 2010] – any graphFor k = Ω(n) : Tight bound – Θ(n/γ).For k < o(n) : Not tight for: line, grid, binary tree, ...
6/12
k -Dissemination with uniform algebraic gossip
![Page 19: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/19.jpg)
Related Work
[Deb et al., 2006] – complete graphFor k ln3 n : Tight bound – Θ(k).
[Mosk-Aoyama and Shah, 2006] – any graph, k = nNot tight bound.
[Borokhovich et al., 2010] – k = nUpper bound – O(∆n) for any graph.Tight bound – Θ(n) for constant max degree graphs.
[Haeupler, 2010] – any graphFor k = Ω(n) : Tight bound – Θ(n/γ).For k < o(n) : Not tight for: line, grid, binary tree, ...
6/12
k -Dissemination with uniform algebraic gossip
![Page 20: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/20.jpg)
Related Work
[Deb et al., 2006] – complete graphFor k ln3 n : Tight bound – Θ(k).
[Mosk-Aoyama and Shah, 2006] – any graph, k = nNot tight bound.
[Borokhovich et al., 2010] – k = nUpper bound – O(∆n) for any graph.Tight bound – Θ(n) for constant max degree graphs.
[Haeupler, 2010] – any graphFor k = Ω(n) : Tight bound – Θ(n/γ).For k < o(n) : Not tight for: line, grid, binary tree, ...
6/12
k -Dissemination with uniform algebraic gossip
![Page 21: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/21.jpg)
Related Work
[Deb et al., 2006] – complete graphFor k ln3 n : Tight bound – Θ(k).
[Mosk-Aoyama and Shah, 2006] – any graph, k = nNot tight bound.
[Borokhovich et al., 2010] – k = nUpper bound – O(∆n) for any graph.Tight bound – Θ(n) for constant max degree graphs.
[Haeupler, 2010] – any graphFor k = Ω(n) : Tight bound – Θ(n/γ).For k < o(n) : Not tight for: line, grid, binary tree, ...
6/12
k -Dissemination with uniform algebraic gossip
![Page 22: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/22.jpg)
Results
7/12
two main results
![Page 23: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/23.jpg)
1st Result
7/12
k -Dissemination
Uniform algebraic gossip
Any graph
O(∆(k + log n + D))
max degree diameter
Const max degree graphs
Θ(k + D)
Optimal!
Results hold for
push, pull, exch
Barbell graph
Ω(nk)
Bad!
v
u
![Page 24: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/24.jpg)
1st Result
7/12
k -Dissemination
Uniform algebraic gossip
Any graph
O(∆(k + log n + D))
max degree diameter
Const max degree graphs
Θ(k + D)
Optimal!
Results hold for
push, pull, exch
Barbell graph
Ω(nk)
Bad!
v
u
![Page 25: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/25.jpg)
1st Result
7/12
k -Dissemination
Uniform algebraic gossip
Any graph
O(∆(k + log n + D))
max degree diameter
Const max degree graphs
Θ(k + D)
Optimal!
Results hold for
push, pull, exch
Barbell graph
Ω(nk)
Bad!
v
u
![Page 26: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/26.jpg)
1st Result
7/12
k -Dissemination
Uniform algebraic gossip
Any graph
O(∆(k + log n + D))
max degree diameter
Const max degree graphs
Θ(k + D)
Optimal!
Results hold for
push, pull, exch
Barbell graph
Ω(nk)
Bad!
v
u
![Page 27: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/27.jpg)
1st Result
7/12
k -Dissemination
Uniform algebraic gossip
Any graph
O(∆(k + log n + D))
max degree diameter
Const max degree graphs
Θ(k + D)
Optimal!
Results hold for
push, pull, exch
Barbell graph
Ω(nk)
Bad!
v
u
![Page 28: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/28.jpg)
2nd Result
8/12
k -Dissemination
Tree based algebraic gossip
v1 v2
v3 v4
v5
v6
v7
v8 v1 v2
v3 v4
v5
v6
v7
v8
Phase 1Span tree algo S
Phase
2
Algebraic gossiponly with parent
t(S)
O(∆(k + log n + D)
O(t(S)+k +log n+d(S))
Any graph, k = Ω(n)
Θ(n)
Optimal!
![Page 29: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/29.jpg)
2nd Result
8/12
k -Dissemination
Tree based algebraic gossip
v1 v2
v3 v4
v5
v6
v7
v8
v1 v2
v3 v4
v5
v6
v7
v8
Phase 1Span tree algo S
Phase
2
Algebraic gossiponly with parent
t(S)
O(∆(k + log n + D)
O(t(S)+k +log n+d(S))
Any graph, k = Ω(n)
Θ(n)
Optimal!
![Page 30: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/30.jpg)
2nd Result
8/12
k -Dissemination
Tree based algebraic gossip
v1 v2
v3 v4
v5
v6
v7
v8 v1 v2
v3 v4
v5
v6
v7
v8
Phase 1Span tree algo S
Phase
2
Algebraic gossiponly with parent
t(S)
O(∆(k + log n + D)
O(t(S)+k +log n+d(S))
Any graph, k = Ω(n)
Θ(n)
Optimal!
![Page 31: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/31.jpg)
2nd Result
8/12
k -Dissemination
Tree based algebraic gossip
v1 v2
v3 v4
v5
v6
v7
v8 v1 v2
v3 v4
v5
v6
v7
v8
Phase 1Span tree algo S
Phase
2
Algebraic gossiponly with parent
t(S)
O(∆(k + log n + D)
O(t(S)+k +log n+d(S))
Any graph, k = Ω(n)
Θ(n)
Optimal!
![Page 32: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/32.jpg)
2nd Result
8/12
k -Dissemination
Tree based algebraic gossip
v1 v2
v3 v4
v5
v6
v7
v8 v1 v2
v3 v4
v5
v6
v7
v8
Phase 1Span tree algo S
Phase
2
Algebraic gossiponly with parent
t(S)
O(∆(k + log n + D)
O(t(S)+k +log n+d(S))
Any graph, k = Ω(n)
Θ(n)
Optimal!
![Page 33: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/33.jpg)
Proof Overview
9/12
k -Dissemination with uniform algebraic gossip
O(∆(k + log n + D))
![Page 34: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/34.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 35: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/35.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 36: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/36.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 37: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/37.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 38: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/38.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 39: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/39.jpg)
Converting a Graph to a System of Queues
9/12
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥?
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆
x1 x2
x3 x4
x5
x6
v1 v2
v3 v4
v5
v6
v7
v8
p ≥ 1∆ (1 − 1
q )
x1 x2
x3 x4
x5
x6
initial graph with an arbitrary node v4
when v4 will finish the protocol?
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
BFS spanning tree rooted at v4
we ignore the messages coming from other edges
v1 v2
v3 v4
v5
v6
v7
v8
x1 x2
x3
x4
x5
x6
customers are helpful messages
customers increase node’s rank
a node need k helpful messages to finish
v5 chooses v6 w.p. ≥ 1∆
message is helpful w.p. ≥ (1 − 1q )
service time is ∼ Geom(p)
![Page 40: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/40.jpg)
Exponential Servers Instead of Geometric
10/12
v1 v2
v3 v4
v5
v6
v7
v8
p
p
p
p
p
p
p
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
µ = p
If X ∼ Geom(p), and Y ∼ Exp(p), then: Pr(Y > t) ≥ Pr(X > t)
exponential server is slower than geometric
we replace servers, thus increasing the stopping time
![Page 41: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/41.jpg)
Exponential Servers Instead of Geometric
10/12
v1 v2
v3 v4
v5
v6
v7
v8
p
p
p
p
p
p
p
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
µ = p
If X ∼ Geom(p), and Y ∼ Exp(p), then: Pr(Y > t) ≥ Pr(X > t)
exponential server is slower than geometric
we replace servers, thus increasing the stopping time
![Page 42: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/42.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2 v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 43: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/43.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2 v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 44: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/44.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2
v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 45: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/45.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2
v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 46: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/46.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2
v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 47: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/47.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2
v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 48: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/48.jpg)
Line is Slower Than Tree
11/12
v1 v2
v3 v4
v5
v6
v7
v8
µ
µ
µ
µ
µ
µ
µ
v4µ µ µ
v4µ µ µλ = µ
2
v4µ µ µλ = µ
2
When does the
last customer
leave the system?
Reduce tree to line
Take all customers out and use Jackson theorem
Inter-arrival time ∼Exp( µ2 )
All arrive after O((k + log n)/µ)
Time to cross MM1 ∼Exp( µ2 )
Time to cross D queues O((D + log n)/µ)
Total: O((k + log n + D)/µ) = O(∆(k + log n + D)) rounds
![Page 49: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/49.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 50: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/50.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 51: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/51.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 52: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/52.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 53: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/53.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 54: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/54.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 55: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/55.jpg)
Summary
12/12
k -Dissemination problem
Uniform algebraic gossip Tree based algebraic gossip
Any graph
O(∆(k + log n + D))
Const max degree graphs
Θ(k + D)Optim
al!
Any graph
O(t(S) + k + log n + d(S))
Any graph, k = Ω(n)
Θ(n)Optim
al!
Open questions
Optimal cases What S to use?
THANK YOU!
![Page 56: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/56.jpg)
Algebraic Gossip – Overhead?
12/12
1,0,2,7 | 250
x1 + 2x3 + 7x4 = 250
ai ∈ Fq∑
aixi ∈ Frq
Initial value size: r log q bits.
Overhead (coefficients): k log q bits.
Bits efficiency: r log qk log q+r log q = r
k+r → 1.
nodes store equations in a matrix form:
1 0 2 72 0 0 41 1 0 00 2 1 5
·
x1x2x3x4
=
2504578308
ai ∈ Fq xi ∈ Frq
∑aixi ∈ Fr
q
![Page 57: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/57.jpg)
Algebraic Gossip – Overhead?
12/12
1,0,2,7 | 250
x1 + 2x3 + 7x4 = 250
ai ∈ Fq∑
aixi ∈ Frq
Initial value size: r log q bits.
Overhead (coefficients): k log q bits.
Bits efficiency: r log qk log q+r log q = r
k+r → 1.
nodes store equations in a matrix form:
1 0 2 72 0 0 41 1 0 00 2 1 5
·
x1x2x3x4
=
2504578308
ai ∈ Fq xi ∈ Frq
∑aixi ∈ Fr
q
![Page 58: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/58.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 59: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/59.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 60: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/60.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 61: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/61.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 62: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/62.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 63: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/63.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 64: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/64.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 65: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/65.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 66: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/66.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 67: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/67.jpg)
Algebraic Gossip – Example
12/12
v1
v2 v3
x1 = a
x3 = cx2 = b
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 0 00 0 0
·
x1x2x3
=
a00
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
325
random coefficients
0,2,0 | 2b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
0 0 00 0 00 0 1
·
x1x2x3
=
00c
562
random coefficients
5,12,0 | 5a + 12b
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
v1
v2 v3
1 0 00 2 00 0 0
·
x1x2x3
=
a2b0
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
436
random coefficients
20,48,6 | 20a + 48b + 6c
v1
v2 v3
1 0 00 2 0
20 48 6
·
x1x2x3
=
a2b
20a + 48b + 6c
0 0 00 1 00 0 0
·
x1x2x3
=
0b0
5 12 00 0 00 0 1
·
x1x2x3
=
5a + 12b0c
⇒x1 = ax2 = bx3 = c
![Page 68: michaelbor.github.iomichaelbor.github.io/presentation_podc2011.pdf · So, Why Algebraic Gossip is Faster? 4/12 A B Pr(helpful) = 1 k [x 1;x2;x 3;:::xk] [x ;x2=?;x ;:::xk] Without](https://reader035.pdfslide.us/reader035/viewer/2022071020/5fd481f80632f6111e516116/html5/thumbnails/68.jpg)
Borokhovich, M., Avin, C., and Lotker, Z. (2010).Tight bounds for algebraic gossip on graphs.In 2010 IEEE International Symposium on InformationTheory Proceedings (ISIT), pages 1758 –1762.
Deb, S., Médard, M., and Choute, C. (2006).Algebraic gossip: a network coding approach to optimalmultiple rumor mongering.IEEE Transactions on Information Theory,52(6):2486–2507.
Haeupler, B. (2010).Analyzing Network Coding Gossip Made Easy.To appear in the 43rd ACM Symposium on Theory ofComputing (STOC), 2011.
Mosk-Aoyama, D. and Shah, D. (2006).Information dissemination via network coding.In ISIT, pages 1748–1752.
12/12
