olivier beaumont, lionel eyraud-dubois, shailesh …...olivier beaumont, lionel eyraud-dubois,...

Overlay networks maximizing throughput

Olivier Beaumont, Lionel Eyraud-Dubois, Shailesh Kumar Agrawal

Cepage team, LaBRI, Bordeaux, France

RO Working GroupMay 24, 2011

Introduction

Outline

1 Introduction

2 Successive algorithms

3 Simulations

4 Conclusions

L. Eyraud-Dubois (Cepage) Overlay networks maximizing throughput 2/ 18

Introduction

Introduction

In this talk: broadcast/streaming operation

One source node holds (or generates) a message

All nodes must receive the complete message

Steady-state: quantity of data per time unit

Goal: optimize throughput

Keep things reasonable: degree constraint

N1 N2 N3

Nn

Ni

N0

bn

d3d2b1d1

b3b2

bidi dn

b0

d0


Introduction

An example

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2


Introduction

An example

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

Best tree: T = 1


Introduction

An example

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

10.5

0.5

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

1

0.51.5

Best DAG: T = 1.5


Introduction

An example

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

1

1

Optimal: T = 2


Introduction

Precise model

An instance

n nodes, with output bandwidth bi and maximal out-degree di

node N0 is the master node that holds the data

A solution (Trees)

A weighted set of spanning trees (wk, Tk) rooted at N0

χk(Nj ,Ni) = 1 iff there is an edge from Nj to Ni in tree Tk

∀j,∑

k

∑i χk(Nj ,Ni)wk ≤ bj (capacity constraint at node j)

∀j,∑

imaxk χk(Nj ,Ni) ≤ dj (degree constraint at node j)

Maximize T =∑

k wk


Introduction

Precise model

An instance

n nodes, with output bandwidth bi and maximal out-degree di

node N0 is the master node that holds the data

A solution (Flows)

Flow f ij from node Nj to Ni

∀j,∑

i fij ≤ bj (capacity constraint at node j)

∀j,∣∣∣{i, f ij > 0

}∣∣∣ ≤ dj (degree constraint at node j)

Maximize T = minj mincut(N0,Nj)


Introduction

Complexity

3-Partition

3p integers ai such that∑

i ai = pT

Partition into p sets Sl such that∑i∈Sl

ai = T

Reduction

p “server” nodes, bj = 2T and dj = 4

3p “client” nodes, bj+p = T − aj anddj+p = 1

1 “terminal” node, b4p = 0, d4p = 0

b0 = b1 = 2T

N0 N1

a2 + a5+a6 = T

T − a1

T − a2a2

a1


Successive algorithms

Outline

1 Introduction

2 Successive algorithmsAcyclic AlgorithmWith cycles

3 Simulations

4 Conclusions


Successive algorithms

Upper bound

If S has throughput T

Node Ni uses at most Xi = min(bi, Tdi)

Total received rate: nT

Thus∑n

i=0min(bi, Tdi) ≥ nTOf course, T ≤ b0

Our algorithms

Inputs: an instance, and a goal throughput T

Output: a solution with resource augmentation (additionalconnections allowed)


Successive algorithms Acyclic Algorithm

Acyclic algorithm

If∑n−1

i=0 min(bi, Tdi) ≥ nT

Order nodes by capacity : X1 ≥ X2 ≥ · · · ≥ Xn

Each node k sends throughput T to as many nodes as possible, inconsecutive order

N0 N1 N2 N3 N4 N5

Provides a valid solution

b0 ≥ TSort by Xi =⇒ ∀k,

∑ki=0Xi ≥ (k + 1)T

Since Xk ≤ Tdk, the outdegree of Nk is at most dk + 1


Successive algorithms With cycles

General case:∑n

i=0Xi ≥ nT

N0 N1 N2 N3 N4 N5

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

1

1

Acyclic algorithm until k0 such that∑k0

i=0Xi < (k0 + 1)T

Recursively build partial solutions in whichI All nodes up to Nk are servedI Only node Nk has remaining bandwidth

Use the source and Nk0−1 to serve Nk0 and Nk0+1

Then for all k, Nk+1 is served by Nk and Nk−1

Final outdegree of Ni: oi ≤ max(di + 2, 4)

Acyclic solution: oi ≤ di + 1

Degree of the source and of Nk0−1 is increased by 1

Nk has edges to Nk−2, Nk−1, Nk+1 and Nk+2.



General case:∑n

i=0Xi ≥ nT

N0 N1 N2 N3 N4 N5

N1

b1 = 1

N2

b2 = 1

N0

b0 = 2

11

1

1

Acyclic algorithm until k0 such that∑k0

i=0Xi < (k0 + 1)TRecursively build partial solutions in which

I All nodes up to Nk are servedI Only node Nk has remaining bandwidth

Use the source and Nk0−1 to serve Nk0 and Nk0+1

Then for all k, Nk+1 is served by Nk and Nk−1

Final outdegree of Ni: oi ≤ max(di + 2, 4)

Acyclic solution: oi ≤ di + 1

Degree of the source and of Nk0−1 is increased by 1

Nk has edges to Nk−2, Nk−1, Nk+1 and Nk+2.



A running example

N2N1

∀i > 0, bi =78

b0 =54

N0

34

114



A running example

N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

114



A running example

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

114

18

78



A running example

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

78



A running example

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

78

34



A running example

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

34



A running example

N6

38

383

8

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

58



A running example

N7

18

12

N6

38

383

8

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

12

12



A running example

N8

123

8

18

N7

18

12

N6

38

383

8

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

12

12



A running example

N9

12

14

14

12

N8

123

8

18

N7

18

12

N6

38

383

8

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1

12



A running example

N10

18

38

12

12

12

N9

12

14

14

12

N8

123

8

18

N7

18

12

N6

38

383

8

58

14 1

4

N5

18

18

18

N4N3

18

78

N2N1

∀i > 0, bi =78

b0 =54

N0

34

1


Simulations

Outline

1 Introduction


3 Simulations

4 Conclusions


Simulations

Comparison of different solutions

Unconstrained solution

Best achievable throughput without degree constraints:∑

i bin

Best Tree

In a tree of throughput T , flow through all edges must be T . Counting

the edges yield∑

imin(di,⌊biT

⌋) ≥ n.

Best Acyclic

Computed by the Acyclic algorithm

Cyclic

Throughput when adding cycles


Simulations

Experimental setting

Random instance generation

Outgoing bandwidths generated from the data of XtremLabproject

Nodes degrees are homogeneous

Complementary CDF of the data used

0.0001

0.001

0.01

0.1

1

1000 10000 100000

PR(X

>x)

x


Simulations

Results: comparisons to Cyclic

0

0.2

0.4

0.6

0.8

1

1 10 100 1000 10000

Thr

ough

put r

atio

Number of nodes

Ratio to Cycle

DAG d=3DAG d=5

DAG d=10Tree d=3Tree d=5

Tree d=10


Simulations

Results: Cyclic vs Unconstrained

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

Thr

ough

put r

atio

Output degree

Cycle ratio against Optimal

N=10N=100

N=1000


Conclusions

Outline

1 Introduction


3 Simulations

4 Conclusions


Conclusions

Summary

Theoretical study of the problem: optimal resource augmentationalgorithm

In practice:I a low degree is enough to reach a high throughputI an acyclic solution is very reasonable

I once the overlay is computed, there exist distributed algorithms toperform the broadcast

Going further

Worst-case approximation ratio of Acyclic ?

Study the robustness of our algorithms

Design on-line and/or distributed versions


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