Gossip-Based Computation of Aggregation Information

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Gossip-Based Computation of Aggregation Information. David Kempe Alin Dobra Johannes Gehrke Presented by Hao Zhou. Content. Introduction Gossip-based Algorithm Analyze Gossip-based Algorithm. Introduction. Peer to peer network Unstructured network Gnutella, Napster Structured network

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Gossip-Based Computation of Aggregation Information

David Kempe

Alin Dobra

Johannes Gehrke

Presented by Hao Zhou

Content

Introduction

Gossip-based Algorithm

Analyze Gossip-based Algorithm

Introduction

Peer to peer network

Unstructured network

Gnutella, Napster

Structured network

DHT-based systems

such as Pastry, Chord, Tepastry, CAN

Fast: O (log n)

Can exactly find a publishing object in a gigantic network space

Gossip-based Algorithm

• But if we want to get the aggregation information for the whole network
• Such as sum value, average value
• Our objective is to calculatethe average value of Xavg =(x1+x2+x3…+x12)/12
• Gossip-based algorithm
• Objective: let the estimation average value close to Xavg for every node

X2

X3

X1

X11

X4

X10

X12

X5

X9

X6

X8

X7

Gossip-based Algorithm

• Xavg = (X1+X2+X3+X4)/4 is a real average value in a peer to peer network
• Xeavg is the estimated average value for the P2P network in a node

(X4+x2)/2

• time=0,
• Xeavg1=X1, Xeavg2=x2, Xeavg3=x3, Xeavg4=x4
• Time=1, Randomly pick up another node
• Xeavg1=X1/ 2, Xeavg2=(X4+x2)/ 2 Xeavg3=(X2+X3)/ 2 Xeavg4= (X1+X3+X4)/ 2

X1

X2/2

(X1+x1+x3+x4)/4

X1/2

X2/2

X2

X1/2

X1/2

(X2+x2+x3+x4)/4

X3

X4/2

(X2+x3)/2

X3/2

X4

X4/2

(X1+x3+x4)/4

(X1+x3+x4)/2

X3/2

(X2+x2+x3+x4)/4

• Time = 2,
• Xeavg1=(X1+X1+X3+X4)/ 4, Xeavg2=(X2+X2+X3+X4)/4, Xeavg3=(X2+X2+X3+X4)/ 4, Xeavg4=(X1+X3+X4)/ 4,

Gossip-based Algorithm

• After m rounds/iterations, Xeavg is very close to Xavg
• We can see Xeavg as Xavg

Converge Speed

Define a variance error= | Xeavg-Xavg |

Our objective is to make the variance close to 0

Calculate the converge speed of this variance

In every round, the variance drops to less than half its previous value

var(t+1) = ( ) var(t)

Xeavg

Xavg

Analyze Gossip-based Algorithm

Gossip-based algorithm is an approximation method

We can control the accuracy

Xeavg never = Xavg, but Xeavg can be very close to Xavg

When variance error=| Xeavg – Xavg| <= ε, we can say Xeavg is Xavg.

Analyze Gossip-based Algorithm

Roughly say, after O(logn+log(1/ ε)) rounds, can we say variance error <= ε in every node

Maybe there are broken network connections

Analyze Gossip-based Algorithm

We have to control the percentage of nodes who obtain err<=ε

We say with probability at least 1-δ,

after O(logn+log(1/ε)+log(1/δ)) rounds,

The err=|Xeavg – Xavg| <= ε

Their contribution:

The diffusion speed of uniform gossip is O(logn+log(1/ε)+log(1/δ)) , with probability at least 1- δ, and variance error <= ε

Algorithm is very simple

Converge speed is very fast