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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

Gossip-Based Computation of Aggregation Information

David Kempe

Alin Dobra

Johannes Gehrke

Presented by Hao Zhou

content

Content

Introduction

Gossip-based Algorithm

Analyze Gossip-based Algorithm

introduction

Introduction

Peer to peer network

Unstructured network

Gnutella, Napster

Structured network

DHT-based systems

such as Pastry, Chord, Tepastry, CAN

Advantages of DHT-based systems

Fast: O (log n)

Can exactly find a publishing object in a gigantic network space

gossip based algorithm

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
  • Disadvantage of DHT-based systems
  • 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 algorithm5

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 algorithm6

Gossip-based Algorithm

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

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

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 algorithm9

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 algorithm10

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

advantages of gossip algorithm

Advantages of Gossip Algorithm

Algorithm is very simple

Converge speed is very fast

Can automatically adjust itself

Nodes join the network

Nodes leave the network

disadvantages of gossip algorithm

Disadvantages of Gossip Algorithm

From their theory, we know after O(logn+log(1/ε)+ log(1/δ)) rounds,

the estimation average value in a local node can be see as a global average value.

But in practice, If we do not know the size of the network, how do we know how many rounds a estimation average value is close enough to the real average value.