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

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Content

• Introduction
• Gossip-based Algorithm
• Analyze Gossip-based Algorithm

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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

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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 calculate the average value
  of Xavg (x1x2x3x12)/12
• Disadvantage of DHT-based systems
• Gossip-based algorithm
• Objective let the estimation average value close
  to Xavg for every node

X3
X2
X1
X4
X11
X10
X12
X5
X9
X6
X8
X7
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Gossip-based Algorithm

• Xavg (X1X2X3X4)/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

(X4x2)/2

• time0,
• Xeavg1X1, Xeavg2x2, Xeavg3x3, Xeavg4x4
• Time1, Randomly pick up another node
• Xeavg1X1/ 2, Xeavg2(X4x2)/ 2 Xeavg3(X2X3)/ 2
  Xeavg4 (X1X3X4)/ 2

X1
X2/2
(X1x1x3x4)/4
X2
X2/2
X1/2
X1/2
X1/2
(X2x2x3x4)/4
X3
X4/2
(X2x3)/2
X3/2
X4
X4/2
(X1x3x4)/4
X3/2
(X1x3x4)/2
(X2x2x3x4)/4

• Time 2,
• Xeavg1(X1X1X3X4)/ 4, Xeavg2(X2X2X3X4)/4,
  Xeavg3(X2X2X3X4)/ 4, Xeavg4(X1X3X4)/ 4,

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Gossip-based Algorithm

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

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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(t1) ( ) var(t)

Xeavg
Xavg
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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 lt e, we can
  say Xeavg is Xavg.

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Analyze Gossip-based Algorithm

• Roughly say, after O(lognlog(1/ e)) rounds, can
  we say variance error lt e in every node
• Maybe there are broken network connections

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Analyze Gossip-based Algorithm

• We have to control the percentage of nodes who
  obtain errlte
• We say with probability at least 1-d,
• after O(lognlog(1/e)log(1/d)) rounds,
• The errXeavg Xavg lt e
• Their contribution
• The diffusion speed of uniform gossip is
  O(lognlog(1/e)log(1/d)) , with probability at
  least 1- d, and variance error lt e

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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

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Disadvantages of Gossip Algorithm

• From their theory, we know after O(lognlog(1/e)
  log(1/d)) 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.

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• Thank you !

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