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Bit-efficient consensus

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Title: Efficient adaptive dispensing against omission failures Author: Micha Strojnowski Last modified by: Micha Strojnowski Created Date: 11/19/2005 2:12:21 PM – PowerPoint PPT presentation

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Title: Bit-efficient consensus


1
Bit-efficient consensus
  • Michal Strojnowski

2
The Byzantine Generals Problem
  • n entities
  • Each has its value
  • They communicate using only point-to-point
    messages
  • The goal is to choose one of this values, and
    make it known to everyone

3
The Byzantine Generals Problem
  • n entities
  • Each has its value
  • They communicate using only point-to-point
    messages
  • The goal is to choose one of this values, and
    make it known to everyone
  • Simplest solution one leader sends its value to
    everyone

4
Traitors
  • In the Byzantine Generals Problem, we assumed any
    malicious behavior of traitors
  • In this case, no protocol is resilient to n/3
    such aberrations

5
Traitors
  • In the Byzantine Generals Problem, we assumed any
    malicious behavior of traitors
  • In this case, no protocol is resilient to n/3
    such aberrations

6
Traitors
  • In the Byzantine Generals Problem, we assumed any
    malicious behavior of traitors
  • In this case, no protocol is resilient to n/3
    such aberrations

7
Crashes
  • Assume that only possible violation of protocol
    is a crash (i.e. death of general)
  • Consensus is possible for any number of crashes,
    but still not easy to achieve
  • t crashes require t1 rounds of communication
  • (in asynchronous case problem is unsolvable)
  • Most protocols use big number of lengthy messages

8
Lowering the number of messages
  • Define a communication graph, and send messages
    through it edges.
  • Ensure that there exists a big, strongly
    connected subset of correct nodes
  • Devise a protocol in which the final value is
    easy to obtain

9
Communicators
  • Degree of every node is d
  • In each subset of 4n/d nodes, there exists a
    subset in which each node has k neighbors
  • Each such subset has at least n/4d vertices
  • Each two sets of n/4d vertices are connected

10
Family of communicators
degree d2 diameter (log n)/2
degree d diameter log n

degree n diameter 1
11
Preliminaries
  • 1 is a dominant value. Any node that receives 1,
    change its own value to 1.
  • Depending on value, we will call node
  • either 1-node or 0-node.
  • We will be using only 1-bit messages.

12
Algorithm (case tltn/2)
  • Each 1-node sends its value to its neighbors. If
    less than k nodes reply, it switches to the next
    communicator.
  • After log n rounds of communication without
    switching, node makes its decision. After that,
    it only answers to received messages.
  • 0-nodes start the same procedure t2log n rounds
    later

13
Analysis
  • Why wait t rounds?

14
Analysis
  • Why wait t rounds?

15
Analysis
  • Why wait t rounds?

16
Analysis
  • Why wait t rounds?

17
Analysis
  • Why wait t rounds?

18
Analysis
  • Why wait t rounds?

19
Analysis
  • If all 1-nodes die out, the final decision is 0
  • If not, in any round without failure,
    neighborhood of
  • 1-nodes expands k/2 times, until it reaches size
    n/2
  • Any node eventually communicates with the set of
  • 1-nodes.
  • Every node sends O(log2n) messages before making
    its decision.
  • For tgtn/2, we use sequence of log n epoches.

20
Conclusions
  • It is possible to achieve consensus in tO(log2n)
    rounds, using messages of total length O(n log3n)
    bits.
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