The Dynamic AndOr Quorum System

1
The Dynamic And-Or Quorum System

• Uri Nadav
• Tel-Aviv University
• joint work with
• Moni Naor
• The Weizmann Institute of Science

DISC 05, Krakow
2
Quorum Systems

• A quorum system is an intersecting family of sets
  
• Formal definition
• U Universe
• F ½ 2U
• 8 A,B 2 F AÅ B ?
• F is called a quorum system. A,B are called
  quorum sets
• Examples
• Majority all sets that contain more than half
  of the elements
• Dictatorship all sets that contain a specific
  element
• Many more

3
Quorum Systems - Applications

• The universe is a set of processors
• Mutual exclusion
• To enter critical section, get permission from a
  quorum
• Intersection property guaranties mutual
  exclusion. GB85
• Replicated database
• Writer Write to a quorum
• Reader Read from a quorum
• Intersection property guaranties effective
  search
• More.

4
Main Challenge

• Dynamic Network
• Processors join and leave
• And-Or quorum system
• Does the job!

5
The And-Or Tree Quorum System NW94

• Universe Leaves of binary tree
• A recursive procedure to choose leaves

AND choose two
OR choose one
AND choose two
OR choose one

• Dual procedure interchange OR ?? AND
• Quorum union of both sets
• The intersection property follows from a known
  fact about circuits (minterm Å maxterm ? )

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Properties of the And-Or NW94

• Optimal load
• Probability distribution for choosing quorums
• Access probability of the busiest processor
• High availability (random faults)
• Processor crashes independently with probability
  p
• At least one quorum survives (all its processors)

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Finding a live quorum - Probe Complexity

• Random faults model
• How many processors queried before a live quorum
  is found?
• Non adaptive algorithms (predefined set of
  processors)
• Can be implemented in a single step (access
  parallel)
• Adaptive algorithms more efficient
• Our work And-Or has optimal probe complexity

8
Our contribution

• Static network
• Non-adaptive and adaptive algorithms
• Both match a lower bound on probe complexity
• Non-adaptive O(n½ log(n))
• Adaptive O(n½)
• Very fast adaptive alg(O(loglog(n)))
• Compared to quorum systems with same load,
  availability
• Dynamic And-Or
• Dynamic P2P version, processors join and leave
• Scales well
• Keep good properties of And-Or

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Probe Complexity Non adaptive algorithm

• If a quorum of subtrees are alive then the whole
  tree is alive
• Fail probability of subtree decays polynomialy

2loglog n

• With high probability the chosen set includes a
  live quorum
• Probe complexity O(n½log n)
• Matches a lower bound NW03

10
Adaptive Algorithm

• Naively choose a quorum and correct OR choices
  locally

X
X
?

• Usually, no need to climb high
• With high probability O(n½) probes suffice
• Trees are dependent - Chernoff bound cannot be
  applied automatically
• Can run in parallel 2loglog (n) rounds

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Dynamic Quorum System

• The universe constantly changes
• New challenges
• Integrity
• Intersection property
• Combinatorial structure and properties
• Locality
• Local way to access a quorum
• Existing constructions
• Dynamic Paths NW03
• Dynamic Probabilistic Quorum System AM03

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Dynamic And-Or System

• Two constructions
• Using Mankus ID management scheme M04
• Implements a dynamic balanced binary tree
• Load, availability and probe complexity remain
  valid for balanced trees
• Using Naor-Wieder Distance Halving NetworkNW03
• Better couples the quorum system with network
  edges
• Saves in message-complexity when considering
  actual network implementation

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The Distance Halving Graph NW03

• continuous graph
• Nodes 0,1) interval
• Edges Left and right outgoing edges

x
x/2
(x1)/2
0
1
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The Distance Halving Network NW03

• Discretization, given a partition by processors
• Each point is covered by exactly one processor
• A quorum in the universe of processors is a set
  that covers a quorum in the universe of points
• Connect two processors if their respective cells
  are connected in the continuous graph

x
x/2
(x1)/2
0
1
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Integrity and System Adaptation

• Dynamic network processors Join/Leave!
• All processors keep an estimation of log(n)
• And-Or tree height is always set to log(n)
• When tree height changes members of quorums
  gossip themselves to children/parent in the
  continuous graph
• AND gate, to both children, OR gate to random
  child

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Properties of the dynamic And-Or
(load, availability, probe complexity)

• Problem in analysis
• Leaves in the tree are dependent
• Balanced Network
• Processors cover almost an equal share
• Constant number of leaves per processor
• Limited dependency
• Technique Dominating product measure Ligget et
  al

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Comparison
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Open Questions

• Lower bound to the adaptive probe complexity
• In one round, we need at least ?(n½ log n) probes
  NW03
• In loglog (n) rounds, And-Or requires only O(n½)
  probes
• Close the gap

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Summary

• And-Or Quorum System
• Non-adaptive, Adaptive Algorithms for finding
  live quorum
• Optimal
• Adaptive case Excellent time complexity
• Adaptation to a dynamic overlay network
• Simple rules of adaptation
• Natural gossip protocol
• Optimal Load, probe complexity and high
  availability

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Disc 2005

• Questions, Comments?