Self Stabilization Classical Results and Beyond

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Self Stabilization Classical Results
and Beyond

• Elad Schiller
• CTI (Grece)

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Self-StabilizationShlomi DolevMIT Press , 2000
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Talk Outline

• Definitions of Computational Model
• Self-Stabilization Requirements
• Complexity Measures
• Example
• Spanning-Tree Construction
• Example
• Self-Stabilising Group Communication in Mobile
  Ad-Hoc Networks
• The mobile ad hoc settings and random walk
• Estimating the number of nodes
• Services implementation
• Conclutions

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The Distributed System Model

• A Distributed System is modeled by processors ?
• set of n state machines communicate with each
  other

• Denote
• Pi - the ith processor
• neighbor of Pi ?
• a processor that can communicate with it

Node i Processor i
Link Pilt-gtPj Pi can communicate with Pj
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Asynchronous Distributed Systems Shared Memory

• Processors communicate
• by the use of shared communication registers
• The configuration will be denoted by c
  (s1,s2,,sn,r1,2,r1,3,ri,j,rn,n-1) where si
  State of Pi ri,j Content of communication
  register i

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The distributed System A Computation Step

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The Interleaving model

• At each given time
• only 1 processor executes a computation step
• Every state transition of a processor
• is due to a communication-step execution
• A step will be denoted by a
• c1 a? c2 denotes the fact that
• c2 can be reached from c1 by a single step a

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The distributed System more definitions

• Step a is applicable to configuration c
• iff ? c c a? c
• An execution E (c1,a1,c2,a2,)
• an alternating sequence such that ci-1 a ? ci
  (igt1)
• A fair execution
• every step that is applicable infinitely often is
  executed infinitely often

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Legal Behavior

• A desired legal behavior is a set of executions
• denoted LE

c
A self-stabilizing system can be started in any
arbitrary configuration and will eventually
exhibit a desired legal behavior
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Time complexity

• Asynchronous round (round) in an execution E
• 1st round is the shortest prefix E of E
• such that each processor executes at least 1 step
  in E, EEE.
• The number of rounds time complexity

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Time complexity (Cont.)

• A Self-Stabilizing algorithm is usually a do
  forever loop
• Asynchronous cycle (cycle) in an execution E
• 1st cycle is the shortest prefix E of E
• such that each processor executes at least 1
  complete iteration of its do forever loop in E,
  EEE.
• Note each cycle spans O(?) rounds
• O(?) is the number of steps required to execute 1
  iteration
• where ? is an upper bound on the number of
  neighbors of Pi

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Talk Outline

• Definitions of Computational Model
• Self-Stabilization Requirements
• Complexity Measures
• Example
• Spanning-Tree Construction
• Example
• Self-Stabilising Group Comm. in Mobile Ad-Hoc
  Net.
• The mobile ad hoc settings and random walk
• Estimating the number of nodes
• Services implementation
• Conclutions

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Spanning-Tree Construction

• The root writes 0 to all its neighbors
• The rest each processor
• chooses the minimal distance of its neighbors,
• adds 1 and updates its neighbors

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Spanning-Tree, System and code

• The output tree is encoded by means of the
  registers

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Spanning-Tree Algorithm for Pi
Root do forever for m 1 to ? do
write rim ?0,0? Other do forever for m
1 to ? do write lrmi read(rmi) FirstFound
false dist 1 min?lrmi.dis ?1 ? m ? ? ? for
m 1 to ? do if not FirstFound and lrmi.dis
dist -1 write rim ?1,dist? FirstFound
true else write rim ?0,dist?

• of processors neighbors
• i the writing processorm for whom the data is
  written

lrji (local register ji) the last value of rji
read by Pi
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Spanning-Tree Application
RUN
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Spanning-Tree, is Self-Stabilizing

• The legal task ST
• every configuration encodes a BFS tree
• of the communication graph
• Definitions
• A floating distance is a value in rij.dis
• smaller than the distance of Pi from the root
• The smallest floating distance in configuration c
  
• is the smallest value among the floating distance

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Spanning-Tree, is Self-Stabilizing

• (Lemma 1) For every k gt 0 and for every
  configuration that follows ? 4k? rounds, it
  holds that
• If there exists a floating distance, then
• the value of the smallest floating distance is at
  least k
• The value in the registers of every processor
  that is within distance k from the root is equal
  to its distance from the root

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Spanning-Tree, is Self-Stabilizing

• Note
• once a value in the register of every processor
  is equal to its distance from the root,
• a processor Pi chooses its parent to be the
  parent in the first BFS tree.
• this implies that
• The algorithm presented is
• Self-Stabilizing for ST

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Talk Outline

• Definitions of Computational Model
• Self-Stabilization Requirements
• Complexity Measures
• Example
• Spanning-Tree Construction
• Example
• Self-Stabilising Group Communication in Mobile
  Ad-Hoc Networks
• The mobile ad hoc settings and random walk
• Estimating the number of nodes
• Services implementation
• Conclutions

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Random walk for self-stabilising group
communication in ad-hoc networks

• With Shlomi Dolev BGU Jennifer Welch TAMU
• IEEE Transactions on Mobile Computing,
• SRDS02 and PODC02

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Mobile Ad Hoc Networks

• Wireless networking, mobile computing
• No pre-existing infrastructure
• Solely rely on wireless links
• Computers re-location, connect, disconnect
• Unlikely not to have transmission faults
• Unlikely not to have transient faults
• Violate the assumption made by the system designer

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Coping with Chaos by Chaos

• Flooding?
• Distributed structure (DAG in TORA)?
• Forcing nodes to move faster than others,
  according to a specific pattern?
• Random walk of a single agent!

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Contributions

• We handle topology changes automatically
• Estimating the number of participating nodes
• User benefit
• Dont cope with lower level complications

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Random Walk of an Agent

• Ensures that if an agent exists
• it covers the system with high probability
• Time-outs are used
• to guarantee the existence of at least one agent
• If several agents exist
• random walk ensures collisions
• Self-stabilizes to work correctly

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Fixed communication graph

• Frequency of changes w.r.t.
• Communication radius
• Speed of nodes / an agent
• Cover time
• The meeting/cover time of a fixed graph
• O(n3)

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Complete Random Change

• Graph can totally change between agent moves
• The choice of movement can be view as
• A choice of the current neighbors
• Then a choice from the neighbors set
• Like a random walk on a complete graph
• Cover time for a complete graph is
• O(n lg n)

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Estimating n (simplified ver.)

• Problem the complexity is O(N3 lg N)
• Replacement of N by n is wanted!
• The agent list the nodes it visits
• Order by most recently visited
• Remove nodes that are not visited
• O(t3 lg t) agent moves
• t is the order in the list
• Output the size of the list as n

ID Mov 585 0 471 1 354 2 784 5 958
87 473 91
gap
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Membership Service

• Establish a view
• Whenever a node joins/leaves the group
• A node may voluntarily join/leave the system
• A node is removed from the members list
• If not visited for a large number of agent moves

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Multicast Service

• The agent carries message history
• Deliver messages by their sending order
• Add a best effort "victory" traversal
• Safe delivery indication
• View agreement

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Conclusion

• Our new approach copes with
• Uncertainty nature of ad-hoc networks
• Probabilistic group communication
• Membership, and multicast
• Requirements meet with high probability
• Performance tuning by varying the probability