Distributed Algorithms Chapter 3 Phase Synchronization on undirected trees

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Distributed AlgorithmsChapter 3Phase
Synchronization on undirected trees
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The problem
Given a undirected tree of sites.

• A E
• C D
• B F G
• H

each site a alternates between the states busy
and pending.
Def. a site a is in its k-th phase if a is its
k-th time busy. The algorithm is to
guarantee if two sites are busy at the same
time, they are in the same phase.
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algorithmic idea
Each phase synchronization is started by the busy
leaves, l Each l sends a message to its unique
neighbour and goes pend.

• A E
• C D
• B F G
• H

a busy site informed by all but one neighbour,
informs the remaining neighbour and goes pending
a pending site informed by the remaining
neighbour, informs all other neighbours and goes
busy.
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The algorithm
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)

(A,0) (B,0) (C,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0)

(B,0) (C,0)
(A,B,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0) a(B,C,0)

(C,0)
(A,B,0) (B,C,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0) a(B,C,0) a(C,B,0)

(A,B,0) (B,C,0) (C,B,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0) a(B,C,0) a(C,B,0) b(C,B,0)

(C,1)
(A,B,0) (B,C,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0) a(B,C,0) a(C,B,0) b(C,B,0)

(C,1)
(A,B,0) (B,C,0)
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A surprizing observation

• a network
• A__________ B __________C
• occurrence sequence orde of variables (x,y,i)
• a(A,B,0) a(B,C,0) a(C,B,0) b(C,B,0) a(C,B,1)

two copies of the token (B,C) at messages
(A,B,0) (B,C,0) (C,B,1)
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A non trivial invariant

• Let a, b, a, b, U ? ? ? ?(U ? U) with
• a(u,n) def 2n r(u) b(u,n) def 2n
  r(u)
• a(u,n) def 2n r(u) b(u,n) def 2n
  r(u)
• a place invariant
• a(busy) messages b(pr1,3(pending))
• a(busy) messages b(pr1,3(pending))

helps to prove if busy.(u,i) and busy.(v,j)
then ij. if busy.(u,i) then eventually
busy.(u,i1)
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Distributed Algorithms End of Chapter 3Phase
Synchronization on undirected trees