Distributed%20Maintenance%20of%20Spanning%20Tree%20using%20Labeled%20Tree%20Encoding

1
Distributed Maintenance of Spanning Tree using
Labeled Tree Encoding

• Vijay K. Garg
• Anurag Agarwal
• PDSL Lab
• University of Texas at Austin

2
Outline

• Previous work and System model
• Core and Non-core strategy
• Nevilles code
• Self-stabilizing spanning tree algorithm
• Conclusion

3
Motivation

• Maintaining spanning trees in distributed fashion
• Broadcast
• Convergecast
• Self Stabilization Dijkstra 74 is a powerful
  fault-tolerance paradigm
• Design algorithms to tolerate transient data
  faults
• Despite faults, algorithm converges to a good
  state

4
Previous Work

• Many self-stabilizing algorithms for spanning
  trees
• Breadth-first spanning tree DIM90, AK93
• Depth-first spanning tree CD94
• Minimum spanning tree AS97
• Our work makes stronger assumptions but achieves
  better bounds

5
Comparison with Previous Work

• Popular model assumes all communication registers
  can be read/written in one time step
• In a completely connected topology, it amounts to
  doing O(n) work in one time step
• Our model assumes processes take one
  communication step
• In our model, the previous algorithms would have
  at least O(n) time complexity

6
System Model

• System with n nodes labeled 1 n
• Nodes form a completely connected graph
• Topology is static
• Computation step
• Internal computation
• One communication event
• A message is ready to be delivered in one time
  step

7
Core and Non-Core Strategy for Self
Stabilization

• Maintain Core and Non-Core data structures
• Core structures are always correct
• Non-core structures can be derived from Core
  structures

Non-Core Structure
Core Structure
Index of permutation 1 n!
Permutation
8
Core and Non-Core strategy for Self
Stabilization

• Strategy Always assume Non-Core structures got
  corrupted and align it with Core structures

9
Core and Non-Core strategy for Self
Stabilization

• Strategy Always assume Non-Core structures got
  corrupted and align it with Core structures

10
Core and Non-Core strategy for Self
Stabilization

• Strategy Always assume Non-Core structures got
  corrupted and align it with Core structures
• Challenge lies in efficient detection and
  correction

n 4
Non-Core Structure
Core Structure
Index of permutation 2
Permutation
1
11
Nevilles Code Neville 53

• Similar to Prufer code
• Each labeled tree with n nodes has one to one
  correspondence with a Nevilles code
• Code is a sequence of n - 2 numbers from the set
  1,,n
• codei denotes the ith number in the code
  sequence

12
Nevilles Code Example
8
6
3
Code 7768338
4
7
5
1
2
13
Spanning Tree ? Nevilles Code

• x least node with degree 1
• for i 1 to n-1
• codei parentx
• Delete edge between x and parentx
• if (degreeparentx 1 parentx ? n)
• x parentx
• else
• x least node with degree 1

14
Nevilles Code Example

• x least node with degree 1
• for i 1 to n-1
• codei parentx
• Delete edge between x and parentx
• if (degreeparentx 1 parentx ? n)
• x parentx
• else
• x least node with degree 1

8
6
3
4
7
5
1
2
code 7
code 77
code 776
code 7768
code 7768338
x 1
x 2
x 7
x 6
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Self Stabilization using Nevilles code

• Need to maintain parent (Non-core) for each
  node
• Auxiliary data structures for efficiency
• codei Nevilles code
• fi Iteration in which node i is chosen as x
• zi last occurrence of node i in code
• Node i maintains ith components of data
  structures
• Put constraints on these data structures so that
  the parent pointers give a valid tree

16
Constraints

• Three constraint sets provide different
  guarantees on the structure of the resulting
  spanning tree with respect to the tree generated
  by Nevilles code

Spanning Tree (R)
Isomorphic (C)
Identical
Efficiency
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Constraints for R

• (R1) For all i codefi parenti
• Follows from the code building procedure

• Node 7 was chosen as x in iteration 3. So f7
  3
• codef7 code3 6 parent7

code 7768338
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Constraints for R

• Simple restrictions on the range of the
  structures
• (R2) For 1 i n 2 1 codei n
• and coden 1 n
• (R3) (i) For 1 i n 1 1 fi n 1
• (R4) For all i zi max j such that codej
  i
• Definition of z

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Constraints for R

• (R5) For all i zi ? 0 ? fi zi 1
• Captures preference given to parent when its
  degree becomes one

• Node 7 occurs last in code at position 2. Hence,
  z7 2.
• Also, f7 3.
• f7 z7 1

code 7768338
20
Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 2 3
2 3 4
3 0 1
4 0 1
5 2 5
2
3
5
4
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Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 2 3
2 3 4
3 0 1
4 0 1
5 2 5
4
2
3
(E1)
code ?
5
4
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Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 2 3
2 3 4
3 0 1
4 0 1
5 2 5
2
3
E1 violated !
(E1)
5
4
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Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 0 3
2 3 4
3 0 1
4 0 1
5 2 5
2
3
(E1)
5
4
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Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 0 3
2 3 4
3 0 1
4 0 1
5 2 5
2
3
check z 3
z max 0,3,4
(E2)
check z 4
5
4
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Maintaining R - Constraint R4

• For all i zi maximum j such that codej i
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

z code
1 4 3
2 3 4
3 0 1
4 0 1
5 2 5
2
3
(E2)
5
4
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Maintaining R - Other Constraints

• Local checks Can be checked and corrected
  without contacting any other node
• (R2) , (R3) (i), (R5)
• (R1) For all i codefi parenti
• Inquire node fi to get codefi and match
  with parenti
• On mismatch, reset parenti to agree with
  codefi

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Analysis of Algorithm for maintaining R

• Theorem The algorithm requires O(1) time per
  node and O(1) messages per node on average in one
  cycle
• Theorem The algorithm stabilizes in O(d) time,
  where d is the upper bound on the number of times
  a node appears in the code
• With high probability, a random code assignment
  would have d O(log n/ log log n)

28
Conclusion

• Self stabilization algorithm for spanning tree
• Requires O(1) messages per node on average
• Provides fast stabilization
• Allows changing root node and systematic
  modification of the tree

29
Future Work

• Remove the restriction on topology and labels
• Apply the strategy of core and non-core states to
  other problems

30
Questions ?
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Nevilles code ? Spanning Tree

• x least node with degree 1
• for i 1 to n-1
• parentx codei
• degreex-- degreeparentx--
• If (degreeparentx 1)
• x codei
• else
• x least node with degree 1

32

• Round vs Bounded Delivery Time
• Round Every process takes atleast one step
• Definition allows one process to send/receive
  multiple messages in one time unit

33
Self Stabilization using Nevilles code

• Need to maintain parent for each node
• Auxiliary data structures for efficient detection
• codei Nevilles code
• fi Iteration in which node i was selected as
  x
• zi last occurrence of node i in code
• Node i maintains ith components of data
  structures
• Put constraints on these data structures so that
  the parent pointers give a valid tree

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Constraints

• (R1) For all i codefi parenti
• (R2) 1 lt i lt n-2, 1 lt codei lt n
• and coden 1 n
• (R3) (i) 1 lt fi lt n 1
• (ii) f is a permutation on 1n
• (R4) For all i zi max. j such that codej
  i
• (R5) For all i zi ! 0 gt fi zi 1

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Two sets of constraints

• R R1, R2, R3(1), R4, R5
• Resulting spanning tree may differ from the one
  given by code in the leaves
• Self-stabilization is easier and more efficient
• C R1, R2, R3, R4, R5
• Resulting spanning tree is isomorphic to the one
  given by code
• Self-stabilization is harder and becomes
  inefficient

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One interesting constraint

• For all i zi maximum j such that codej I
• Split the constraint into two different
  constraints
• (E1) zi ? 0 ? codezi i
• (E2) codej i ? zi j

For (E1), node i queries the node zi to get
codezi and matches it against i For (E2),
every node j with codej i sends a message to
node i containing j Node i then sets zi max
zi, j
37
References

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