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
Neville’s 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
- Add references plus time comparisons

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
- Make round a separate bullet : highlight this fact

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
Core Structure
Non-Core Structure
Index of permutation 2
Permutation
n = 4 1 4 3
Talk about the other case as well – probably add a slide here explaining it

9. “Core and Non-Core” strategy for Self Stabilization
Strategy: Always assume Non-Core structures got corrupted and align it with Core structures
Core Structure
Non-Core Structure
Index of permutation 2
Permutation
n = 4 1 4 3
Talk about the other case as well – probably add a slide here explaining it 1 2 3 4

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 2 3 4 1 2 4 3 1

11. Neville’s Code [Neville 53]
Similar to Prufer code
Each labeled tree with n nodes has one to one correspondence with a Neville’s code
Code is a sequence of n - 2 numbers from the set {1,…,n}
code[i] denotes the ith number in the code sequence

12. Neville’s Code: Example8 6 3
Code = 7 5 4
Highlight the fact that we are not deleting the nodes 5 and 4 2 1

13. Spanning Tree → Neville’s Code
x = least node with degree 1
for i = 1 to n-1
code[i] = parent[x]
Delete edge between x and parent[x]
if (degree[parent[x]] = 1 && parent[x] ≠ n)
x = parent[x]
else
Change symbols for not equals etc

14. Neville’s Code: Example
x = least node with degree 1
for i = 1 to n-1
code[i] = parent[x]
Delete edge between x and parent[x]
if (degree[parent[x]] = 1 && parent[x] ≠ n)
x = parent[x]
else 8 6 3 7 5 4
Highlight the fact that we are not deleting the nodes 5 and 4 2 1
code = 7
code =
code = 77
code = 776
code = 7768
x = 6
x = 2
x = 1
x = 7

15. Self Stabilization using Neville’s code
Need to maintain “parent” (Non-core) for each node
Auxiliary data structures for efficiency
code[i] : Neville’s code
f[i] : Iteration in which node i is chosen as “x”
z[i] : 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 Neville’s code
Spanning Tree (R)
Isomorphic (C)
Identical
Efficiency

17. Constraints for R (R1) For all i: code[f[i]] = parent[i]
Follows from the code building procedure 8 6 3 4 7 1 2 5
Node 7 was chosen as “x” in iteration 3. So f[7] = 3
code[f[7]] = code[3] = 6 = parent[7]
code =

18. Constraints for R Simple restrictions on the range of the structures
(R2) For 1 ≤ i ≤ n – 2: 1 ≤ code[i] ≤ n
and code[n – 1] = n
(R3) (i) For 1 ≤ i ≤ n – 1: 1 ≤ f[i] ≤ n – 1
(R4) For all i: z[i] = max j such that code[j] = i
Definition of z

19. Constraints for R (R5) For all i: z[i] ≠ 0  f[i] = z[i] + 1
Captures preference given to parent when its degree becomes one 8 6 3 4 7 1 2 5
Node 7 occurs last in code at position 2. Hence, z[7] = 2.
Also, f[7] = 3.
f[7] = z[7] + 1
code =

20. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j z
code 1 2 3 4 5 2 3 1 5 4

21. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j 4 z
code 1 2 3 4 5 2 3
(E1)
code ? 1 5 4

22. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j z
code 1 2 3 4 5 2 3
E1 violated !
(E1) 1 5 4

23. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j z
code 1 3 2 4 5 2 3
(E1) 1 5 4

24. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j z
code 1 3 2 4 5 2 3
check z ≥ 3
z = max {0,3,4}
(E2) 1
check z ≥ 4 5 4

25. Maintaining R - Constraint R4
For all i: z[i] = maximum j such that code[j] = i
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j z
code 1 4 3 2 5 2 3
(E2) 1 5 4

26. Maintaining R - Other Constraints
Local checks: Can be checked and corrected without contacting any other node
(R2) , (R3) (i), (R5)
(R1) For all i: code[f[i]] = parent[i]
Inquire node f[i] to get code[f[i]] and match with parent[i]
On mismatch, reset parent[i] to agree with code[f[i]]

27. 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 ?

31. Neville’s code → Spanning Tree
x = least node with degree 1
for i = 1 to n-1
parent[x] = code[i]
degree[x]--; degree[parent[x]]--;
If (degree[parent[x]] == 1)
x = code[i]
else

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 Neville’s code
Need to maintain “parent” for each node
Auxiliary data structures for efficient detection
code[i] : Neville’s code
f[i] : Iteration in which node i was selected as x
z[i] : 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

34. Constraints (R1) For all i: code[f[i]] = parent[i]
(R2) 1 <= i <= n-2, 1 <= code[i] <= n
and code[n – 1] = n
(R3) (i) 1 <= f[i] <= n – 1
(ii) f is a permutation on [1…n]
(R4) For all i: z[i] = max. j such that code[j] = i
(R5) For all i: z[i] != 0 => f[i] = z[i] + 1

35. 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

36. One interesting constraint
For all i: z[i] = maximum j such that code[j] = I
Split the constraint into two different constraints
(E1) z[i] ≠ 0  code[z[i]] = i
(E2) code[j] = i  z[i] ≥ j
For (E1), node i queries the node z[i] to get code[z[i]] and matches it against i
For (E2),
every node j with code[j] = i sends a message to node i containing j
Node i then sets z[i] = max { z[i], j }

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