Distributed Maintenance of Spanning Tree using Labeled Tree Encoding

1

Distributed Maintenance of Spanning Tree using Labeled Tree Encoding

Vijay K. GargAnurag Agarwal

PDSL LabUniversity 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

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

Core Structure Non-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 2 4 3

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 2 4 31 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

Core Structure Non-Core Structure

Index of permutation

2

Permutation

n = 4

1 2 4 31 1 2 3 4

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: Example

Code = 7768338

8

6 3

47

12

5

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 x = least node with degree 1

14

Neville’s Code: Example

code = 7code = 77code = 776code = 7768code = 7768338

8

6 3

47

12

5

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 x = least node with degree 1

x = 1x = 2x = 7x = 6

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

47

12

5

Node 7 was chosen as “x” in iteration 3. So f[7] = 3

code[f[7]] = code[3] = 6 = parent[7]

code = 7768338

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

47

12

5

Node 7 occurs last in code at position 2. Hence, z[7] = 2.Also, f[7] = 3.

f[7] = z[7] + 1

code = 7768338

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

2

5

1

3

4

z code

1 2 3

2 3 4

3 0 1

4 0 1

5 2 5

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

2

5

1

3

4

z code

1 2 3

2 3 4

3 0 1

4 0 1

5 2 5

(E1) code ?

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

2

5

1

3

4

z code

1 2 3

2 3 4

3 0 1

4 0 1

5 2 5

(E1)E1

violated !

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

2

5

1

3

4

z code

1 0 3

2 3 4

3 0 1

4 0 1

5 2 5

(E1)

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

2

5

1

3

4

z code

1 0 3

2 3 4

3 0 1

4 0 1

5 2 5

(E2)

check z ≥ 3

check z ≥ 4

z = max {0,3,4}

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

2

5

1

3

4

z code

1 4 3

2 3 4

3 0 1

4 0 1

5 2 5

(E2)

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 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 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 iFor (E2),

every node j with code[j] = i sends a message to node i containing jNode i then sets z[i] = max { z[i], j }

37

References1. Y. Afek, S. Kutten, and M. Yung. Memory-efficient self stabilizing protocols for general networks.In Proc. of the 4th Int’l Workshop on Distributed Algorithms, pages 15–28. Springer-Verlag, 1991.2. S. Aggarwal and S. Kutten. Time optimal self-stabilizing spanning tree algorithm. In Proc.of the 13th Conference on Foundations of Software Technology and Theoretical ComputerScience, pages 400–410, 1993.3. G. Antonoiu and P. Srimani. Distributed self-stabilizing algorithm for minimum spanningtree construction. In European Conference on Parallel Processing, pages 480–487, 1997.4. A. Arora and M. Gouda. Distributed reset. IEEE Transactions on Computers, 43(9):1026–1038, 1994.5. B. Awerbuch, B. Patt-Shamir, and G. Varghese. Self-stabilization by local checking andcorrection (extended abstract). In IEEE Symposium on Foundations of Computer Science,pages 268–277, 1991.6. Z. Collin and S. Dolev. Self-stabilizing depth-first search. Information Processing Letters,49(6):297–301, 1994.8. E. W. Dijkstra. Self-stabilizing systems in spite of distributed control. Communications ofthe ACM, 17:643–644, 1974.9. S. Dolev, A. Israeli, and S. Moran. Self-stabilization of dynamic systems assuming onlyread/write atomicity. In Proc. of the ninth annual ACM symposium on Principles of DistributedComputing, pages 103–117. ACM Press, 1990.10. S. Huang and N. Chen. A self stabilizing algorithm for constructing breadth first trees.Information Processing Letters, 41:109–117, 1992.11. C. Johnen. Memory efficient, self-stabilizing algorithm to construct bfs spanning trees. InProc. of the sixteenth annual ACM symposium on Principles of Distributed Computing, page288. ACM Press, 1997.12. E. H. Neville. The codifying of tree-structure. Proceedings of Cambridge PhilosophicalSociety, 49:381–385, 1953.