Distributed Maintenance of Spanning Tree using Labeled Tree Encoding Vijay K. Garg Anurag Agarwal PDSL Lab University of Texas at Austin

Outline Previous work and System model “Core” and “Non-core” strategy Neville’s code Self-stabilizing spanning tree algorithm Conclusion

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

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

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

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

“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

“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

“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

“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

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

Neville’s Code: Example 8 6 3 Code = 7768338 7 5 4 Highlight the fact that we are not deleting the nodes 5 and 4 2 1

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

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 = 7768338 code = 77 code = 776 code = 7768 x = 6 x = 2 x = 1 x = 7

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

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

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 = 7768338

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

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 = 7768338

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

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

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

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

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

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

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]]

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)

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

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

Questions ?

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

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

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

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

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

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 }

References 1. 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 Computer Science, pages 400–410, 1993. 3. G. Antonoiu and P. Srimani. Distributed self-stabilizing algorithm for minimum spanning tree 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 and correction (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 of the ACM, 17:643–644, 1974. 9. S. Dolev, A. Israeli, and S. Moran. Self-stabilization of dynamic systems assuming only read/write atomicity. In Proc. of the ninth annual ACM symposium on Principles of Distributed Computing, 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. In Proc. of the sixteenth annual ACM symposium on Principles of Distributed Computing, page 288. ACM Press, 1997. 12. E. H. Neville. The codifying of tree-structure. Proceedings of Cambridge Philosophical Society, 49:381–385, 1953.