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Distributed Maintenance of Spanning Tree using Labeled Tree Encoding Vijay K. Garg Anurag Agarwal PDSL Lab University of Texas at Austin 1 Outline Previous work and System model “Core” and “Non-core” strategy Neville’s code Self-stabilizing spanning tree algorithm Conclusion 2 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 3 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 4 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 5 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 6 “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 Index of permutation Non-Core Structure Permutation 1 … n! 7 “Core and Non-Core” strategy for Self Stabilization Strategy: Always assume Non-Core structures got corrupted and align it with Core structures n=4 Core Structure Non-Core Structure Permutation Index of permutation 2 1 2 4 3 8 “Core and Non-Core” strategy for Self Stabilization Strategy: Always assume Non-Core structures got corrupted and align it with Core structures n=4 Core Structure Non-Core Structure Permutation Index of permutation 2 1 2 4 3 3 4 9 “Core and Non-Core” strategy for Self Stabilization Strategy: Always assume Non-Core structures got corrupted and align it with Core structures n=4 Core Structure Non-Core Structure Permutation Index of permutation 12 1 2 4 3 3 4 Challenge lies in efficient detection and correction 10 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 11 Neville’s Code: Example 8 6 3 Code = 7768338 7 2 5 4 1 12 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 13 Neville’s Code: Example 8 x = least node with degree 1 for i = 1 to n-1 6 3 7 5 4 code[i] = parent[x] Delete edge between x and parent[x] if (degree[parent[x]] = 1 && parent[x] ≠ n) x = parent[x] else 2 1 x = least node with degree 1 code = 7768338 7 77 776 7768 x=6 1 2 7 14 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 15 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 16 Constraints for R (R1) For all i: code[f[i]] = parent[i] Follows from the code building procedure 8 6 Node 7 was chosen as “x” in iteration 3. So f[7] = 3 3 code[f[7]] = code[3] = 6 = parent[7] 7 2 5 1 4 code = 7768338 17 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 18 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 Node 7 occurs last in code at position 2. Hence, z[7] = 2. Also, f[7] = 3. 8 6 3 f[7] = z[7] + 1 7 2 5 1 4 code = 7768338 19 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 3 1 5 4 z code 1 2 3 2 3 4 3 0 1 4 0 1 5 2 5 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 (E1) code ? 5 4 3 1 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 (E1) 3 1 5 E1 violated ! 4 z code 1 2 3 2 3 4 3 0 1 4 0 1 5 2 5 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 (E1) 3 1 5 4 z code 1 0 3 2 3 4 3 0 1 4 0 1 5 2 5 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 3 check z ≥ 3 z = max {0,3,4} (E2) 1 5 check z ≥ 4 4 z code 1 0 3 2 3 4 3 0 1 4 0 1 5 2 5 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 (E2) 3 1 5 4 z code 1 4 3 2 3 4 3 0 1 4 0 1 5 2 5 25 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]] 26 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) 27 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 28 Future Work Remove the restriction on topology and labels Apply the strategy of core and non-core states to other problems 29 Questions ? 30 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 31 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 32 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 33 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 34 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 35 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 } 36 References 1. 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