1. A Distributed Polylogarithmic Time Algorithm for Self-Stabilizing Skip Graphs
Stefan Schmid & Christian Scheideler
Dept. of Computer Science
University of Paderborn
Andrea Richa
Dept. of Computer Science
Arizona State University
Riko Jacob & Hanjo Täubig
Dept. of Computer Science
Technische Universität München

2. Overlay Network
Internet

3. Overlay Network
Basic question: how to organize sites in a scalable and robust overlay network???
Problem: high join/leave activity!

4. Overlay Network
Scalability: Every operation needs at most (poly-)logarithmic time and work

5. Overlay Network
Robustness: can recover from any attack ( self-stabilizing overlay network)

6. Overview Basic notation Prior work Self-stabilizing skip graph
Conclusion

7. Basic Notation Overlay network: directed graph G=(V,E):
V: set of nodes
E  {(v,w) | v,wV}: set of edges
v knows w D B C A

8. Basic Notation State of a node v: Local variables Neighborhood N(v)
Set of actions/rules of the form <label>: <guard>  <commands>
The set of actions is the same for every node and is supposed to be immutable.

9. Basic Notation
The nodes may only know the local state of their direct neighbors.
Only the following overlay commands:
u.insert(v,w): u asks neighbor v to establish edge to neighbor w v v u u w w

10. Basic Notation
The nodes may only know the local state of their direct neighbors.
Only the following overlay commands:
u.move(v,w): u asks neighbor v to establish edge to neighbor w and removes (u,w)
no delete operation! v v u u w w

11. Basic Notation
The nodes may only know the local state of their direct neighbors.
Only the following overlay commands:
u.move(v,w): u asks neighbor v to establish edge to neighbor w and removes (u,w)
sufficient to merge parallel edges v v u u w w

12. Basic Notation
Self-stabilizing overlay network: A network that can get back to a desired topology from any initial state in which the network is still weakly connected.

13. Basic Notation During the self-stabilization process:
Node set is fixed, all nodes reliable
Messages arrive within a time unit
No message loss

14. Basic Notation During the self-stabilization process:
Time proceeds in synchronous rounds.
All actions that are enabled in a round (i.e. their guard is true) are executed.
All local and overlay commands are correctly executed within the given round.
Simplifies analysis!

15. Basic Requirements
Scalability: Only O(polylog(n)) many enabled actions per join or leave operation to get back to legal topology
Robustness: Self-stabilization from any weakly connec-ted state

16. Our Requirements
Scalability: Only O(polylog(n)) many enabled actions per join or leave operation to get back to legal topology
Robustness: Quick self-stabilization from any weakly connected state

17. Overview Basic notation Prior work Self-stabilizing skip graph
Conclusion

18. Prior Work
Self-stabilization goes back to Dijkstra´s seminal work in 1974
Self-stabilizing algorithms known for leader election, various communication protocols, graph theory problems, termination detection, clock synchronization, etc.
Underlying network is usually assumed to be static (if not, then changes are not under control of algorithm but happen in a random or adversarial manner)

19. Prior Work
Static networks allow efficient cross-checking of distributed computation
Development of various techniques that convert conventional algorithms into self-stabilizing algorithms (Awerbuch, Dolev, Herman, Kutten, Patt-Shamir, Varghese,…) v u
St+1(u) = f(St(u),St(v),St(w)) w

20. Prior Work Two options:
find old neighbors
reset
These general techniques do not work (well) for self-stabilizing overlay networks.
Time t
Time t+1 v w u u w´ v´
w´´
St+1(u) = f(St(u),St(v),St(v´))

21. Prior Work Suprisingly little known for self-stabilizing
overlay networks
Chord network [Stoica et al.]: recovery from certain degenerate states. Problem: Chord not locally checkable
Skip graphs [Aspnes & Shah]: recovery from certain degenerate states. Problem: skip graphs not locally checkable as well
Self-stabilizing line/cycle/hypertree: - Angluin, Aspnes, … Cramer & Fuhrmann Shaker & Reeves Dolev & Kat 2007

22. Overview Basic notation Prior work Self-stabilizing skip graph
Conclusion

23. Self-Stabilizing Skip Graph
Each node v has arbitrary unique name v.id and random bit string v.rs
v.id and v.rs are assumed to be immutable
prei(v): first i bits of v.rs
Skip graph rule:
For every node v and iIN0:
v connects to closest successor and predecessor w (w.r.t. v.id ) with prei(w) = prei(v)

24. Self-Stabilizing Skip Graph
Nodes v with v.rs=0…
Nodes v with v.rs=1…

25. Self-Stabilizing Skip Graph
Hierarchical view:
000
001 00 01 10 11 1 
(log n) degree, (log n) diameter, (1) expansion w.h.p.

26. Self-Stabilizing Skip Graph
Problem: original skip graph does not allow local checking of correct topology w v
01..
10..
11..
00..
11..
10..
01..
11..
10..
00..

27. Self-Stabilizing Skip Graph
Problem: original skip graph does not allow local checking of correct topology
Solution: extend connections
For each node v let
succi(v,b), b{0,1}: closest successor of v with prefix prei(v)b
predi(v,b), b{0,1}: closest predecessor of v with prefix prei(v)b
rangei(v)=[minb predi(v,b), maxb succi(v,b)]
v connects to all nodes wrangei(v)with prei(w) = prei(v)
Result: SKIP+
Skip graph: rangei(v)=[predi(v),succi(v)]

28. Self-Stabilizing Skip Graph
(log n) degree, (log n) diameter, (1) expansion w.h.p.

29. Self-Stabilizing Skip Graph
edge (u,v) stable: SKIP+ edge from viewpoint of u
otherwise, (u,v) is temporary
flag F(v): indicator in u whether (u,v) stable
Rule 1a: create reverse edges u v u v
stable

30. Self-Stabilizing Skip Graph
Rule 1b/c: introduce stable edges Rule 2: forward temporary edges v v u u b c w w
w.id  rangei(v) v v u u
temp
stable
w (longer prefix match) w

31. Self-Stabilizing Skip Graph
Rule 3a: introduce all u changes set of stable neighbors: initiates insert(v,w) for all neighbors v,w of u Rule 3b: linearize v1 v2 v3 vk v1 v2 v3 vk … …
stable
stable
stable u u

32. Self-Stabilizing Skip Graph
Each node u follows six rules:
Rule 1a: create reverse edges to u
Rules 1b and 1c: introduce stable edges between neighbors of u
Rule 2: forward temporary edges
Rule 3a: introduce all when stable neighborhood changes
Rule 3b: linearize stable neighborhood

33. Self-Stabilizing Skip Graph
Theorem 1: For any weakly connected graph, the rules establish SKIP+ in O(log2 n) rounds w.h.p. Theorem 2: Any join or leave operation in a perfect SKIP+ requires at most O(log4 n) work.

34. Self-Stabilizing Skip Graph
Proof of Theorem 1:
Bottom-up phase: connected components are formed for every prefix
000
001 00 01 10 11 1 

35. Self-Stabilizing Skip Graph
Proof of Theorem 1:
Top-down phase: components sorted

36. Conclusions Many interesting fronts to work on in context
of self-stabilizing overlay networks:
Show O(log n) runtime bound
bound/restrict number of enabled actions to polylog at each node per round
self-stabilizing networks under adversarial behavior
self-preserving networks
self-optimizing networks

37. Questions?