1. Competitive Self-Stabilizing k-Clustering
Yvan Rivierre
Joint work with
Ajoy K. Datta, Stéphane Devismes,
Karel Heurtefeux, and Lawrence L. Larmore
Submitted to ICDCS
Co-authors:
Ajoy K. Datta
Stéphane Devismes
Karel Heurtefeux
Lawrence L. Larmore

2. Competitive Self-Stabilizing k-Clustering
Yvan Rivierre
Joint work with
Ajoy K. Datta, Stéphane Devismes,
Karel Heurtefeux, and Lawrence L. Larmore
Submitted to ICDCS
Co-authors:
Ajoy K. Datta
Stéphane Devismes
Karel Heurtefeux
Lawrence L. Larmore

3. Introduction to k-Clustering
Hierarchical organization of processes

4. Introduction to k-Clustering
Hierarchical organization of processes

5. Introduction to k-Clustering
Hierarchical organization of processes

6. Introduction to k-Clustering
Hierarchical organization of processes

7. Self-Stabilization
Illegitimate states
Legitimate states

8. Convergence
Illegitimate states
Legitimate states

9. Closure
Illegitimate states
Legitimate states

10. Finite Time of Stabilization
System States
transient
faults
transient
faults
Time

11. Competitiveness Finding a minimum k-clustering set is NP-hard.
An algorithm is X-competitive if it builds a k-clustering of size at most X times the smallest possible number of k-clusters.
|Clr| ≤ X |Min|

12. Unit Disk Graphs K-clustering for scalability
Self-stabilization for both

13. Quasi Unit Disk Graphs λ ≥ 1 1 K-clustering for scalability
Self-stabilization for both 1

14. Contribution A k-clustering algorithm that is self-stabilizing,
for identified networks,
in the shared memory model,
building at most O(n/k) k-clusters,
7.2552k+O(1)-competitive in UDGs,
and λ²k+O(λ)-competitive in QUDGs.

15. Related Work
Self-stabilizing k-Clustering algorithms (Datta & al., 2009) (Caron & al., 2010)
8k+O(1)-competitive k-Clustering algorithm (Fernandess & Malkhi, 2002)
Related to the k-dominating set problem (plenty of solutions for k = 1)

16. Algorithm: General Overview
Combination of self-stabilizing algorithms:
Spanning Tree Construction (Huang & Chen, 1992)
MIS Tree Build (imp. Fernandess & Malkhi, 2002)
® for the sole purpose of competitiveness
k-Clusterheads Selection
k-Clustering Construction
Parallel execution of both algorithms
BFST
MIST
CHS(k)
CC(k)

17. k-Clusterheads Selection: α

18. k-Clusterheads Selection: α

19. k-Clusterheads Selection: α
2k hops k

20. k-Clusterheads Selection: α
2k hops k k 1

21. k-Clusterheads Selection: α? 2k k
k+1 k
2k hops k k 1 k

22. k-Clusterheads Selection: α? 2k
Tall nodes
k+1 k
2k hops k 1
Short nodes k

23. k-Clusterheads Selection: α
α hops

24. k-Clusterheads Selection: α
α hops

25. Inductive computation of α?
α=?

26. Inductive computation of α?
α=?
A = MaxShort { α < k }
B = MinTall { α ≥ k }
α=A
α=B
α=0
α=k
(0 ≤ A < k ≤ B ≤ 2k)
α=0

27. Inductive computation of α1 1
α=?
A = MaxShort { α < k }
B = MinTall { α ≥ k }
α=A
α=B A
A+B+2 B
α=0
α=k
(0 ≤ A < k ≤ B ≤ 2k)
α=0

28. Inductive computation of α
A + B + 2 ≤ 2k
α=B+1
A = MaxShort { α < k }
B = MinTall { α ≥ k }
α=A
α=B
A+B+2
α=0
α=k
(0 ≤ A < k ≤ B ≤ 2k)
α=0

29. Inductive computation of α
A + B + 2 > 2k
α=A+1
A = MaxShort { α < k }
B = MinTall { α ≥ k }
α=A
α=B
A+B+2
α=0
α=k
(0 ≤ A < k ≤ B ≤ 2k)
α=0

30. 7.2552k+0(1)-competitive in UDGs

31. 7.2552k+0(1)-competitive in UDGs? k k k

32. 7.2552k+0(1)-competitive in UDGs? k k
k + 1 processes k

33. 7.2552k+0(1)-competitive in UDGs
MIS Tree ? k k
k + 1 processes k

34. 7.2552k+0(1)-competitive in UDGs1 2 3 4
height
MIS Tree
(MIS = Maximal
Independent Set) ? k k
k + 1 processes k

35. 7.2552k+0(1)-competitive in UDGs1 2 3 4
height
(MIS = Maximal
Independent Set) ? k k
⌈k /2⌉ proc. ∈ MIS k

36. 7.2552k+0(1)-competitive in UDGs1 2 3 4
height
(|Clr| - 1) ⌈k/2⌉ ≤ |MIS| - 1
(MIS = Maximal
Independent Set) ? k k
⌈k /2⌉ proc. ∈ MIS k

37. 7.2552k+0(1)-competitive in UDGs1 2 3 4
height
(|Clr| - 1) ⌈k/2⌉ ≤ |MIS| - 1
(MIS = Maximal
Independent Set)
|IS| ≤ ((2π/√3)k² + πk + 1) |Opt| ? k k
⌈k /2⌉ proc. ∈ MIS k

38. 7.2552k+0(1)-competitive in UDGs1 2 3 4
height
(|Clr| - 1) ⌈k/2⌉ ≤ |MIS| - 1
(MIS = Maximal
Independent Set)
|IS| ≤ ((2π/√3)k² + πk + 1) |Opt| ? k k
⌈k /2⌉ proc. ∈ MIS k
|Clr| ≤ 1 + ((4π/√3)k + 2π) |Opt| » k+0(1) |Opt|

39. Overall Stabilization in O(n) rounds
Stabilization Time D
BFS Tree
in O(D) rounds 2D
MIS Tree
in O(n) rounds
k-Clustering
Overall Stabilization in O(n) rounds

40. Overall Stabilization in O(n) rounds
Stabilization Time D
BFS Tree
in O(D) rounds 2D
MIS Tree
in O(n) rounds
k-Clustering
Overall Stabilization in O(n) rounds
Time Bottleneck is MIS Tree Construction
Can we improve it?

41. MIS Tree & Nick's Class
Nick's Class (NC): set of all problems solved in //, in O(loga n) time, using O(nb) processors.
NC ⊆ P: proving NC = P is considered to be as difficult as proving NP = P

42. MIS Tree & Nick's Class
A problem A ∈ P is P-complete, if given any problem B ∈ P, ∃ a NC-reduction of B to A.
If ∃ a P-complete problem ∈ NC, then NC = P

43. MIS Tree & Nick's Class Usual conjecture: P-NC ¹ ∅
That implies: ∀ A is P-complete, A ∈ P-NC
∀ A ∈ P-NC, A is inherently sequential ⇒ distributed algorithm in W(n)

44. MIS Tree Problem is P-complete
Proof:
Circuit Value Problem (CV) is P-complete
We give an NC-reduction of CV to MIS Tree

45. Conclusion Self-stabilizing k-clustering algorithm which
stabilizes in O(n) rounds,
uses O(log n) space per process,
builds at most O(n/k) k-clusters,
is competitive in UDGs and QUDGs.
MIS Tree construction is P-complete

46. Thanks for your attention.

47. MIS Tree 1 2 3 4
height
Spanning tree such that the set of processes located at even levels is an MIS (Maximal Independent Set).