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Self-stabilizing k-Independent Dominating Set Construction

Self-stabilizing k-Independent Dominating Set Construction. Colette JOHNEN University of Bordeaux, LaBRI, France Johnen@labri.fr www.labri.fr/~johnen. Content. Motivation and Definitions Related Works ( Fast k-Independent Dominating set) FID algorithm

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Self-stabilizing k-Independent Dominating Set Construction

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  1. Self-stabilizing k-Independent Dominating Set Construction Colette JOHNEN University of Bordeaux, LaBRI, France Johnen@labri.fr www.labri.fr/~johnen

  2. Content • Motivation and Definitions • Related Works • (Fast k-Independent Dominating set) FIDalgorithm • (Simple k-Independent Dominating set) SIDalgorithm • Conclusion

  3. The Motivation is the Clustering-Building hierachical-structure on top of infrastructure-less network - • To part network nodes into non-overlapping groups called clusters • Each cluster has a single head, node acting has cluster coordinator • In k-hop cluster, each node is at distance at most k of its head

  4. D : a k-dominating set All nodes are at distance at most k of a node of D ( D is also named a distance-k dominating set) node of D: k=4

  5. D : a k-dominating set All nodes are at distance at most k of a node of D ( D is also named a distance-k dominating set) node of D: k=4

  6. D : a k-dominating set All nodes are at distance at most k of a node of D ( D is also named a distance-k dominating set) node of D: k=4

  7. I : a k-independent set The distance between two nodes of I is greater than k ( I is also named a distance-k independent set) node of I: k=4

  8. kID : a k-independent dominating set All nodes of V/kID are at distance at most k of a node of kID and the distance between two nodes of kID is greater than k node of kID: k=4

  9. kID : a k-independent dominating set All nodes of V/kID are at distance at most k of a node of kID and the distance between two nodes of kID is greater than k Theorem : the size of any k-independent dominating set is at most max(1,n.(k+2)/2 -1) nodes node of kID: k=4

  10. Self-stabilizing 1-independent dominating set construction • [GHJS 03] in WAPDCM : central deamon • [XGHS 03] in IWDC : synchronous deamon • [MFGT 05] in WWAN : using network criterion to select the heads • [KM 06] in IPDPS : SS algorithm with safe convergence • [JN 06] in OPODIS : robust SS algorithm • [DT06] in OPODIS : probabilistic algorithm

  11. MIS tree BFS tree Competitive self-stabilizing k-clustering[DLDHR 12] • The size of k-dominating set is bounded by 1+ (n-1) /(k+1) • Algo : to build a MIS tree rooted at the elected node, then the k-dominating set on of the tree

  12. Competitive self-stabilizing k-clustering[DLDHR 12] 0 0 0 2 -1 1 -3 -4 -3 3 4 -2 -4 2 -1 -4 -2 -3 1 k=4 12 Building of the k-dominating set on of the MIS tree ANR Displexity

  13. self-stabilizing small k-dominating sets[DLDHR 13] k=4 13 • The size of k-dominating set is bounded by n /(k+1) • Algo : to build a BFS tree rooted at the elected node, and then to build the k-dominating set the BFS tree ANR Displexity

  14. 4 2 3 1 [0,0,0,0,1] [0,0,0,1,1] [0,1,1,1,1] [0,0,1,1,1] 0 4 2 1 3 [1,0,0,0,0] [1,0,0,0,1] [1,0,0,1,1] [1,1,1,1,1] [1,0,1,1,1] 1 0 4 2 3 [1,0,0,0,0] [1,0,0,0,1] [1,0,0,1,1] [1,0,1,1,1] [1,1,1,1,1] self-stabilizing small k-dominating sets[DLDHR 13] Building of the k-dominating set on the BFS tree [3,3,3,3,3] 0 k=4

  15. self-stabilizing small k-dominating sets[DLDHR 13] colorDom=1 4 2 3 colorDom=1 colorDom=1 0 4 2 3 colorDom=1 colorDom=1 0 4 2 colorDom=1 3 [3,3,3,3,3] 0 1 colorDom=1 colorDom=1 1 colorDom=1 1 colorDom=1 k=4 15 ANR Displexity Building of the k-dominating set on the BFS tree

  16. Silent Self-Stabilizing size bounded k-dominating set construction

  17. Content • Motivation and definition • Related Works • (Fast k-Independent Dominating set) FIDalgorithm • variable : one table of size k+1 • 3 rules • SIDalgorithm • Conclusion

  18. FID variable on node v Dom[] is a table of size k+1 dom[i]isthe identifier of the largestheadhaving a path of lengthibetweenitself and v if such a headexistsotherwisedom[i] =  vbelongs to thek-independentdominating set iffdom[0] = idv - v is a head 90 [90, , 90, , 90] 89 [, 90, , 90, ] Node identifier [the contain of dom]

  19. Election rule of FID : a ordinary node having no knowledge of a head in its k-neighborhood having a larger identifier than its own one : becomes a head 90 [, 89, 88,77, 86] 90 [90, , 88, 77, 86] 89 [, 88, 77, 86, ] node identifier [the contain of dom] k=4

  20. Resignation rule of FID : a head having knowledge of a head in its k-neighborhood having a larger identifier than its own one : becomes ordinary 81 [81, , 81, , ] 81 [, , 81, 87, ] 79 [, 81,87, , ] Node identifier [the contain of dom] k=4

  21. Updating rule of FID : for i >0, dom[i](v) = max(dom[i-1](u) | u being a v’s neighbor) 88 [, , 90, , ] 85 [, , 88, , ] 84 [, 85, 89, , ] 83 [, , 88,, ] 72 [, 87, 81, , ] 84 [, , 87,90, ] k=4

  22. Content • Motivation and definition • Related Works • (Fast k-Independent Dominating set) FIDalgorithm • Exampleunder the synchonousschedule • SIDalgorithm • Conclusion

  23. Every node is a Head 88 [88, , , , ] 90 [90, , , ,  ] 89 [89, , , , ] 77 [77, , , , ] 86 [86, , , , ] 84 [84, , , , ] 85 [85, , , , ] 80 [80, , , , ] 83 [83, , , , ] 82 [90, , , , ] 79 [79, , , , ] 81 [81, , , , ] 87 [87, , , , ] 72 [72, , , , ] 66 [66, , , , ] Id the contain of dom[] k=4

  24. 5 Heads 88 [, 89 90 [90,  89 [, 90 77 [, 88 86 [86,  84 [, 88 85 [85,  80 [, 85 83 [ , 84 82 [, 86 79 [, 81 81 [81,  87 [87,  72 [, 87 66 [, 87 Id for i >1, d[i]=  k=4

  25. 3 Heads 88 [, , 90 90 [90, , 90 89 [, 90 89 77 [, 86, 89 86 [, , 88 84 [, 85, 89 85 [, , 88 80 [, 85,  83 [, , 88 82 [, 86, 87 79 [, 81, 87 81 [81, , 81 87 [87, , 87 72 [, 87, 81 66 [ ,87, 86 Id for i >2, d[i]=  k=4

  26. 2 terminal Heads 88 [, , 90, 89 90 [90, , 90, 89 89 [, 90, , 90 77 [, , , 90 86 [, , 86, 89 84 [, , 87, 90 85 [, , 85, 89 80 [, , , 88 83 [, , 86 ,89 82 [, , , 87, 88 79 [ ,81, 87, 81 81 [, , 81, 87 87 [87, , 87, 86 72 [, 87, 81, 87 66 [ ,87, 86, 87 Id for i = 4, d[i]=  k=4

  27. 2 terminal Heads 88 [, , 90, 87 90 [90 89 [, 90 77 [, , , 90, 87] 86 [, , , 87, 90] 84 [, , 87, 90 85 [, , , 87,90] 80 [, , , 85, 89] 83 [, , , 87, 90] 82 [, , , 87 79 [ , , 87 81 [, , 81, 87 87 [87 72 [, 87 66 [ ,87 Id [a prefix of dom k=4

  28. 2 terminal Heads 88 [, , 90, 87 90 [90 89 [, 90 77 [, , , 90, 87] 86 [, , , 87, 90] 84 [, , 87, 90 85 [, , , 87, 90] 80 [, , , , 87] 83 [, , , 87, 90] 82 [, , , 87 79 [ , , 87 81 [, , , 87 87 [87 72 [, 87 66 [ ,87 Id [a prefix of dom stable k=4

  29. Proofs of FID* • FID is a silent self-stabilizing algorithm building k-independent dominating set • any terminal configuration is legitimate • all computations are finite (ad absurdio argumentation) • the convergence time : 4n+k rounds * In tech. Report RR 1472-13 available in www.labri.fr/~johnen

  30. Convergence Time

  31. Memory Space

  32. Content • Motivation and definition • Related Works • FIDalgorithm • (Simple k-Independent Dominating set) SIDalgorithm • 2 shared variables • 3 rules • Conclusion

  33. SID variables on node v firstHead(v)=(d1, idu)-iduisthe identifier of the closesthead to v,and d1istheir distance vbelongsto the k-independentdominating set ifffirstHead(v)=(0, idv)- v is a head If vhas a single head in its k-neighborhood secondHead(v)=  OtherwisesecondHead(v)= (d2, idw) - idwis the identifier of the second closesthead to v,and d2istheir distance

  34. Updatingruleof SID : if needed, a node v updatesfirstHead(v) andsecondHead(v) 88 (3,77) (3,80) 84 (2,77) (2,80) 83 (2,66) (3,77) 85 (3,77) (3,80) 84 (3,66) (4,77) 72 (2,66) (3,77) id firstHead secondHead k=4

  35. Election rule of SID : a ordinary node having no knowledge of a head in its k-neighborhood : becomes a head 90 (3,77)(4,72) 89 (4,77) (4,80) 90 (0,90)  id firstHead secondHead k=4

  36. Resignation rule of SID : a head having knowledge of a head in its k-neighborhood having a smallest identifier than its own one : becomes ordinary 84 (1,72) (1,83) 72 (2,66) (2,81) 72 (0,72)  79 (1,72) (1,81) 87 (1,66) (1,72) id firstHead secondHead k=4

  37. Content • Motivation and definition • Related Works • FIDalgorithm • (Simple k-Independent Dominating set) SIDalgorithm • Exampleunder the synchronousschedule • Conclusion

  38. Every node is a Head 88 (0,88)  90 (0,90) 89 (0,89)  77 (0,77)  86 (0,86)  84 (0,84)  85 (0,85)  80 (0,80)  83 (0,83)  82 (0,82)  72 (0,82)  79 (0,79)  81 (0,81)  87 (0,87)  66 (0,66)  id firstHead secondHead k=4

  39. 4 Heads 88 (1,77) (1,84) 90 (1,89) 89 (1,88) (1,90) 77 (0,77)  86 (1,77) (1,82) 84 (1,72) (1,83) 85 (1,80) (1,84) 80 (0,80)  83 (1,82) (1,84) 82 (1,66) (1,83) 72 (0,72)  79 (1,72) (1,81) 81 (1,79)  87 (1,66) (1,72) 66 (0,66)  id firstHead secondHead k=4

  40. 3 Heads 88 (1,77) (2,72) 90 (2,88) 89 (2,77) (2,84) 77 (0,77)  86 (1,77) (2,66) 84 (1,72) (2,77) 85 (1,80) (2,72) 80 (0,80)  83 (2,66) (2,72) 82 (1,66) (2,77) 72 (2,66) (2,81) 79 (1,72)  81 (2,72)  87 (1,66) (1,72) 66 (0,66)  id firstHead secondHead k=4

  41. 1 Head 88 (1,77) (2,72) 90 (3,77)(3,84) 89 (2,77) (3,72) 77 (3,66) (3,72) 86 (1,77) (2,66) 84 (2,77) (2,80) 85 (1,80) (2,72) 80 (3,72)  83 (2,66) (2,72) 82 (1,66) (2,77) 72 (2,66) (3,77) 79 (3,66) (3,72) 81 (2,72) (3,84) 87 (1,66) (3,81) 66 (0,66)  id firstHead secondHead k=4

  42. 1 Head 88 (3,77) (3,80) 90 (3,77)(4,72) 89 (2,77) (3,72) 77 (3,66) (3,72) 86 (2,66) (3,77) 84 (2,77) (2,80) 85 (3,77) (3,80) 80 (3,72)  83 (2,66) (3,77) 82 (1,66) (2,77) 72 (2,66) (3,77) 79 (3,66) (3,72) 81 (4,66) (4,72) 87 (1,66) (4,77) 66 (0,66)  id firstHead secondHead k=4

  43. 1 Head 88 (3,77) (3,80) 90 (3,77)(4,72) 89 (4,77) (4,80) 77 (3,66) (3,80) 86 (2,66) (3,77) 84 (3,66) (4,77) 85 (3,77) (3,80) 80 (4,77)  83 (2,66) (3,77) 82 (1,66) (4,77) 72 (2,66) (3,77) 79 (3,66) (4,77) 81 (4,66) (4,72) 87 (1,66) (4,77) 66 (0,66)  id firstHead secondHead k=4

  44. 2 Heads 88 (4,66)  90 (0,90)  89 (4,77) (4,80) 77 (3,66)  86 (2,66)  84 (3,66) (4,77) 85 (4,66)  80 (4,77)  83 (2,66)  82 (1,66) (4,77) 72 (2,66)  79 (3,66) (4,77) 81 (4,66)  87 (1,66) (4,77) 66 (0,66)  id firstHead secondHead k=4

  45. 3 terminal Heads 88 (4,66)  90 (0,90)  89 (1,90)  77 (3,66)  86 (2,66)  84 (3,66)  85 (4,66)  80 (0,80)  83 (2,66)  82 (1,66)  72 (2,66)  79 (3,66)  81 (4,66)  87 (1,66)  66 (0,66)  id firstHead secondHead k=4

  46. 3 terminal Heads 88 (2,90) (4,66) 90 (0,90)  89 (1,90)  77 (3,66)  86 (2,66)  84 (3,66)  85 (1,80) (4,66) 80 (0,80)  83 (2,66)  82 (1,66)  72 (2,66)  79 (3,66)  81 (4,66)  87 (1,66)  66 (0,66)  id firstHead secondHead k=4

  47. 3 terminal Heads 88 (2,90) (4,66) 90 (0,90)  89 (1,90)  77 (3,66) (3,90) 86 (2,66)  84 (2,80) (3,66) 85 (1,80) (4,66) 80 (0,80)  83 (2,66)  82 (1,66)  72 (2,66)  79 (3,66)  81 (4,66)  87 (1,66)  66 (0,66)  id firstHead secondHead k=4

  48. 3 terminal Heads 88 (2,90) (3,80) 90 (0,90)  89 (1,90)  77 (3,66) (3,90) 86 (2,66) (4,90) 84 (2,80) (3,66) 85 (1,80) (4,66) 80 (0,80)  83 (2,66) (3,80) 82 (1,66)  72 (2,66) (3,80) 79 (3,66)  81 (4,66)  87 (1,66)  66 (0,66)  id firstHead secondHead k=4

  49. 3 terminal Heads 88 (2,90) (3,80) 90 (0,90)  89 (1,90) (4,80) 77 (3,66) (3,90) 86 (2,66) (4,90) 84 (2,80) (3,66) 85 (1,80) (4,66) 80 (0,80)  83 (2,66) (3,80) 82 (1,66) (4,80) 72 (2,66) (3,80) 79 (3,66) (4,80) 81 (4,66)  87 (1,66) (4,80) 66 (0,66)  id firstHead secondHead k=4

  50. Proofs of SID* • SID is a silent self-stabilizingalgorithmbuilding k-independentdominating set • any terminal configuration islegitimate • all computations are finite (ad absurdio argumentation) • Open question : To establish the convergence time * In tech. Report RR 1473-13 available in www.labri.fr/~johnen

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