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STABILIZATION & LOCALITY

STABILIZATION & LOCALITY. Shay Kutten Technion, Israel. Recall: Traditional methods are global. * Dijkstra- 1 fault- O( n ) Time ; f faults- o( fn ) time * [Katz, Perry]- 1 st general method, 1 fault- O( n ) time (1) self stab bcast freezees all nodes

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STABILIZATION & LOCALITY

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  1. STABILIZATION & LOCALITY Shay Kutten Technion, Israel

  2. Recall: Traditional methods are global * Dijkstra- 1 fault-O(n) Time; f faults- o(fn) time * [Katz, Perry]- 1st general method, 1 fault- O(n) time (1) self stab bcast freezees all nodes (2) global self-stab snapshot to a leaser (3) leader checks global state. (4) Leader initializes every node (if faulty) (5) bcast unfreezes nodes

  3. Recall: Traditional methods are global(continuted) * Global reset (general) methods, 1 fault- O(n) time [Afek, Kutten, Yung], [Awerbuch, Patt-Shamir, Vargheses], [Awerbuch, K., Mansour, Patt-Shamir, Varghese] (1) bcast freezes all nodes (2) reset to a specific initial state [Dolev, Herman] superstabilizing global reset: (2’) reset to a state “nearest” to current state (still 1 fault- O(n) time)

  4. Another look at Dijkstra’s algorithm

  5. 8 8 8 8 8 8 8 8 8 8 7 = 8 8 7 8 7 token 7 8 7 7  8 Global effect example leader

  6. 8 8 8 8 8 8 8 8 leader 8 8 7 = 8 8 7 token 8 7 7 8 8 7 Global effect example

  7. 8 8 8 8 8 8 8 8 leader 8 8 7 = 8 8 7 token 8 7 8 8 8 7 Global effect example

  8. 7 8 8 8 8 8 8 8 8 leader 8 8 8 7 8 7 token 7 8 7 Global effect example

  9. 7 7 8 8 8 8 8 8 leader 8 8 8 7 8 7 token 7 8 7 Global effect example

  10. 7 7 8 7 8 8 8 8 leader 8 8 8 7 8 7 token 7 8 7 Global effect example

  11. 7 7 8 7 8 7 8 8 leader 8 8 8 7 8 7 token 7 8 7 Global effect example

  12. The effect of one fault circles whole ring 7 7 8 7 8 7 8 7 leader 8 8 8 7 8 7 token 7 8 7 Global effect sample

  13. General purpose computer A B Fast stupid switch E C D Fast route from A to C, passing only B’s stupid switch, not B’s general purpose computer. Possible since route is preset. Another example of self stab in industry

  14. A B E C D In bcast how does A’s stupid switch detect that it already received the bcast Another example of self stab in Industry

  15. A B E C D (non- self stab) solution- stupid switch forwards only over ports Ports of links marked tree Another example of self stab in industry

  16. A B fault E C D (non- self stab) solution- (stupid) switch forwards only over ports Ports of links marked tree. Vulnerable to state fault: suppose the “tree” is really a cycle. Another example of self stab in industry

  17. root A pulse pulse E B C D Industrial solutions to the self stab problem(1) Digital’s LAN bridges solution

  18. root A pulse pulse E B pulse pulse C D Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges Solution

  19. root A pulse pulse E B pulse pulse C D pulse Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges Solution

  20. root A 6 6 E B C D Hop counter decreased. When hop counter reaches 0, discard message. Industrial Solutions to the self stab problem(2) IBM’s ATM Solution

  21. root A 6 6 E B 5 5 C D Hop counter decreased. When hop counter reaches 0, discard message. Industrial Solutions to the self stab problem (2) IBM’s ATM Solution

  22. root A E B 5 5 C D 4 4 4 Hop counter decreased. When hop counter reaches 0, discard message. Industrial Solutions to the self stab problem (2) IBM’s ATM Solution

  23. root A E B 3 3 3 C D 4 4 4 3 Hop counter decreased. When hop counter reaches 0, discard message. Industrial Solutions to the self stab problem(1) Digital’s LAN Bridges Solution

  24. root A 2 2 E B 2 3 3 2 C D 3 Hop counter decreased. When hop counter reaches 0, discard message. Industrial solutions to the self stab problem(1) Digital’s LAN bridges solution

  25. Scallability • Industrial solutions are global. They do not scale well • Conventional self stab protocols are global. They do not scale well

  26. (Local “checking” [Awerbuch, Patt-Shamir, Varghese]) root 0 1 14 2 14 13 3 A cycle will be detected by a node seeing Only its parent’s state and its own state. 12 4 5 11 6 10 7 Local detection [Afek, Kutten, Yung]

  27. 2(A) 1(A) D B root A 0(C) C 0(A) root G E 2(A) 1(A) H F 1(C) 0(A) Local checking of a spanning tree, with a root

  28. Any graph marking function is locally checkable • Any algorithm global state is locally checkable • For any bit complexity Cfunction with local checking bit complexity O(C) • Some complexities for interesting marking functions are known • Many other problems are open Local checking of other functions

  29. Goal: (unknown)f faults O(f) time for correction Idea: Diameter of faults is f. Prevent faults expansion, shrink faulty area. 0 1 14 2 14 13 3 fault 12 4 22 11 23 25 24 5 10 6 7 Possibly consistent faults From local checking to local correction

  30. Goal: (unknown)f faults O(f) time for correction Idea: Diameter of faults is f. Prevent faults expansion, shrink faulty area. 0 1 14 2 14 13 3 fault 12 4 22 11 23 (“impossible” if faulty majority) 25 24 5 10 6 7 Possibly consistent faults From local checking to local correction

  31. For example: need to prevent expansion of Faulty area: recall Dijkstra’s algorithm

  32. 7 7 8 7 8 7 8 8 leader 8 8 8 7 8 7 token 7 8 7 f faults create “gap” or “bump” of length f.

  33. T T T f faults create “gap” or “bump” of length f.

  34. val2 val1 Adversary spoils (ones) minority of replicated val1 and of states. Alg. Recovers val1 everywhere. A B val5 D C E val3 val4 Another example of expanding faulty area: Stable Value Problem

  35. val1 val1 A B val5 D C E val1 val1 val’1 val1 val’1 Stable Value Problem Adversary spoils (ones) minority of replicated val1 and of states. Alg. Recovers val1 everywhere.

  36. val2 val1 A B val5 D C E val3 val4 Reducing a general problem to the Stable Value Problem B can compute any func(val1, val2, val3,val4, val5) if it receives every correct vali.

  37. val1 val1 A B “Simple” since E’s vote can be spoiled in D on the way to B. val1 D C E val1 val1 “Simple” but global solution:consensus voting

  38. val1 A B Time = O(diameter) even for one fault. Desired: few faults short time val1 D C E val1 val1 “Simple” but global solution:consensus voting val1

  39. Local voting does not solve Minority faulty notes but local majority everywhere

  40. In O(f) time get >2f votes so majority is non faulty A 2f Idea 1: For (unknown) f faults get votes (values) from radius 2f

  41. In O(f) time get >2f authentic votes so majority is non faulty A 2f C C’s vote at A is authentic (though faulty) B’s vote at A is NOT authentic B Idea 1: For (unknown) f faults get votes (values) from radius 2f

  42. 1 D A B C 1 E Bcast (vote sending) spreading faults problem under state faults

  43. 1 A said: “1” D A B C 1 E Bcast (vote sending) spreading faults problem under state faults

  44. 1 A said: “1” D A B C A said: “0” 1 E Bcast (vote sending) spreading faults problem under state faults

  45. 1 A said: “1” D A B C A said: “0” 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

  46. 1 A said: “1” D A B C A said: “0” 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

  47. 1 A said: “1” D A B C A said: “0” 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

  48. 1 A said: “1” D A B C A said: “0” 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

  49. 1 A said: “1” D A B C A said: “0” 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

  50. Actually, A said: “1” 1 A said: “1” D A B C 1 A said: “0” E Bcast (vote sending) spreading faults problem under state faults

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