Wait-Free Computability for General Tasks

Transcription

1 Wait-Free Computability for General Tasks Companion slides for Distributed Computing Through Combinatorial Topology Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum 1

2 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Conditions

3 Review Star ¾ Star(¾,K) is the complex of facets of K containing ¾ Complex Distributed 3 Computing through

4 Review Facet Facet A facet is a simplex of maximal dimension not a facet Distributed 4 Computing through

5 Review Open Star Star o (¾,K) union of interiors of simplexes containing ¾ Point Set Distributed 5 Computing through

6 Review Link Link(¾,K) is the complex of simplices of Star(¾,K) not containing ¾ Complex Distributed 6 Computing through

7 Review A simplicial map Á is rigid if dim Á(¾) = dim ¾. 12-Mar-15 7

8 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Conditions

9 The Hourglass task I O

10 Single-Process Executions P,0 P R Q R,0 Q,0 I O

11 P and R only (P and Q Symmetric) P P R R I O

12 Q and R only I O

13 Claim: Hourglass satisfies conditions of fundamental theorem But has no wait-free immediate snapshot protocol! 12-Mar-15 13

14 Claim: Hourglass satisfies conditions of fundamental theorem But has no wait-free immediate snapshot protocol! 12-Mar-15 14

15 homotopy: I O O I carried by

16 Claim: Hourglass satisfies conditions of fundamental theorem But has no wait-free immediate snapshot protocol! 12-Mar-15 16

17 Claim: The Hourglass task solves 2-set agreement Which has no wait-free read-write protocol. 12-Mar-15 17

18 Protocol: Write input value to announce array Run Hourglass task 12-Mar-15 18

19 P P P P,1 R or Q R Q R Q Find non-null announce[] value Look in announce[] array

20 What Went Wrong? Theorem A colorless (I,O, ) has a wait-free immediate snapshot protocol iff there is a continuous map f: I O... carried by 12-Mar-15 20

21 One Direction is OK Theorem If (I,O, ) has a wait-free read-write protocol then there is a continuous map f: I O... carried by 12-Mar-15 21

22 The Other Direction Fails Theorem? If there is a continuous map f: I O... carried by then does (I,O, ) have a wait-free IS protocol? 12-Mar-15 22

23 Review f: I O... Á: Bary N I O Protocol Simplicial approximation Repeated snapshot 12-Mar-15 23

24 Review f: I O... Á: Bary N I O Protocol Simplicial approximation Not colorpreserving Repeated snapshot Another process s output? 12-Mar-15 24

25 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Conditions

26 Theorem A colorless (I,O, ) has a wait-free immediate snapshot protocol iff there is a continuous map f: I O... carried by How can we adapt this theorem to colored tasks? 12-Mar-15 26

27 Fundamental Theorem for Colored Tasks Theorem (I,O, ) has a wait-free read-write protocol iff I has a chromatic subdivision Div I & color-preserving simplicial map Á: Div I O carried by 12-Mar-15 27

28 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I O

29 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I O

30 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I O

31 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I Not a colorless task! O

32 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I O

33 Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 I No simplicial map: I O carried by P,0 Q,0 O

34 Q,1 P,1 Q,1 P,1 Á P,0 Q,0 P,0 Q,0 Div I O

35 // code for P T decide(t input) { announce[p] = input; if (input == 1) return 1; else if (announce[q]!= 1) return 0 } else return 1 // code for Q T decide(t input) { announce[p] = input; if (input == 0) return 0; else if (announce[p]!= 0) return 1 else return 0 } 12-Mar-15 35

36 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Conditions

37 Protocol ) Map protocol δ I (I) O decision map input complex protocol complex output complex carried by 12-Mar-15 37

38 Protocol ) Map (I) δ O subdivision of input complex carried by 12-Mar-15 38

39 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Conditions

40 Task (I,O, ) ¾ (¾) I O 12-Mar-15 40

41 chromatic simplex ¾ chromatic subdivision Div ¾ 12-Mar-15 41

42 ¾ (¾) I O 12-Mar-15 42

43 Theorem says If there is a chromatic subdivision ¾ (¾) I O 12-Mar-15 43

44 Theorem says If there is a chromatic subdivision and a simplicial map à carried by à ¾ (¾) I O 12-Mar-15 44

45 Theorem says If there is a chromatic subdivision and a simplicial map à carried by à ¾ (¾) I 12-Mar O then there is a wait-free IS protocol!

46 Let s start with something easier 12-Mar-15 46

47 Let s start with a special case If there is a simplicial map Ã: Ch N ¾ (¾) Ã Ch N ¾ I (¾) O 12-Mar-15 47

48 Let s start with something easier If there is a simplicial map Á: Ch N ¾ (¾) Á Ch N ¾ (¾) I 12-Mar O then there is a wait-free IS protocol!

49 Protocol Iterated immediate snapshot I Ch N I Mar-15

50 For any chromatic subdivision Div ¾ If there is a color and carrier-preserving simplicial map Á: Ch N ¾ Div ¾ Á Ch N ¾ Div ¾ 12-Mar-15 50

51 Geometric construction Inductively divide boundary 12-Mar-15 51

52 Geometric construction 12-Mar-15 Displace vetexes from barycenter 52

53 Geometric construction 12-Mar-15 53

54 Geometric construction Mesh(Ch ¾) is max diameter of a simplex 12-Mar-15 54

55 Subdivision shrinks mesh mesh(ch ¾) c mesh(¾) for some 0 < c < 1 12-Mar-15 55

56 Open cover 12-Mar-15 56

57 Lesbesgue Number 12-Mar-15 57

58 Open stars form an open cover for a complex ostar(v) 12-Mar-15 58

59 Intersection Lemma Vertexes lie on a common simplex iff their open stars intersect 12-Mar-15 59

60 Pick N large enough that each (closed) star of Ch N ¾ has diameter less than each star of Ch N ¾ lies in a open star of Div ¾ 12-Mar-15 60

61 Defines a vertex map. Simplicial by intersection lemma. 12-Mar-15 61

62 We have just proved the Simplicial Approximation Theorem Á Ch N ¾ Div ¾ There is a carrier-preserving simplicial map Á: Ch N ¾ Div ¾ Not necessarily color-preserving! 12-Mar-15 62

63 An open-star cover is chromatic if every simplex of Ch N ¾ is covered by open stars of of the same color. 12-Mar-15 63

64 If the open-star cover is chromatic. Then the simplicial map. Is color preserving! Must show that covering can be made chromatic 12-Mar-15 64

65 Open Cover Fail Two simplexes conflict If colors disjoint, but polyhedrons overlap. cannot map to same color 12-Mar-15 65

66 Open Cover Fail An open-star cover is chromatic iff there are no conflicting simplexes. We will show how to eliminate conflicting simplexes 12-Mar-15 66

67 Carriers 12-Mar-15 67

68 Perturbation 12-Mar-15 68

69 Room for perturbation Star contains ² ball in carrier around vertex 12-Mar-15 69

70 Room for perturbation Can perturb to any point within ² ball in carrier and still have subdivision 12-Mar-15 70

71 Open Cover Fail ½ has q+1 colors has p+1 colors 12-Mar-15 71

72 Simplexes lie in hyperplane of dimension p+q (because they overlap) 12-Mar-15 72

73 Some vertex has carrier of dimension p+q+1 (because there are p+q+2 colors) 12-Mar-15 73

74 Can perturb vertex within (p+q+1)- dimension ² ball 12-Mar-15 74

75 Can perturb vertex within (p+q+1)- dimension ² ball Out of the hyperplane 12-Mar-15 75

76 Repeat until star diameter < Lebesgue number: Construct Ch Ch N-1 * ¾ Perturb to Ch N * ¾ So open-star cover is chromatic Construct color-preserving simplicial map 12-Mar-15 76

77 Div ¾ Ã (¾) Given 12-Mar-15 77

78 Constructed Á Ch N ¾ Div ¾ Ã (¾) Given 12-Mar-15 78

79 Iterated immediate snapshot here Á Ch N ¾ Div ¾ Ã (¾) Yields protocol here! 12-Mar-15 79

80 Road Map Inherently colored tasks Solvability for colored tasks Protocol ) map Map ) protocol A Sufficient Topological Condition

81 Link-Connected O is link-connected if for each 2 O, link(,o) is (n dim )-connected. 12-Mar-15 81

82 not a fan not link-connected. 12-Mar-15 82

83 Theorem If, for all ¾ 2 I, (¾) is ((dim ¾)-1)-connected, and O is link-connected then (I,O, ) has a wait-free IS protocol 12-Mar-15 83

84 Proof Strategy If, for all ¾ 2 I, (¾) is ((dim ¾)-1)-connected, and O is link-connected, there exists subdivision Div & color-preserving simplicial map ¹: Div I! O carried by. 12-Mar-15 84

85 Proof Strategy If, for all ¾ 2 I, (¾) is ((dim ¾)-1)-connected, and O is link-connected, there exists subdivision Div & color-preserving simplicial map ¹: Div I! O carried by. 12-Mar-15 85

86 Lemma rigid & color-preserving on boundary means color-preserving everywhere suppose we have a rigid simplicial map Á: Div ¾ O that is color-preserving on Div ¾ then Á is color-preserving on Div ¾ 12-Mar-15 86

87 Lemma If O is link-connected can extend rigid simplicial map Á n-1 : skel n-1 I O to a rigid simplicial map Á n : Div I O where Div skel n-1 I = skel n-1 I 12-Mar-15 87

88 Induction Base: n = 1 Á hinge I collapses O Á div I does not collapse O

89 Induction Step: Á does not collapse (n-1)-simplexes Á I O

90 Div I O

91 does not collapse Á Div I O exploit connected link

92 Summary Inductively use connectivity of (¾) link-connectivity of O to construct a color-preserving simplicial map Á: Div I Ocarried by. protocol follows from main theorem 12-Mar-15 92

93 This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 Unported License. 93