1. Manifold Protocols TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AA Companion slides for Distributed Computing Through Combinatorial Topology Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

2. Kinds of Results 2 Impossibility Separation Task T cannot be solved in model M Task T cannot be solved in model M Task T can be solved by a protocol for T’, but not vice-versa Distributed Computing Through Combinatorial Topology

3. Road Map 3 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results Distributed Computing Through Combinatorial Topology

4. Road Map 4 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results Distributed Computing Through Combinatorial Topology

5. Simplicial Complex 5 Distributed Computing Through Combinatorial Topology

6. 6 Manifolds Every (n-1)-simplex a face of two n-simplexes Every (n-1)-simplex a face of two n-simplexes Distributed Computing Through Combinatorial Topology

7. 7 Manifolds Every (n-1)-simplex a face of two n-simplexes Every (n-1)-simplex a face of two n-simplexes technically, a pseudo-manifold Distributed Computing Through Combinatorial Topology

8. 8 Manifold with Boundary Internal (n-1)-simplex a face of two n-simplexes Internal (n-1)-simplex a face of two n-simplexes Boundary (n-1)-simplex a face of one n-simplex Boundary (n-1)-simplex a face of one n-simplex Boundary  C

9. Why Manifolds? 9 Nice combinatorial properties Many useful theorems Easy to prove certain claims True, most complexes are not manifolds …. Still a good place to start. Distributed Computing Through Combinatorial Topology

10. Road Map 10 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results Distributed Computing Through Combinatorial Topology

11. Immediate Snapshot Executions 11 Restricted form of Read-Write memory Protocol complexes are manifolds (Equivalent to regular R-W memory) But we will not prove it yet. Distributed Computing Through Combinatorial Topology

12. 000 1 Write Single-writer, multi-reader variables 12

13. Distributed Computing Through Combinatorial Topology 100 100 Snapshot Single-writer, multi-reader variables 13

14. Immediate Snapshot Executions 14 Pick a set of processes write together snapshot together Repeat with another set Distributed Computing Through Combinatorial Topology

15. Example Executions PQR write snap write snap write snap P??PQ?PQR 15 Each process writes, then takes snapshot Each process has a view time Distributed Computing Through Combinatorial Topology

16. Example Executions PQR write snap write snap write snap P??PQR 16 Moving last process one round earlier, Changes this view from PQ? to PQR PQR write snap write snap write snap P??PQ?PQR Distributed Computing Through Combinatorial Topology

17. PQR write snap write snap PQR PQR write snap write snap PQR PQR write snap write snap write snap P??PQR Example Executions 17 Moving last processes one round earlier, Changes this view from P?? to PQR Distributed Computing Through Combinatorial Topology

18. Protocol Complex Process (color) & view 18Distributed Computing Through Combinatorial Topology

19. Protocol Complex PQR write snap write snap write snap PQR write snap PQR write snap write snap 19Distributed Computing Through Combinatorial Topology

20. Standard Chromatic Subdivision 20 Distributed Computing Through Combinatorial Topology

21. Vertex (P i, ¾ i ) Combinatorial Definition (I) 21 Process name view ¾ i µ ¾ Input simplex ¾ Protocol complex Ch(¾) Distributed Computing Through Combinatorial Topology

22. Combinatorial Definition (II) 22 Each process's write appears in its own view P i 2 names(¾ i ). Snapshots are ordered ¾ i µ ¾ j or vice-versa Each snapshot ordered immediately after write if P i 2 names(¾ j ), then ¾ i µ ¾ j. If P j saw P i write … then P j ‘s snapshot not later than P j ‘s.

23. Manifold Theorem 23 If the input complex I is a manifold … Distributed Computing Through Combinatorial Topology

24. Manifold Theorem 24 … so is the immediate snapshot protocol complex Ch( I ). Distributed Computing Through Combinatorial Topology

25. Proof of Manifold Theorem 25 Let ¿ n-1 be an (n-1)- simplex of Ch( I ). Must prove that ¿ n-1 is a face of exactly one or two n-simplexes of Ch( I ). Distributed Computing Through Combinatorial Topology

26. Without Loss of Generality … 26 Simplex of I seen by last process Where ¾ i µ ¾ i+1 for 0 · i < n Re-index processes in execution order (ignoring ties) Distributed Computing Through Combinatorial Topology

27. Proof Strategy 27 Count the ways we can extend… Where ¾ n is an n-simplex of Ch(¾) ¿ n-1 = {(P 0, ¾ 0 ), …, (P n-1, ¾ n-1 )} ¿ n = {(P 0, ¾ 0 ), …, (P n-1, ¾ n-1 i, (P n, ¾ n )} Distributed Computing Through Combinatorial Topology

28. Cases 28 dim n-1 or dim n Boundary or internal 3 cases Distributed Computing Through Combinatorial Topology

29. 29 ¿ n-1¿ n-1 ¾n-1¾n-1 Boundary (n-1)-simplex Case One Distributed Computing Through Combinatorial Topology

30. 30 Exactly one n-simplex ¾ of I contains ¾ n-1 ¾ ¾n-1¾n-1 ¿n-1¿n-1 Case One Distributed Computing Through Combinatorial Topology

31. 31 Exactly one n-simplex ¿ n of Ch( I ) contains ¿ n-1 ¿ n = ¿ n-1 [ h P n, ¾ i ¾ Case One Distributed Computing Through Combinatorial Topology

32. 32 ¿n-1¿n-1 ¾n-1¾n-1 Internal (n-1)-simplex Case Two Distributed Computing Through Combinatorial Topology

33. 33 Exactly two n-simplexes ¾ 0 & ¾ 1 of I contain ¾ n-1 ¾0¾0 ¾1¾1 Case Two Distributed Computing Through Combinatorial Topology

34. 34 Exactly 2 n-simplexes ¿ 0 n & ¿ 1 n of Ch( I ) contain ¿ n-1 ¿ n 0 = ¿ n-1 [ h P n, ¾ 0 i ¿ n 1 = ¿ n-1 [ h P n, ¾ 1 i ¾0¾0 ¾1¾1 Case Three Distributed Computing Through Combinatorial Topology

35. 35 ¿n-1¿n-1 ¾n-1¾n-1 n-simplex Case Three Distributed Computing Through Combinatorial Topology

36. Suppose 36 P n 2 names( ¾ k ) (P k saw P n ’s write) P n  names( ¾ k-1 ) (P k-1 did not see P n ’s write) Special case: k=0, ¾ -1 = ; ¾n µ ¾k¾n µ ¾k ¾n µ ¾k¾n µ ¾k ) ) ¾k-1 ½ ¾n¾k-1 ½ ¾n ¾k-1 ½ ¾n¾k-1 ½ ¾n ) ) Either ¾ k-1 ½ ¾ n ½ ¾ k or ¾ k-1 ½ ¾ n = ¾ k Case Three

37. 37 P k-1 did not see P n P k did see P n P n between P k-1 & P k P n with P k This completes the proof. Case Three Distributed Computing Through Combinatorial Topology

38. Manifold Protocols 38 A protocol M ( I ) is a manifold protocol if If I is a manifold, so is M ( I ) M (  I ) =  M ( I ). Example: immediate snapshot Important: closed under composition Distributed Computing Through Combinatorial Topology

39. Road Map 39 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results

40. Distributed Computing Through Combinatorial Topology k-set Agreement: before 32 19 21 40

41. 19 21 41 k-set Agreement: after k = 2 Distributed Computing Through Combinatorial Topology

42. Theorem 42 No Manifold Protocol can solve n-set agreement Including Immediate Snapshot Including read-write memory Distributed Computing Through Combinatorial Topology

43. Sperner Coloring 43Distributed Computing Through Combinatorial Topology

44. Sperner Coloring 44Distributed Computing Through Combinatorial Topology “Corners” have distinct colors

45. Sperner Coloring 45Distributed Computing Through Combinatorial Topology Edge vertexes have corner colors “Corners” have distinct colors

46. Sperner Coloring 46Distributed Computing Through Combinatorial Topology Edge vertexes have corner colors “Corners” have distinct colors Every vertex has face boundary colors

47. Sperner’s Lemma 47Distributed Computing Through Combinatorial Topology In any Sperner coloring, at least one n-simplex has all n+1 colors

48. Sperner Coloring for Manifolds with Boundary (base) 48 One color here Other color here Only those colors in interior Distributed Computing Through Combinatorial Topology

49. Sperner Coloring for Manifolds with Boundary (inductive) 49 M is n-manifold colored with ¦  M = [ i M i Where M i is non-empty (n-1)- manifold Sperner-colored with ¦ n {i } Distributed Computing Through Combinatorial Topology

50. Sperner’s Lemma for Manifolds 50Distributed Computing Through Combinatorial Topology MM Odd number of n- simplexes have all n+1 colors Sperner Coloring on Manifold M

51. Proof of Sperner’s Lemma 51 Induction on n Base One color here Other color here Odd number of changes Distributed Computing Through Combinatorial Topology

52. Induction Step 52 Distributed Computing Through Combinatorial Topology

53. Dual Graph One vertex per n-simplex … 53 Distributed Computing Through Combinatorial Topology

54. Dual Graph One vertex per n-simplex … One “external” vertex 54 Distributed Computing Through Combinatorial Topology

55. Discard One Color 55 n+1 colors “restricted”n colors Distributed Computing Through Combinatorial Topology

56. Edges 56 If two n-simplexes share an (n-1)-face … with restricted n colors … add an edge. Distributed Computing Through Combinatorial Topology

57. Edges 57 Same for external vertex Distributed Computing Through Combinatorial Topology

58. Some Vertexes have Degree Zero 58 Even degree Distributed Computing Through Combinatorial Topology

59. Some Vertexes have Degree Two 59 Even degree Only n restricted colors Distributed Computing Through Combinatorial Topology

60. All n+1 Colors ) Degree 1 60 Odd degree Distributed Computing Through Combinatorial Topology

61. Induction Hypothesis Odd number of n-colored simplexes 61 Distributed Computing Through Combinatorial Topology

62. Induction Hypothesis External vertex has odd degree 62 Distributed Computing Through Combinatorial Topology

63. Odd Degrees 63 One external vertex Even n-simplexes with n+1 colors? All the rest Distributed Computing Through Combinatorial Topology

64. Odd Degrees 64 One external vertex Even n-simplexes with n+1 colors? All the rest Every graph Even number of odd degree vertexes sum of degrees is even… each edge contributes 2 Distributed Computing Through Combinatorial Topology

65. Odd Degrees 65 One external vertex Even n-simplexes with n+1 colors? All the rest Even number of odd degree vertexes Distributed Computing Through Combinatorial Topology

66. Parity 66 One external vertex Even number of odd degree vertexes n-simplexes with n+1 colors? Even Odd Distributed Computing Through Combinatorial Topology

67. Parity 67 One external vertex Even number of odd degree vertexes n-simplexes with n+1 colors? Even Odd QED Distributed Computing Through Combinatorial Topology

68. No Manifold Task can solve n-Set Agreement 68 Assume protocol exists: Run manifold task protocol Choose value based on vertex Idea: Color vertex with “winning” process name … Distributed Computing Through Combinatorial Topology

69. 69 Manifold Task for n-Set Agreement manifold Only P wins Only Q and R win Sperner coloring

70. 70 Manifold Task for n-Set Agreement manifold Sperner coloring Distributed Computing Through Combinatorial Topology

71. 71 Manifold Task for n-Set Agreement manifold Sperner coloring n+1 colors Contradiction: at most n can win Execution with n+1 winners Distributed Computing Through Combinatorial Topology

72. Road Map 72 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results Distributed Computing Through Combinatorial Topology

73. Weak Symmetry-Breaking 73 Group 0 Group 1 If all processes participate … At least one process in each group Distributed Computing Through Combinatorial Topology

74. Weak Symmetry-Breaking 74 Group 0 Group 1 If fewer participate … we don’t care. Distributed Computing Through Combinatorial Topology

75. Anonymous Protocols 75 Trivial solution: choose name parity WSB protocol should be anonymous Output depends on … Input … Interleaving … But not name Restriction on protocol, not task!

76. Road Map 76 Manifolds Sperner’s Lemma and k-Set Agreement Immediate Snapshot Model Weak Symmetry-Breaking Separation results

77. Next Step 77 Construct manifold task that solves weak-symmetry-breaking Because it is a manifold, it cannot solve n-set agreement Separation: n-set agreement is harder than WSB Distributed Computing Through Combinatorial Topology

78. A Simplex 78Distributed Computing Through Combinatorial Topology

79. Standard Chromatic Subdivision 79Distributed Computing Through Combinatorial Topology

80. Glue Three Copies Together 80Distributed Computing Through Combinatorial Topology

81. Glue Opposite Edges 81Distributed Computing Through Combinatorial Topology

82. The Moebius Task 82Distributed Computing Through Combinatorial Topology

83. Defines a Manifold Task 83Distributed Computing Through Combinatorial Topology boundary

84. Manifold Task 84Distributed Computing Through Combinatorial Topology Note Sperner coloring on boundary boundary

85. Terminology 85Distributed Computing Through Combinatorial Topology Each face has a central simplex 1-dim 2-dim

86. Subdivided Faces 86Distributed Computing Through Combinatorial Topology external internal

87. Black-and-White Coloring (I) 87Distributed Computing Through Combinatorial Topology Central 2-simplex: White near external face Black elsewhere

88. Black-and-White Coloring (II) 88Distributed Computing Through Combinatorial Topology Black Central simplex of external face: Central simplex of external face:

89. Black-and-White Coloring (III) 89Distributed Computing Through Combinatorial Topology All others White

90. Moebius Solves WSB 90Distributed Computing Through Combinatorial Topology

91. Moebius Solves WSB 91Distributed Computing Through Combinatorial Topology

92. Moebius Solves WSB 92Distributed Computing Through Combinatorial Topology Every n-simplex has both black & white colors. Boundary coloring is symmetric!

93. General Construction 93 n+1 “copies” » 0, … » n of Ch(¾) »0»0 »0»0 »n»n »n»n … … »1»1 »1»1 (picture not exact!) n = 2N Distributed Computing Through Combinatorial Topology

94. General Construction 94 »0»0 »0»0 »i-1»i-1 »i-1»i-1 … … n = 2N »i»i »i»i »i+1»i+1 »i+1»i+1 Face i is external »k»k »k»k … … … … »n»n »n»n Distributed Computing Through Combinatorial Topology

95. General Construction 95 »0»0 »0»0 »i-1»i-1 »i-1»i-1 … … n = 2N »i»i »i»i »i+1»i+1 »i+1»i+1 Face i is external »k»k »k»k … … … … »n»n »n»n 2N others

96. Distributed Computing Through Combinatorial Topology General Construction 96 »0»0 »0»0 »i-1»i-1 »i-1»i-1 … … n = 2N »i»i »i»i »i+1»i+1 »i+1»i+1 »k»k »k»k … … … … »n»n »n»n Identify Faces j Rank j+N out of 2N Face i is external

97. Distributed Computing Through Combinatorial Topology General Construction 97 »0»0 »0»0 »i-1»i-1 »i-1»i-1 … … n = 2N »i»i »i»i »i+1»i+1 »i+1»i+1 »k»k »k»k … … … … »n»n »n»n Symmetric! Rank j+N out of 2N Face i is external Why it’s a (non-orientable) manifold Why it’s a (non-orientable) manifold Why it works only for even dimensions

98. Open Problem 98 Generalize to odd dimensions … or find counterexample. Distributed Computing Through Combinatorial Topology

99. Black-and-White Coloring (I) 99Distributed Computing Through Combinatorial Topology Central n-simplex: White near external face Black elsewhere

100. Black-and-White Coloring (II) 100Distributed Computing Through Combinatorial Topology Black Central m-simplexes of external face for m > 0 Central m-simplexes of external face for m > 0

101. Black-and-White Coloring (III) 101Distributed Computing Through Combinatorial Topology All others White

102. No Monochromatic n-simplexes 102Distributed Computing Through Combinatorial Topology Central n-simplex By construction

103. No Monochromatic n-simplexes 103Distributed Computing Through Combinatorial Topology Intersects internal face Black at n-center White at edge

104. No Monochromatic n-simplexes 104Distributed Computing Through Combinatorial Topology Intersects external face White at n-center Black at m-center, 0 < m < n

105. Progress 105Distributed Computing Through Combinatorial Topology Weak Symmetry-Breaking Moebius Task no yes Set Agreement

106. Next Step 106Distributed Computing Through Combinatorial Topology Weak Symmetry-Breaking Moebius Task no yes Anonymous Set Agreement

107. 107 Group 1 Choose name with anonymous n-Set agreement Choose name with anonymous n-Set agreement My name written? My name written? Group 0 no · n+1 Write name Read names yes Set Agreement  WSB Distributed Computing Through Combinatorial Topology

108. Choose name with anonymous n-Set agreement Choose name with anonymous n-Set agreement Proof 108 Group 1 My name written? My name written? Group 0 no · n+1 Write name Read names yes Set Agreement  WSB If all n+1 participate … First name written joins Group 0 Some name not chosen, it joins Group 1 Distributed Computing Through Combinatorial Topology

109. Choose name with anonymous n-Set agreement Choose name with anonymous n-Set agreement Proof 109 Group 1 My name written? My name written? Group 0 no · n+1 Write name Read names yes Set Agreement  WSB Protocol is anonymous … Because we use anonymous set agreement “black box” (Any set agreement protocol can be made anonymous) Distributed Computing Through Combinatorial Topology

110. Conclusions 110 Some tasks harder than others … n-set agreement solves weak-symmetry breaking But not vice-versa Distributed Computing Through Combinatorial Topology

111. Remarks 111 Combinatorial and algorithmic arguments complement one another Combinatorial: what we can’t do Algorithmic: what we can do Distributed Computing Through Combinatorial Topology

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