1. Wait-Free Computability for General Tasks
Companion slides for
Distributed Computing
Through Combinatorial Topology
Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
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2. Road Map Inherently colored tasks Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions

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

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

5. Review
Open Star
Staro(¾,K) union of interiors of simplexes containing ¾
Point Set
Distributed Computing through Combinatorial Topology

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

7. A simplicial map Á is rigid if
Review
A simplicial map Á is rigid if
dim Á(¾) = dim ¾.
18-Nov-18

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 ExecutionsR 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. Hourglass satisfies conditions of fundamental theorem …
Claim:
Hourglass satisfies conditions of fundamental theorem …
But has no wait-free immediate snapshot protocol!
18-Nov-18

14. Hourglass satisfies conditions of fundamental theorem …
Claim:
Hourglass satisfies conditions of fundamental theorem …
But has no wait-free immediate snapshot protocol!
18-Nov-18

15. homotopy: |I |  |O| O I
carried by ¢

16. Hourglass satisfies conditions of fundamental theorem …
Claim:
Hourglass satisfies conditions of fundamental theorem …
But has no wait-free immediate snapshot protocol!
18-Nov-18

17. The Hourglass task solves 2-set agreement …
Claim:
The Hourglass task solves 2-set agreement …
Which has no wait-free read-write protocol.
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18. Write input value to announce array …
Protocol:
Write input value to announce array …
Run Hourglass task …
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19. Look in announce[] array …P P P
Find non-null announce[] value
P,1
R or Q R Q R Q
Look in announce[] array …

20. What Went Wrong? f: |I|  |O|... carried by ¢
Theorem
A colorless (I,O,¢) has a wait-free immediate snapshot protocol iff there is a continuous map …
f: |I|  |O|...
We will prove a simple theorem that completely characterizes when it is possible to solve a colorless task in wait-free read-write memory.
The theorem states that a colorless task $(\cI^*,\cO^*,\Delta^*)$ has a wait-free read-write protocol if and only if there is a continuous map between their polyhedrons,
carried by ¢
18-Nov-18

21. One Direction is OK f: |I|  |O|... carried by ¢
Theorem
If (I,O,¢) has a wait-free read-write protocol …
then there is a continuous map …
We will prove a simple theorem that completely characterizes when it is possible to solve a colorless task in wait-free read-write memory.
The theorem states that a colorless task $(\cI^*,\cO^*,\Delta^*)$ has a wait-free read-write protocol if and only if there is a continuous map between their polyhedrons,
f: |I|  |O|...
carried by ¢
18-Nov-18

22. The Other Direction Fails
Theorem?
If there is a continuous map …
f: |I|  |O|...
carried by ¢ …
We will prove a simple theorem that completely characterizes when it is possible to solve a colorless task in wait-free read-write memory.
The theorem states that a colorless task $(\cI^*,\cO^*,\Delta^*)$ has a wait-free read-write protocol if and only if there is a continuous map between their polyhedrons,
then does (I,O,¢) have a wait-free IS protocol?
18-Nov-18

23. Simplicial approximation
Review
f: |I|  |O|...
Simplicial approximation
Á: BaryN I  O
Repeated snapshot
Protocol
18-Nov-18

24. Review f: |I|  |O|... Á: BaryN I  O Simplicial approximation
Not color-preserving
Á: BaryN I  O
Repeated snapshot
Another process’s output?
Protocol
18-Nov-18

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

26. How can we adapt this theorem to colored tasks?
A colorless (I,O,¢) has a wait-free immediate snapshot protocol iff there is a continuous map …
f: |I|  |O|...
carried by ¢
We will prove a simple theorem that completely characterizes when it is possible to solve a colorless task in wait-free read-write memory.
The theorem states that a colorless task $(\cI^*,\cO^*,\Delta^*)$ has a wait-free read-write protocol if and only if there is a continuous map between their polyhedrons,
How can we adapt this theorem to colored tasks?
18-Nov-18

27. Fundamental Theorem for Colored Tasks
(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…
We will prove a simple theorem that completely characterizes when it is possible to solve a colorless task in wait-free read-write memory.
The theorem states that a colorless task $(\cI^*,\cO^*,\Delta^*)$ has a wait-free read-write protocol if and only if there is a continuous map between their polyhedrons,
carried by ¢
18-Nov-18

28. ¢ Quasi-Consensus Q,1 P,1 Q,1 P,1 P,0 Q,0 P,0 Q,0 I O
Each of $P_0$ and $P_1$ is given a binary input. If both have input
$v$, then both must decide $v$. If they have mixed inputs, then
either they agree, or $P_1$ may decide $0$ and $P_0$ may decide 1 (but
not vice-versa).
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
Each of $P_0$ and $P_1$ is given a binary input. If both have input
$v$, then both must decide $v$. If they have mixed inputs, then
either they agree, or $P_1$ may decide $0$ and $P_0$ may decide 1 (but
not vice-versa).
P,0
Q,0
P,0
Q,0 I O

30. ¢ Quasi-Consensus P,0 Q,0 P,1 Q,1 Q,1 P,1 P,0 Q,0 I O
Each of $P_0$ and $P_1$ is given a binary input. If both have input $v$, then both must decide $v$. If they have mixed inputs, then either they agree, or $P_1$ may decide $0$ and $P_0$ may decide 1 (but
not vice-versa).
P,0
Q,0 I O

31. ¢ Not a colorless task! Quasi-Consensus P,0 Q,0 P,1 Q,1 Q,1 P,1 P,0

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

33. ¢ No simplicial map: I  O carried by ¢ Quasi-Consensus P,0 Q,0 P,1
It is easily seen that there is no simplicial map carried by $\Delta$ from the input complex to the output complex.
The vertexes of input simplex $\set{ang{P,0},\ang{Q,1}}$
$\ang{P,0}$ and $\ang{Q,1}$,
but there is no single output simplex containing both vertexes.
P,0
Q,0
No simplicial map:
I  O
carried by ¢ I O

34. Q,1
P,1
Q,1
P,1 Á
Nevertheless, there is a map satisfying the conditions of the theorem from a subdivision of the input complex. If input simplex {<P,0>,<Q,1>} is subdivided as shown then it can be ``folded'' around the output complex, allowing input vertexes <P,0> and <Q,1> to be mapped to their counterparts in the output complex.
P,0
Q,0
P,0
Q,0
Div I O

35. Code is asymmetric! // 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 is asymmetric!
// 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 }
Here is a simple protocol for quasi-consensus. If P has input 0 and Q has input 1, then this protocol admits three distinct executions: one in which both decide 0, one in which both decide 1, and one in which Q decides 0 and P decides 1.These three executions correspond to the three simplexes in the subdivision of {<P,0>,<Q,1}>.
18-Nov-18

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

37. Protocol ) Map I ¥(I) O protocol decision map ¥ δ input complex
output
complex
carried by
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38. Protocol ) Map ¥(I) O δ subdivision carried by of input ¢ complex
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39. Road Map Inherently colored tasks Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions

40. ¾ ¢(¾) I O Task (I,O,¢) 18-Nov-18
Recall that a task is given by an input complex \cI, an output complex \cO, and a relation \Delta that maps each input simplex \sigma to a subcomplex of \cO.
¢(¾) I O
18-Nov-18

41. chromatic subdivision Div ¾
chromatic simplex ¾
Recall that a that a simplex $\sigma = \set{\vs_0, \ldots, \vs_n}$ is
\emph{chromatic} if each vertex is labeled with a distinct color,
and a \emph{chromatic subdivision} $\Div \sigma$ is a subdivision of $\sigma$
where (1) each simplex of the subdivision is chromatic,
and (2) for each $\tau \subset \sigma$,
each vertex in $\Div \tau$ is labeled with a color from $\tau$.
chromatic subdivision Div ¾
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42. The theorem we want to prove says that if there exists a chromatic subdivision Div \cI of the input complex, then there is an algorithm.
¢(¾) I O
18-Nov-18

43. If there is a chromatic subdivision …
Theorem says …
If there is a chromatic subdivision …
The theorem we want to prove says that if there exists a chromatic subdivision Div \cI of the input complex, then there is an algorithm.
¢(¾) I O
18-Nov-18

44. Ã ¾ ¢(¾) I O Theorem says … If there is a chromatic subdivision …
and a simplicial map à carried by ¢ … Ã
The theorem we want to prove says that if there exists a chromatic subdivision Div \cI of the input complex, then there is an algorithm.
¢(¾) I O
18-Nov-18

45. Ã ¾ ¢(¾) I O Theorem says … If there is a chromatic subdivision …
and a simplicial map à carried by ¢ … Ã
The theorem we want to prove says that if there exists a chromatic subdivision Div \cI of the input complex, then there is an algorithm.
¢(¾) I O
… then there is a wait-free IS protocol!
18-Nov-18

46. Let’s start with something easier …
18-Nov-18

47. Ã ChN¾ ¢(¾) I O Let’s start with a special case …
If there is a simplicial map Ã: ChN ¾  ¢(¾) … Ã
ChN¾
¢(¾) I O
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18-Nov-18 47

48. Á ¢(¾) ChN¾ I O Let’s start with something easier …
If there is a simplicial map Á: ChN ¾  ¢(¾) … Á
¢(¾)
ChN¾ I O
… then there is a wait-free IS protocol!
18-Nov-18
18-Nov-18 48

49. Iterated immediate snapshot
Protocol
Iterated immediate snapshot
Informally, the processes start out on vertexes of a single simplex $\sigma$ of $\cI$, and they eventually halt on vertexes of a single simplex of $\Delta(\sigma) \subseteq \cI$.
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50. For any chromatic subdivision Div ¾ …
If there is a color and carrier-preserving simplicial map Á: ChN ¾  Div ¾ … Á
Div ¾
ChN¾
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51. Geometric construction
Inductively divide boundary
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52. Geometric construction
Displace vetexes from barycenter
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53. Geometric construction
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54. Geometric construction
Mesh(Ch ¾) is max diameter of a simplex
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55. Subdivision shrinks mesh
mesh(Ch ¾) · c mesh(¾) for some 0 < c < 1
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56. Open cover
An \emph{open cover} $\bbU$ for a complex $\cK$ is a finite collection of open sets
$U_0, \ldots, U_k$ such that $\cK \subseteq \cup_{i=0}^k U_i$.
18-Nov-18

57. Lesbesgue Number ¸ 18-Nov-18
If $U_0, \ldots, U_k$ an open cover for $\cK$, there exists a $\lambda > 0$ (called a \emph{Lebesgue number}) such that any set of diameter less than $\lambda$ lies in a single $U_i$.
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58. Open stars form an open cover for a complex
ostar(v)
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59. Vertexes lie on a common simplex iff their open stars intersect
Intersection Lemma
Vertexes lie on a common simplex iff their open stars intersect
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60. … each star of ChN ¾ lies in a open star of Div ¾
Pick N large enough that each (closed) star of ChN ¾ has diameter less than ¸ …
… each star of ChN ¾ lies in a open star of Div ¾
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61. Simplicial by intersection lemma.
Defines a vertex map ….
Simplicial by intersection lemma. Á
18-Nov-18

62. Á Div ¾ ChN¾ Á: ChN ¾  Div ¾ …
We have just proved the Simplicial Approximation Theorem Á
Div ¾
ChN¾
There is a carrier-preserving simplicial map
Á: ChN ¾  Div ¾ …
Not necessarily color-preserving!
18-Nov-18
18-Nov-18 62

63. An open-star cover is chromatic if every simplex ¿ of
ChN ¾ is covered by open stars of of the same color.
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64. If the open-star cover is chromatic ….
Then the simplicial map ….
Is color preserving! Á
Must show that covering can be made chromatic …
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65. Open Cover Fail Two simplexes conflict … If colors disjoint, but …
polyhedrons overlap.
cannot map to same color
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66. We will show how to eliminate conflicting simplexes
Open Cover Fail
An open-star cover is chromatic iff there are no conflicting simplexes.
We will show how to eliminate conflicting simplexes
18-Nov-18

67. Carriers
Let $\Div \sigma$ be a subdivision of $\sigma$. For any simplex $\tau \in \Div \sigma$, the \emph{carrier} of $\tau$ in $\Div \sigma$ is the smallest face $\kappa$ of $\sigma$ such that $\tau$ is in $\Div \kappa$.
18-Nov-18

68. Perturbation
18-Nov-18

69. Star contains ² ball in carrier around vertex
Room for perturbation
Star contains ² ball in carrier around vertex
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70. Room for perturbation
Can perturb to any point within ² ball in carrier and still have subdivision
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71. Open Cover Fail
½ has q+1 colors
¿ has p+1 colors
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72. Simplexes lie in hyperplane of dimension p+q (because they overlap)
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73. Some vertex has carrier of dimension p+q+1
(because there are p+q+2 colors)
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74. Can perturb vertex within (p+q+1)-dimension ² ball …
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75. Can perturb vertex within (p+q+1)-dimension ² ball …
Out of the hyperplane
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76. Repeat until star diameter < Lebesgue number:
Construct Ch ChN-1* ¾
Perturb to ChN* ¾
So open-star cover is chromatic
Construct color-preserving simplicial map
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77. Div ¾
¢(¾) Ã
Given
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18-Nov-18 77

78. ChN¾
Constructed Á
Div ¾
¢(¾) Ã
Given
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18-Nov-18 78

79. Iterated immediate snapshot here …
ChN¾ Á
Div ¾
¢(¾) Ã
Yields protocol here!
18-Nov-18
18-Nov-18 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,
A geometric complex is subdivided by partitioning each of its simplexes into smaller simplexes without changing the complex's polyhedron.
O is link-connected if for each ¿ 2 O,
link(¿,O) is (n dim ¿)-connected.
18-Nov-18

82. not a fan
not link-connected.
18-Nov-18

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
18-Nov-18

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 ¢.
18-Nov-18

85. Proof Strategy If, for all ¾ 2 I, ¢(¾) is ((dim ¾)-1)-connected, and
so protocol exists
by colored theorem
O is link-connected,
there exists subdivision Div
& color-preserving simplicial map
¹: Div I ! O carried by ¢.
18-Nov-18

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
18-Nov-18

87. Lemma If O is link-connected … can extend rigid simplicial map
Án-1: skeln-1 I  O
to a rigid simplicial map
Án: Div I  O
where Div skeln-1 I = skeln-1 I
18-Nov-18

88. Á O I O Á’ div I Induction Base: n = 1 collapses hinge does not
collapse0.eps Á’
div I
does not
collapse

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

90. Div I
collapse2.eps O

91. Div I
does not
collapse Á’ O
exploit connected
link

92. Summary connectivity of ¢(¾) Inductively use … link-connectivity of O
to construct a color-preserving simplicial map
Á: Div I  Ocarried by ¢.
protocol follows from main theorem
18-Nov-18

93.          
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