Transcript pptx

Wait-Free Computability for
General Tasks
Companion slides for
Distributed Computing
Through Combinatorial Topology
Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions
Review
Star
¾
Star(¾,K) is the complex of facets of K containing ¾
Complex
Distributed
3
Computing through
Review
Facet
Facet
not a
facet
A facet is a simplex of maximal dimension
Distributed
4
Computing through
Review
Open Star
Staro(¾,K) union of interiors of simplexes containing ¾
Point Set
Distributed
5
Computing through
Review
Link
Link(¾,K) is the complex of simplices of
Star(¾,K) not containing ¾
Distributed
6
Complex
Computing through
Review
A simplicial map Á is rigid if
dim Á(¾) = dim ¾.
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions
The Hourglass task
I
O
Single-Process Executions
P,0
P
¢
R
Q
R,0
I
Q,0
O
P and R only
(P and Q Symmetric)
P
P
¢
R
R
I
O
Q and R only
¢
I
O
Claim:
Hourglass satisfies conditions
of fundamental theorem …
But has no wait-free immediate
snapshot protocol!
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Claim:
Hourglass satisfies conditions
of fundamental theorem …
But has no wait-free immediate
snapshot protocol!
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homotopy: |I |  |O|
O
I
carried by ¢
Claim:
Hourglass satisfies conditions
of fundamental theorem …
But has no wait-free immediate
snapshot protocol!
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Claim:
The Hourglass task solves
2-set agreement …
Which has no wait-free
read-write protocol.
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Protocol:
Write input value to
announce array …
Run Hourglass task …
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P
P
P
Find non-null
announce[]
value
P,1
R or Q
R
Q
R
Q
Look in announce[] array …
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 ¢
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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 ¢
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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?
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Review
f: |I|  |O|...
Simplicial
approximation
Á: BaryN I  O
Protocol
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Repeated
snapshot
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Review
f: |I|  |O|...
Á: BaryN I  O
Protocol
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Simplicial
Not colorapproximation
preserving
Repeated
Another
snapshot
process’s
output?
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions
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?
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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 ¢
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Quasi-Consensus
Q,1
P,1
Q,1
P,1
P,0
Q,0
¢
P,0
Q,0
I
O
Quasi-Consensus
Q,1
P,1
Q,1
P,1
P,0
Q,0
¢
P,0
Q,0
I
O
Quasi-Consensus
Q,1
P,1
Q,1
P,1
P,0
Q,0
¢
P,0
Q,0
I
O
Quasi-Consensus
Q,1
P,1
Q,1
P,1
P,0
Q,0
¢
P,0
Q,0
I
O
Not a colorless task!
Quasi-Consensus
Q,1
P,1
Q,1
P,1
P,0
Q,0
¢
P,0
Q,0
I
O
Quasi-Consensus
Q,1
P,1
Q,1
P,1
¢
P,0
Q,0
P,0
No
simplicial map:
IO
carried by ¢ O
I
Q,0
Q,1
P,1
Q,1
P,1
P,0
Q,0
Á
P,0
Q,0
Div I
O
// code for P
T decide(T input) {
announce[P] = input;
if (input == 1)
return 1;
else if (announce[Q] != 1)
return 0
else
// code for Q
return 1
T decide(T input) {
}
announce[P] = input;
if (input == 0)
return 0;
else if (announce[P] != 0)
return 1
else
return 0
}
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions
Protocol ) Map
protocol
I
input
complex
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¥
¥(I)
protocol
complex
δ
decision
map
O
output
complex
carried by
¢
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Protocol ) Map
¥(I)
subdivision
of input
complex
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δ
O
carried by
¢
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Conditions
Task (I,O,¢)
¾
I
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¢(¾)
O
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chromatic simplex ¾
chromatic subdivision Div ¾
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¾
I
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¢(¾)
O
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Theorem says …
If there is a chromatic subdivision …
¾
I
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¢(¾)
O
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Theorem says …
If there is a chromatic subdivision …
and a simplicial map à carried by ¢ …
Ã
¾
I
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¢(¾)
O
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Theorem says …
If there is a chromatic subdivision …
and a simplicial map à carried by ¢ …
Ã
¾
I
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¢(¾)
O
… then there is a wait-free IS protocol!
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Let’s start with something easier …
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Let’s start with a special case …
If there is a simplicial map Ã: ChN ¾  ¢(¾) …
Ã
ChN¾
I
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¢(¾)
O
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Let’s start with something easier …
If there is a simplicial map Á: ChN ¾  ¢(¾) …
Á
ChN¾
I
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¢(¾)
O
… then there is a wait-free IS protocol!
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Protocol
Iterated immediate snapshot
N
I
Ch I
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For any chromatic subdivision Div ¾ …
If there is a color and carrier-preserving simplicial
map Á: ChN ¾  Div ¾ …
Á
ChN¾
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Div ¾
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Geometric construction
Inductively divide boundary
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Geometric construction
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Displace vetexes from barycenter
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Geometric construction
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Geometric construction
Mesh(Ch ¾) is max diameter of a simplex
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Subdivision shrinks mesh
mesh(Ch ¾) · c mesh(¾) for some 0 < c < 1
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Open cover
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Lesbesgue Number
¸
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Open stars form an open
cover for a complex
ostar(v)
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Intersection Lemma
Vertexes lie on a
common simplex iff their
open stars intersect
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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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Defines a vertex map ….
Simplicial by intersection lemma.
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We have just proved the
Simplicial Approximation
Theorem
Á
ChN¾
Div ¾
There is a carrier-preserving simplicial map
Á: ChN ¾  Div ¾ …
Not necessarily color-preserving!
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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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If the open-star cover is chromatic ….
Then the simplicial map ….
Is color preserving!
Must show that covering can be
made chromatic …
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Open Cover Fail
Two simplexes conflict …
If colors disjoint, but …
polyhedrons overlap.
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cannot map to same color65
Open Cover Fail
An open-star cover is chromatic iff there are no
conflicting simplexes.
We will show how to eliminate conflicting simplexes
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Carriers
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Perturbation
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Room for perturbation
Star contains ² ball in carrier around vertex
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Room for perturbation
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Can perturb to any point within
² ball in carrier and still have
subdivision
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Open Cover Fail
½ has q+1 colors
¿ has p+1 colors
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Simplexes lie in hyperplane
of dimension p+q
(because they overlap)
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Some vertex has carrier of
dimension p+q+1
(because there are p+q+2 colors)
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Can perturb vertex within (p+q+1)dimension ² ball …
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Can perturb vertex within (p+q+1)dimension ² ball …
Out of the hyperplane
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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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Div ¾
Ã
Given
¢(¾)
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Constructed
Á
ChN¾
Div ¾
Ã
Given
¢(¾)
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Iterated immediate
snapshot here …
Á
ChN¾
Div ¾
Ã
Yields protocol here!
¢(¾)
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Road Map
Inherently colored tasks
Solvability for colored tasks
Protocol ) map
Map ) protocol
A Sufficient Topological Condition
Link-Connected
O is link-connected if for each ¿ 2 O,
link(¿,O) is (n - 2 - dim ¿)-connected.
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not a fan
not link-connected.
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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
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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 ¢.
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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 ¢.
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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
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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
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Induction Base: n = 1
Á
hinge
I
collapses
O
Á’
div I
does not
collapse
O
Induction Step: Á does not collapse (n-1)-simplexes
Á
I
O
Div I
O
does not
collapse
Á’
Div I
O
exploit connected
link
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
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This work is licensed under a
Creative Commons AttributionNoncommercial 3.0 Unported
License.
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