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Slide1
Wait-Free Computability for General Tasks
TexPoint
fonts used in EMF.
Read the
TexPoint manual before you delete this box.: AAAA
Companion slides
forDistributed ComputingThrough Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
1
Slide2Road Map
Inherently colored tasks
Protocol
)
mapSolvability for colored tasks
Map
) protocolA Sufficient Topological Conditions
Slide3Star
Distributed Computing through Combinatorial Topology
3
Star(
¾
,
K
)
is the complex of facets of
K
containing
¾
Complex
¾
Review
Slide4Facet
Distributed Computing through Combinatorial Topology
4
A
facet
is a simplex of maximal dimension
Review
Facet
not a
facet
Slide5Open Star
Distributed Computing through Combinatorial Topology
5
Star
o
(
¾
,
K
)
union of interiors of simplexes containing
¾
Point Set
Review
Slide6Link
Distributed Computing through Combinatorial Topology
6
Link(
¾
,
K
)
is the complex of simplices of
Star(
¾
,
K
)
not containing
¾
Complex
Review
Slide711-Mar-15
7
Review
A simplicial map
Á is rigid
ifdim
Á(¾) = dim ¾.
Slide8Road Map
Inherently colored tasks
Protocol
)
map
Solvability for colored tasks
Map ) protocolA Sufficient Topological Conditions
Slide9I
O
The Hourglass task
Slide10P
Q
R
Q,0
R,0
P,0
¢
Single-Process Executions
I
O
Slide11P
R
R
P
¢
P and R only
(P and Q Symmetric)
I
O
Slide12¢
Q and R only
I
O
Slide1311-Mar-15
13
Claim:
Hourglass satisfies conditions of fundamental theorem …
But has no wait-free immediate snapshot protocol!
Slide14But has no wait-free immediate snapshot protocol!
11-Mar-15
14
Claim:
Hourglass satisfies conditions of fundamental theorem …
Slide15I
O
homotopy
:
|
I
|
|
O
|
carried by
¢
Slide1611-Mar-15
16
Claim:
Hourglass satisfies conditions of fundamental theorem …
But has no wait-free immediate snapshot protocol!
Slide1711-Mar-15
17
Claim:
The Hourglass task solves 2-set agreement …
Which has no wait-free read-write protocol.
Slide1811-Mar-15
18
Protocol:
Write input value to
announce array …
Run Hourglass task …
Slide19P,1
P
Look in
announce
[] array …
P
P
R
Q
R
Q
R or Q
Find non-null
announce
[] value
Slide20Theorem
What Went Wrong?
11-Mar-15
20
A colorless (I
,O,
¢) has a wait-free immediate snapshot protocol iff there is a continuous map …f: |
I
|
|
O
|...
carried by
¢
Slide21Theorem
One Direction is OK
11-Mar-15
21
If (I
,O,
¢) has a wait-free read-write protocol …f: |I|
|
O
|...
carried by
¢
then there is a continuous map …
Slide22Theorem?
The Other Direction Fails
11-Mar-15
22
then does (I
,O,
¢) have a wait-free IS protocol?f: |I|
|
O
|...
carried by
¢
…
If there is a continuous map …
Slide23Review
11-Mar-1523
Á
:
BaryN I
O
f: |I| |O
|...
Protocol
Simplicial approximation
Repeated snapshot
Slide24Review
11-Mar-1524
Á
:
BaryN I
O
f: |I| |O
|...
Protocol
Simplicial approximation
Repeated snapshot
Not color-preserving
Another process’s output?
Slide25Road Map
Inherently colored tasks
Protocol
)
map
Solvability for colored tasks
Map ) protocolA Sufficient Topological Conditions
Slide26Theorem
11-Mar-15
26
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?
Slide27Theorem
Fundamental Theorem for Colored Tasks
11-Mar-15
27
(I
,O,¢
) has a wait-free read-write protocol iff …carried by ¢
I
has a
chromatic
subdivision
Div
I
…
&
color-preserving
simplicial map
Á
: Div
I
O
…
Slide28P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
I
O
¢
Quasi-Consensus
Slide29P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
I
O
¢
Quasi-Consensus
Slide30P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
¢
Quasi-Consensus
I
O
Slide31P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
¢
Quasi-Consensus
I
O
Not a colorless task!
Slide32P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
¢
Quasi-Consensus
I
O
Slide33P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
¢
Quasi-Consensus
I
O
No simplicial map:
I
O
carried by
¢
Slide34P,0
Q,0
P,1
Q,1
P,0
Q,0
P,1
Q,1
Div
I
Á
O
Slide3511-Mar-15
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
}Code is asymmetric!
Slide36Road Map
Inherently colored tasks
Protocol
)
mapSolvability for colored tasks
Map
) protocolA Sufficient Topological Conditions
Slide37Protocol ) Map
11-Mar-15
37
protocol
I¥(
I)
O¥
δ
input
complex
protocol
complex
output
complex
decision
map
carried by
¢
Slide38Protocol ) Map
11-Mar-15
38
¥(
I)O
δ
subdivisionof inputcomplex
carried by
¢
Slide39Road Map
Inherently colored tasks
Protocol
)
map
Solvability for colored tasks
Map ) protocolA Sufficient Topological Conditions
Slide4011-Mar-15
40
Task
(
I,O,¢)
I
O
¢
(
¾
)
¾
Slide4111-Mar-15
41
chromatic simplex
¾
chromatic subdivision
Div
¾
Slide4211-Mar-15
42
I
O
¢
(
¾
)
¾
Slide4311-Mar-15
43
I
O
¢
(
¾
)
¾
Theorem says …
If there is a chromatic subdivision …
Slide4411-Mar-15
44
I
O
¢
(
¾
)
¾
Ã
Theorem says …
If there is a chromatic subdivision …
and a simplicial map
Ã
carried by
¢
…
Slide4511-Mar-15
45
I
O
¢
(
¾
)
¾
Ã
… then there is a wait-free IS protocol!
Theorem says …
If there is a chromatic subdivision …
and a simplicial map
Ã
carried by
¢
…
Slide4611-Mar-15
46
Let’s start with something easier …
Slide4711-Mar-15
47
11-Mar-15
47
I
O
¢
(
¾
)
Ch
N
¾
Ã
Let’s start with a special case …
If there is a simplicial map
Ã
:
Ch
N
¾
¢
(
¾
)
…
Slide4811-Mar-15
48
11-Mar-15
48
I
O
¢
(
¾
)
Á
Let’s start with something easier …
If there is a simplicial map
Á
:
Ch
N
¾
¢
(
¾
)
…
… then there is a wait-free IS protocol!
Ch
N
¾
Slide4949
11-Mar-15
Protocol
Iterated immediate snapshot
Slide5011-Mar-15
50
11-Mar-15
50
Á
For any chromatic subdivision
Div
¾
…
If there is a color and carrier-preserving simplicial map
Á
:
Ch
N
¾
Div
¾
…
Ch
N
¾
Div
¾
Slide51Geometric construction
11-Mar-15
51
Inductively divide boundary
Slide52Geometric construction
11-Mar-15
52
Displace
vetexes
from
barycenter
Slide53Geometric construction
11-Mar-15
53
Slide54Geometric construction
11-Mar-15
54
Mesh(Ch
¾
)
is max diameter of a simplex
Slide55Subdivision shrinks mesh
11-Mar-15
55
mesh(Ch
¾
)
·
c mesh(
¾
)
for some
0 < c < 1
Slide56Open cover
11-Mar-15
56
Slide57Lesbesgue
Number
11-Mar-15
57
¸
Slide58Open stars form an open cover for a complex
11-Mar-15
58
ostar
(
v
)
Slide59Intersection Lemma
11-Mar-15
59
Vertexes lie on a common simplex
iff
their open stars intersect
Slide6011-Mar-15
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
Slide6111-Mar-15
61
Defines a vertex map ….
Á
Simplicial by intersection lemma.
Slide6211-Mar-15
62
11-Mar-15
62
Á
We have just proved the Simplicial Approximation Theorem
There is a carrier-preserving simplicial map
Á
:
Ch
N
¾
Div
¾
…
Ch
N
¾
Div
¾
Not necessarily color-preserving!
Slide6311-Mar-15
63
An open-star cover is
chromatic
if every simplex
¿
of
Ch
N
¾
is covered by open stars of
of
the same color.
Slide6411-Mar-15
64
Then the simplicial map ….
Á
Is color preserving!
If the open-star cover is chromatic ….
Must show that covering can be made chromatic …
Slide65Open Cover Fail
11-Mar-15
65
Two simplexes
conflict
…
If colors disjoint, but …
polyhedrons overlap.
cannot map to same color
Slide66Open Cover Fail
11-Mar-15
66
An open-star cover is chromatic iff there are no conflicting simplexes.
We will show how to eliminate conflicting simplexes
Slide67Carriers
11-Mar-1567
Slide68Perturbation
11-Mar-15
68
Slide69Room for perturbation
11-Mar-15
69
Star contains
²
ball in carrier around vertex
Slide70Room for perturbation
11-Mar-15
70
Can perturb to any point within
²
ball in carrier and still have subdivision
Slide71Open Cover Fail
11-Mar-15
71
¿
has
p
+1
colors
½
has
q
+1
colors
Slide7211-Mar-15
72
Simplexes lie in
hyperplane
of dimension
p+q
(because they overlap)
Slide7311-Mar-15
73
Some vertex has carrier of dimension
p+q
+1(because there are p+q+2
colors)
Slide7411-Mar-15
74
Can perturb vertex within (
p+q
+1)-
dimension
²
ball …
Slide7511-Mar-15
75
Can perturb vertex within (
p+q
+1)-
dimension
²
ball …
Out of the
hyperplane
Slide7611-Mar-15
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
Slide7711-Mar-15
77
11-Mar-15
77
Div
¾
¢
(
¾
)
Ã
Given
Slide7811-Mar-15
78
11-Mar-15
78
Á
Ch
N
¾
Div
¾
¢
(
¾
)
Ã
Constructed
Given
Slide7911-Mar-15
79
11-Mar-15
79
Á
Ch
N
¾
Div
¾
¢
(
¾
)
Ã
Iterated immediate snapshot here …
Yields protocol here!
Slide80Road Map
Inherently colored tasks
Protocol
)
map
Solvability for colored tasks
Map ) protocolA Sufficient Topological Condition
Slide81Link-Connected
11-Mar-1581
O
is
link-connected
if for
each
¿
2
O
,
link
(
¿
,
O
)
is
(
n
- 2 - dim
¿
)
-
connected.
Slide8211-Mar-15
82
not link-connected
.
not a fan
Slide83Theorem
11-Mar-1583
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
Slide84Proof Strategy
11-Mar-15
84
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
¢
.
Slide85Proof Strategy
11-Mar-15
85
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
¢
.
so protocol exists
by colored theorem
Slide86Lemma
11-Mar-1586
suppose we have a rigid simplicial map
that is color-preserving on
Div
¾
then Á is color-preserving on Div
¾
Á
:
Div
¾
O
rigid & color-preserving on boundary
means color-preserving everywhere
Slide87Lemma
11-Mar-1587
If
O
is link-connected …where
Div skel
n-1 I = skeln-1 I
to a rigid simplicial map
can extend rigid simplicial map
Á
n-1
:
skel
n-1
I
O
Á
n
:
Div
I
O
Slide88O
O
Á
Á’
I
div
I
collapses
does not
collapse
hinge
Induction Base:
n
= 1
Slide89Á
O
I
Induction Step:
Á
does not collapse
(
n
-1)
-simplexes
Slide90O
Div
I
Slide91Div
I
O
Á’
exploit connected
link
does not
collapse
Slide92Summary
11-Mar-1592
to construct a color-preserving simplicial map
Á
: Div
I
Ocarried by ¢.
Inductively use …
link-connectivity of
O
connectivity of
¢
(
¾
)
protocol follows from main theorem
Slide9393
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