Transcript Slides (PPTX)

Combinatorial Topology
and
Distributed Computing
Part Two
Elements of Combinatorial
Topology
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Overview
Basic concepts of
Combinatorial
Topology
How they model
distributed & concurrent
computation
Duality between
combinatorial & continuous
mechanisms
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A Vertex
Combinatorial: an element of a set.
Geometric: a point in highdimensional Euclidean Space
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Simplexes
Combinatorial: a set of vertexes.
Geometric: convex hull of points in
general position
1-simplex
0-simplex
dimension
2-simplex
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3-simplex
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Simplicial Complex
Combinatorial: a set of simplexes
simplexes “glued
closeGeometric:
under inclusion.
together” along faces …
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Simplicial Maps
Vertex-to-vertex map …
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Simplicial Map
Vertex-to-vertex map …
that sends simplexes to
simplexes
piece-wise linear map
on geometric simplexes
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Carrier Map
M
Maps simplex …
to subcomplex.
Preserves intersections: M (¾ Å ¿) = M (¾) Å M (¿)
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Vertex = Process State
Process ID (color)
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Value (input or output)
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Simplex = Global State
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Complex = Global States
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Input Complex for Binary
Consensus
0
All possible initial states
0
1
Processes: red, green, blue
0
1
Independently assigned 0 or 1
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Output Complex for Binary
Consensus
All possible final states
0
0
0
1
1
Output values all 0 or all 1
1
Two disconnected simplexes
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Carrier Map for Consensus
All 0 outputs
All 0 inputs
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Carrier Map for Consensus
All 1 inputs
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All 1 outputs
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Carrier Map for Consensus
All 0 outputs
Mixed 0-1 inputs
All 1 outputs
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Task Specification
(I, O, ¢)
Carrier map
Input complex
Output complex
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
broadcast view;
view += messages received;
}
return δ(view)
Finite program
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
Start with input value
broadcast view;
view += messages received;
}
return δ(view)
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
broadcast view;
Run
for
fixed
number
of
rounds
view += messages received;
}
return δ(view)
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
broadcast view;
view += Send
messages
currentreceived;
view to others
}
return δ(view)
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
broadcast view;
view += messages received;
}
Concatenate messages
return δ(view)
received to view
(full-information protocol)
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Protocol
view = my input value;
for (i = 0; i < r; i++) {
broadcast view;
view += messages received;
}
return δ(view)
finally, apply task-specific
decision map to view
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Protocol Complex
Vertex: process ID, view
Complete log of messages
sent & received
Simplex: compatible set of views
Each execution defines a simplex
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Example: Synchronous
Message-Passing
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Round 0
Round 1
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Failures: Fail-Stop
Partial
broadcast
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Single Input: Round Zero
No messages sent
0
View is input value
0
0
Same as input simplex
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Round Zero Protocol Complex
0
No messages sent
0
1
View is input value
0
1
Same as input complex
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Single Input: Round One
00
00
0 00
00
00
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0 00
0 00
00
00
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Single Input: Round One
00
00
0 00
00
00
0 00
0 00
no one fails
00
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Single Input: Round One
00
00
0 00
00
00
0 00
blue fails
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0 00
no one fails
00
00
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Single Input: Round One
red fails
green fails
00
00
0 00
00
00
0 00
blue fails
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0 00
no one fails
00
00
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Protocol Complex: Round One
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Protocol Complex: Round Two
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Protocol Complex Evolution
zero
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one
two
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protocol
complex
Summary
output
complex
input
complex
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d
Δ
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Decision Map
d
Simplicial map,
sending simplexes
to simplexes
Protocol complex
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Output complex
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Lower Bound Strategy
d
Find topological
“obstruction” to
this simplicial map
Protocol complex
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Output complex
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Consensus Example
Subcomplex of
all-0 inputs
Must map
here
0
1
Protocol
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d
0
1
Output
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Consensus Example
0
1
Protocol
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d
Subcomplex of
all-1 inputs
0
1
Must map
Output
here
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Consensus Example
Image under d must
start here ..
0
1
Protocol
Path from
“all-0” to “all-1”
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0
d
1
and end here
Output
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Consensus Example
path
0
d
1
0
?
1
Output
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Consensus Example
Image under d must
start here ..
But this “hole” is
d
an obstruction
Protocol
Path from
“all-0” to “all-1”
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and end here
Output
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Conjecture
A protocol cannot
solve consensus
if its complex is
path-connected
Model-independent!
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If Adversary keeps Protocol
Complex path-connected …
Forever …
Consensus is impossible
For r rounds …
A round-complexity lower bound
For time t …
A time-complexity lower bound
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Another Conjecture
A protocol cannot solve
k-set agreement if its
complex has “no holes”
in dimensions < k
Later!
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This work is licensed under a
Creative Commons AttributionNoncommercial 3.0 Unported
License.
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