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Manifold Protocols
Companion slides forDistributed Computing
Through Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
2
Kinds of Results
Impossibility
Separation
Task T cannot be solvedin model M
Task T can be solved by a protocol for T’, but not vice-versa
Distributed Computing Through Combinatorial Topology
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
Distributed Computing Through Combinatorial Topology
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
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Simplicial Complex
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Manifolds
Every (n-1)-simplex aface of two n-simplexes
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Manifolds
Every (n-1)-simplex aface of two n-simplexes
technically, a pseudo-manifold
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Manifold with BoundaryInternal (n-1)-simplex aface of two n-simplexes
Boundary (n-1)-simplex aface of one n-simplex
Boundary C
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Why Manifolds?
Nice combinatorial properties
Many useful theorems
Easy to prove certain claims
True, most complexes are not manifolds ….
Still a good place to start.
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
Distributed Computing Through Combinatorial Topology
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Immediate Snapshot Executions
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
0 0 0
1
Write
Single-writer, multi-reader variables12
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1 0 0
100Snapshot
Single-writer, multi-reader variables13
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Immediate Snapshot Executions
Pick a set of processes
write together
snapshot together
Repeat with another set
Example Executions
P Q R
write
snap
write
snap
write
snap
P?? PQ? PQR
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Each process writes, then takes snapshot
Each process has a view
time
Distributed Computing Through Combinatorial Topology
Example Executions
P Q R
write
snap
write write
snap snap
write
snap
P?? PQR PQR
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Moving last processone round earlier,
Changes this view from PQ? to PQR
P Q R
write
snap
write
snap
write
snap
P?? PQ? PQR
Distributed Computing Through Combinatorial Topology
P Q R
write write write
snap snap snap
write write
snap snap
PQR PQR PQR
P Q R
write write write
snap snap snap
write write
snap snap
PQR PQR PQR
P Q R
write
snap
write write
snap snap
write
snap
P?? PQR PQR
Example Executions
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Moving last processesone round earlier,
Changes this view from P?? to PQRDistributed Computing Through
Combinatorial Topology
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Protocol ComplexProcess (color)
& view
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Protocol ComplexP Q R
write
snap
write
snap
write
snap
P Q R
write write write
snap snap snap
P Q R
write
snap
write write
snap snap
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Standard Chromatic Subdivision
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Vertex (Pi, ¾i)
Combinatorial Definition (I)
Process name view ¾i µ ¾
Input simplex ¾
Protocol complex Ch(¾)
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Combinatorial Definition (II)
Each process's write appears in its own view
Pi 2 names(¾i).
Snapshots are ordered
¾i µ ¾j or vice-versa
Each snapshot ordered immediately after writeif Pi 2 names(¾j), then ¾i µ ¾j.
If Pj saw Pi write … then Pj‘s snapshot not later than Pj‘s.
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Manifold Theorem
If the input complex I is a manifold …
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Manifold Theorem
… so is the immediate snapshot protocol complex Ch(I).
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Proof of Manifold Theorem
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).
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Without Loss of Generality …
¿n¡ 1 = fhP0;¾0i ; : : :;hPn¡ 1;¾n¡ 1ig
Simplex of I seen by last process
Where ¾i µ ¾i+1 for 0 · i < n
Re-index processes in execution order(ignoring ties)
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Proof Strategy
Count the ways we can extend…
Where ¾n is an n-simplex of Ch(¾)
¿n-1 = {(P0, ¾0), …, (Pn-1, ¾n-1)}
¿n = {(P0, ¾0), …, (Pn-1, ¾n-1i, (Pn, ¾n)}
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Cases
¿n¡ 1 = fhP0;¾0i ; : : :;hPn¡ 1;¾n¡ 1ig
dim n-1 or dim n
Boundary or internal
3 cases
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¿ n-
1
¾n-
1
¿n¡ 1 = fhP0;¾0i ; : : :;hPn¡ 1;¾n¡ 1ig Boundary(n-1)-simplex
Case One
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Exactly one n-simplex ¾ of I contains ¾n-1
¾
¾n-
1
¿n-1
Case One
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Exactly one n-simplex ¿n of Ch(I) contains ¿n-1
¿n = ¿n-1 [ hPn, ¾i
¾
Case One
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¿n-1
¾n-1
¿n¡ 1 = fhP0;¾0i ; : : :;hPn¡ 1;¾n¡ 1ig Internal(n-1)-simplex
Case Two
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Exactly two n-simplexes ¾0 & ¾1 of I contain
¾n-1
¾0
¾1
Case Two
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Exactly 2 n-simplexes ¿0n & ¿1
n of Ch(I) contain ¿n-1
¿n0 = ¿n-1 [ hPn, ¾0i
¿n1 = ¿n-1 [ hPn, ¾1i
¾0
¾1
Case Three
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¿n-1
¾n-1
¿n¡ 1 = fhP0;¾0i ; : : :;hPn¡ 1;¾n¡ 1ig n-simplexCase Three
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Suppose
Pn 2 names(¾k)(Pk saw Pn’s write)
Pn names(¾k-1)(Pk-1 did not see Pn’s write)
Special case: k=0, ¾-1 = ;
¾n µ ¾k)
¾k-1 ½ ¾n)
Either ¾k-1 ½ ¾n ½ ¾k or ¾k-1 ½ ¾n = ¾k
Case Three
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Pk-1 did not see Pn
Pk did see Pn
Pn between Pk-1 & Pk
Pn with Pk
This completes the proof.
Case Three
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Manifold Protocols
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
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
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k-set Agreement: before
32 19
21
40
19 19
21
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k-set Agreement: after
k = 2Distributed Computing Through
Combinatorial Topology
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Theorem
No Manifold Protocol can solve n-set agreement
Including Immediate SnapshotIncluding read-write memory
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Sperner Coloring
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Sperner Coloring
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“Corners” have distinct colors
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Sperner Coloring
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Edge vertexes have corner colors
“Corners” have distinct colors
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Sperner Coloring
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Edge vertexes have corner colors
“Corners” have distinct colors
Every vertex has face boundary colors
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Sperner’s Lemma
Distributed Computing Through Combinatorial Topology
In any Sperner coloring, at least one n-simplex has all n+1 colors
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Sperner Coloring for Manifolds with Boundary (base)
One color hereOther color here
Only those colors in interior
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Sperner Coloring for Manifolds with Boundary (inductive)
M is n-manifold colored with ¦
M = [i Mi
Where Mi is non-empty (n-1)-manifold Sperner-colored with ¦ n {i }
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Sperner’s Lemma for Manifolds
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M
Odd number of n-simplexes have all
n+1 colors
Sperner Coloring on Manifold
M
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Proof of Sperner’s Lemma
Induction on nBase
One color hereOther color here
Odd number of changesDistributed Computing Through
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Induction Step
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Dual Graph
One vertex per n-simplex …
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Dual Graph
One vertex per n-simplex …
One “external” vertex
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Discard One Color
n+1 colors
“restricted”n colors
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Edges
If two n-simplexes share an (n-1)-face …with restricted n colors …
add an edge.
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Edges
Same for external vertex
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Some Vertexes have Degree Zero
Even degreeDistributed Computing Through
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Some Vertexes have Degree Two
Even degree
Only n restricted colors
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All n+1 Colors ) Degree 1
Odd degreeDistributed Computing Through
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Induction Hypothesis
Odd number of n-colored simplexes
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Induction Hypothesis
External vertex has odd degree
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OddDegrees
One external vertex
Even
n-simplexes with n+1 colors?
All the rest
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OddDegrees
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
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OddDegrees
One external vertex
Even
n-simplexes with n+1 colors?
All the rest
Even number of odd degree vertexes
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Parity
One external vertex
Even number of odd degree vertexes
n-simplexes with n+1 colors?Even
Odd
Odd
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Parity
One external vertex
Even number of odd degree vertexes
n-simplexes with n+1 colors?Even
Odd
Odd
QEDDistributed Computing Through
Combinatorial Topology
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No Manifold Task can solve n-Set Agreement
Assume protocol exists: Run manifold task protocol
Choose value based on vertex
Idea:Color vertex with “winning”
process name …
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Manifold Task for n-Set Agreement
manifold
Only P wins
Only Q and R win
Sperner coloring
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Manifold Task for n-Set Agreement
manifold
Sperner coloring
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Manifold Task for n-Set Agreement
manifold
Sperner coloring
n+1 colors
Contradiction: at most n can win
Execution with n+1 winners
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
Distributed Computing Through Combinatorial Topology
Distributed Computing Through Combinatorial Topology
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Weak Symmetry-Breaking
Group 0 Group 1
If all processes participate …
At least one process in each group
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Weak Symmetry-Breaking
Group 0 Group 1
If fewer participate …
we don’t care.
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Anonymous Protocols
Trivial solution: choose name parity
WSB protocol should be anonymous
Output depends on …Input …Interleaving …
But not name
Restriction on protocol, not task!
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Road MapManifolds
Sperner’s Lemma and k-Set Agreement
Immediate Snapshot Model
Weak Symmetry-Breaking
Separation results
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Next Step
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
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A Simplex
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Standard Chromatic Subdivision
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Glue Three Copies Together
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Glue Opposite Edges
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The Moebius Task
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Defines a Manifold Task
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boundary boundary
boundary
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Manifold Task
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Note Sperner coloring on boundary
boundary boundary
boundary
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Terminology
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Each face has a central simplex
1-dim
2-dim
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Subdivided Faces
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external
internalinternal
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Black-and-White Coloring (I)
Distributed Computing Through Combinatorial Topology
Central 2-simplex:
White near external face
Black elsewhere
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Black-and-White Coloring (II)
Distributed Computing Through Combinatorial Topology
Black
Central simplexof external face:
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Black-and-White Coloring (III)
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All others
White
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Moebius Solves WSB
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Moebius Solves WSB
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Moebius Solves WSB
Distributed Computing Through Combinatorial Topology
Every n-simplex has both black & white colors.
Boundary coloring is symmetric!
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General Construction
n+1 “copies” »0, … »n of Ch(¾)
»0 »n…
»1
(picture not exact!)
n = 2N
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General Construction
»0 »i-1…
n = 2N»i
»i+1
Face i is external
»k… …
»n
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General Construction
»0 »i-1…
n = 2N»i
»i+1
Face i is external
»k… …
»n
2N others
96Distributed Computing Through Combinatorial Topology
General Construction
»0 »i-1…
n = 2N»i
»i+1 »k… …
»n
Identify Faces j
Rank j+N out of 2N
Face i is external
97Distributed Computing Through Combinatorial Topology
General Construction
»0 »i-1…
n = 2N»i
»i+1 »k… …
»n
Symmetric!
Rank j+N out of 2N
Face i is external
Why it’s a(non-orientable)
manifold Why it works only for even dimensions
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Open Problem
Generalize to odd dimensions …
or find counterexample.
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Black-and-White Coloring (I)
Distributed Computing Through Combinatorial Topology
Central n-simplex:
White near external face
Black elsewhere
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Black-and-White Coloring (II)
Distributed Computing Through Combinatorial Topology
Black
Central m-simplexes
of external face for m > 0
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Black-and-White Coloring (III)
Distributed Computing Through Combinatorial Topology
All others
White
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No Monochromaticn-simplexes
Distributed Computing Through Combinatorial Topology
Central n-simplexBy construction
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No Monochromaticn-simplexes
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Intersects internal face Black at n-center
White at edge
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No Monochromaticn-simplexes
Distributed Computing Through Combinatorial Topology
Intersects external face White at n-center
Black at m-center, 0 < m < n
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Progress
Distributed Computing Through Combinatorial Topology
Weak Symmetry-Breaking
Moebius Taskno
yes
Set Agreement
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Next Step
Distributed Computing Through Combinatorial Topology
Weak Symmetry-Breaking
Moebius Taskno
yes
Anonymous Set Agreement
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Group 1
Choose name with anonymous
n-Set agreement
My namewritten? Group 0
no
· n+1
Write name
Read names
yes
Set Agreement WSB
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Choose name with anonymous
n-Set agreement
Proof
Group 1
My namewritten? Group 0
no
· n+1
Write name
Read names
yes
Set Agreement WSB
If all n+1 participate …First name written
joins Group 0Some name not chosen, it joins
Group 1
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Choose name with anonymous
n-Set agreement
Proof
Group 1
My namewritten? 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)
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Conclusions
Some tasks harder than others …
n-set agreement solves weak-symmetry breaking
But not vice-versa
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Remarks
Combinatorial and algorithmic arguments complement one
anotherCombinatorial: what we can’t do
Algorithmic: what we can do
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