Transcript cs.brown.edu

Two-Process Systems
Companion slides for
Distributed Computing
Through Combinatorial Topology
Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
Distributed Computing through
Combinatorial Topology
Two-Process Systems
Two-process systems can
be captured by elementary
graph theory
gentle introduction to more
general structures needed
later for larger systems
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2
Combinatorial Topology
Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Combinatorial Topology
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Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Combinatorial Topology
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A Vertex
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Combinatorial Topology
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A Vertex
Combinatorial: an element of a set.
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A Vertex
Combinatorial: an element of a set.
Geometric: a point in Euclidean Space
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Combinatorial Topology
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An Edge
8
An Edge
Combinatorial: a set of two vertexes.
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An Edge
Combinatorial: a set of two vertexes.
Geometric: line segment joining two points
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A Graph
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A Graph
Combinatorial: a set of sets of vertices.
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A Graph
Combinatorial: a set of sets of vertices.
Geometric: points joined by line segments
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Graphs
finite set V with a collection
G of subsets of V,
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Graphs
vertices
finite set V with a collection
G of subsets of V,
simplices
(singular: simplex)
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Graphs
finite set V with a collection
G of subsets of V,
(1) If X 2 G, then |X| · 2
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Graphs
finite set V with a collection
G of subsets of V,
(1) If X 2 G, then |X| · 2
vertex: |X| = 1
edge: |X|= 2
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Graphs
finite set V with a collection
G of subsets of V,
(1) If X 2 G, then |X| · 2
(2) for all v 2 V, {v} 2 G
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Graphs
finite set V with a collection
G of subsets of V,
(1) If X 2 G, then |X| · 2
(2) for all v 2 V, {v} 2 G
(3) for all X 2 G, and Y ½ X, Y 2 G
19
Dimension
dimension 0
dimension 1
dim(X) = |X|-1.
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Pure Graphs
pure of dim 0
pure of dim 1
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Graph Coloring
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Graph Coloring
Â: G ! C
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Graph Coloring
Â: G ! C
for each edge (s0, s1) 2 G, Â(s0)  Â(s1).
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Graph Coloring
Â: G ! C
for each edge (s0, s1) 2 G, Â(s0)  Â(s1).
chromatic graphs
usually process names
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Graph Labeling
0
1
0
1
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Graph Labeling
0
1
f: G ! L
0
1
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Graph Labeling
0
1
0
1
f: G ! L
usually values from some domain
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Labeled Chromatic Graph
1
0
1
0
name(s) = Â(s)
view(s) = f(s)
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Simplicial Maps
Vertex-to-vertex map …
that also sends edges to edges.
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Rigid Simplicial Maps
A simplicial map can send
an edge to a vertex …
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Rigid Simplicial Maps
A simplicial map can send
an edge to a vertex …
A simplicial map that sends
distinct vertices to distinct
vertices is rigid.
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A Path Between two Vertices
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A Path Between two Vertices
A graph is connected if
there is a path between
every pair of vertices
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Not Connected
A graph is connected if
there is a path between
every pair of vertices
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Theorem
Theorem
The image of a connected
graph under a simplicial map
is connected.
Á
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Carrier Maps
For graphs G, H, a carrier map
©: G ! 2H
©
Carries each simplex of G to a subgraph of H …
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Carrier Maps
For graphs G, H, a carrier map
©: G ! 2H
©
Carries each simplex of G to a subgraph of H …
satisfying monotonicity:
for allDistributed
¾,¿2G,
if ¾µ¿, then ©(¾)µ©(¿).38
Computing through
Combinatorial Topology
Strict Carrier Maps
Monotonicity
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
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Strict Carrier Maps
Monotonicity
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
Equivalent to …
©(¾Å¿) µ ©(¾) Å ©(¿)
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Strict Carrier Maps
Monotonicity
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
Equivalent to …
©(¾Å¿) µ ©(¾) Å ©(¿)
Definition
© is strict if ©(¾Å¿) = ©(¾) Å ©(¿)
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Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Two Processes
Hello! I’m
Hello! I’m
Alice
Bob
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Informal Task Definition
Processes start with input values …
They communicate …
They halt with output values …
legal for those inputs.
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Formal Task Definition
Input graph I
all possible assignments of input values
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Formal Task Definition
Input graph I
all possible assignments of input values
Output graph O
all possible assignments of output values
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Formal Task Definition
Input graph I
all possible assignments of input values
Output graph O
all possible assignments of output values
Carrier map ¢: I ! 2O
all possible assignments of output values
for each input
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Task Input Graph: Consensus
0
1
I
0
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Task Input Graph
0
1
I
0
1
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Task Input Graph
1
1
Pure
Colored by process names
Labeled by input values
0
0
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Task Output Graph
1
1
O
0
0
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Task Carrier Map
1
1
¢: I ! 2O
I
0
1
0
O
0
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0
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Task Carrier Map
1
1
¢: I ! 2O
I
0
1
0
1
O
0
0
If Bob runs alone with input 1 …
then he decides output 1.
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Task Carrier Map
1
1
¢: I ! 2O
I
0
1
0
1
O
0
0
If Bob and Alice both have input 1 …
then they both decide output 1.
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Task Carrier Map
1
1
¢: I ! 2O
I
0
1
0
1
O
0
0
If Bob has 1 and Alice 0 …
then they must agree, on either one.
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Example: Coordinated Attack
Alice
Bob
Enemy
Alice and Bob win
If they both attack
together
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Input Graph
Attack at dawn!
I
Attack at noon!
Indifferent
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Output Graph
failed!
dawn!
noon!
O
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Carrier Map
I
¢
O
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Carrier Map
dawn!
I
¢
dawn!
O
failed!
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Carrier Map
noon!
I
¢
O
failed!
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Combinatorial Topology
noon!
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Carrier Map
dawn!
I
¢
dawn!
O
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Example: Coordinated Attack
Alice
Bob
Enemy
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Example: Coordinated Attack
Alice
Bob
Enemy
Alice and Bob realize that
they do not need to agree
on an exact time …
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Example: Coordinated Attack
Alice
Bob
Enemy
Alice and Bob realize that
they do not need to agree
on an exact time …
they will win if attack times
are sufficiently close.
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Coordinated Attack Graphs
I
0
1
¢
0
1/5
2/5
3/5
4/5
1
O
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Coordinated Attack Graphs
I
0
1
¢
0
1/5
2/5
3/5
4/5
1
O
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Coordinated Attack Graphs
I
0
1
¢
0
1/5
2/5
3/5
4/5
1
O
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Coordinated Attack Graphs
I
0
1
¢
0
1/5
2/5
3/5
4/5
1
O
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Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Protocols
Models of Computation
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int := 0 to L do
mem[A] := view;
view := view + mem[A] + mem[B];
return δ(view)
Finite program
Bob’s protocol is symmetric
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Alice’s Protocol
shared mem array 0..1 of Value
view:
:= mymemory
input value;
sharedValue
two-element
for i: int := 0 to L do
mem[A] := view;
view := view + mem[B];
return δ(view)
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int :=
0 with
to input
L dovalue
Start
mem[A] := view;
view := view + mem[B];
return δ(view)
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int := 0 to L do
mem[A] := view;
Run for L layers
view := view + mem[B];
return δ(view)
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int := 0 to L do
mem[A] := view;
view := view + mem[B];
return Alice
δ(view)
writes her value, read Bob’s
value, and concatenate it to my view
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int := 0 to L do
mem[A] := view;
view := view + mem[B];
return Alice
δ(view)
writes her value, read Bob’s
value, and concatenate it to my view
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Alice’s Protocol
shared mem array 0..1 of Value
view: Value := my input value;
for i: int := 0 to L do
mem[A] := view;
view := view + mem[B];
return δ(view)
finally, apply task-specific
decision map to view
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Formal Protocol Definition
Input graph I
all possible assignments of input values
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Formal Protocol Definition
Input graph I
all possible assignments of input values
Protocol graph P
all possible process views after execution
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Formal Protocol Definition
Input graph I
all possible assignments of input values
Protocol graph P
all possible process views after execution
Carrier map ¥: I ! 2P
all possible assignments of views
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One-Round Protocol Graph
I
0
1
¥
0?
01
01
?1
P
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One-Round Protocol Graph
P
0?
01
01
?1
Colored by process names
Labeled with final views
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One-Round Protocol Graph
P
0?
01
01
?1
Alice finishes before Bob
starts, doesn’t see his value
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One-Round Protocol Graph
P
0?
01
01
?1
Alice and Bob run together,
she sees his value.
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One-Round Protocol Graph
P
0?
01
01
?1
Alice finishes, then Bob starts
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One-Round Protocol Graph
P
0?
01
01
?1
Alice and Bob run together
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One-Round Protocol Graph
P
0?
01
01
?1
Bob can’t tell whether Alice saw him
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Execution Carrier Map
I
0
1
¥
0?
01
01
?1
P
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Execution Carrier Map
I
0
¥: I ! 2P
0?
1
¥
01
01
P
?1
strict carrier map
¥(¾) Å ¥(¿) = ¥(¾ Å90 ¿)
Distributed Computing through
Combinatorial Topology
The Decision Map
0?
01
01
?1
2/3
1
Protocol graph
δδ
0
1/3
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Output graph
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All Together
0
1
I
¥
¢
0?
01
01
?1
P
0
1/3
2/3
1
O
δ
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Definition
Decision map δ is carried by carrier map ¢ if
for each input vertex s,
δ(¥(s)) µ ¢(s)
for each input edge ¾,
δ(¥(¾)) µ ¢(¾).
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Meaning
δ(¥(s)) µ ¢(s)
process strarts in state s
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Meaning
δ(¥(s)) µ ¢(s)
runs the protocol to completion
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Meaning
δ(¥(s)) µ ¢(s)
makes a decision …
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Meaning
δ(¥(s)) µ ¢(s)
decision is permitted by task carrier map
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Solving a Task
Definition
The protocol (I,P,¥) solves the task (I, O, ¢)
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Solving a Task
Definition
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
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Solving a Task
Definition
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
a simplicial decision map
δ:P ! O
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Solving a Task
Definition
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
a simplicial decision map
δ:P ! O
such that δ is carried by ¢.
(δ agrees with ¢)
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Layered Read-Write Model
102
Layered Read-Write Protocol
shared mem array 0..1,0..L of Value
view: Value := my input value;
for i: int := 0 to L do
mem[i][A] := view;
view := view + mem[A] + mem[B];
return δ(view)
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Layered Read-Write Protocol
shared mem array 0..1,0..L of Value
view: Value := my input value;
for i: int := 0 to L do
mem[i][A] := view;
As before,
run for+L mem[A]
layers + mem[B];
view
:= view
return δ(view)
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Layered Read-Write Protocol
shared mem array 0..1,0..L of Value
view: Value := my input value;
for i: int := 0 to L do
mem[i][A] := view;
view := view + mem[A] + mem[B];
return δ(view)
Each layer uses a distinct, “clean” memory
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Layered R-W Protocol Graph
I
0
1
¥
0?
01
01
?1
P
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Layered R-W Protocol Graph
I
¥
P
P is always a subdivision of I
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Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Alice’s 1/3-Agreement
Protocol
mem[A] := 0
other := mem[B]
if other == ? then
decide 0
else
decide 1/3
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Alice’s 1/3-Agreement
Protocol
mem[A] := 0
if mem[B] == ? then
Alice writes her value to memory
decide 0
else
decide 1/3
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Alice’s 1/3-Agreement
Protocol
mem[A] := 0
if mem[B] == ? then
decide 0
Ifelse
she doesn’t see Bob’s value, decide her own.
decide 1/3
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Alice’s 1/3-Agreement
Protocol
mem[A] := 0
if mem[B] == ? then
decide 0
else
decide 1/3
If she see’s Bob’s value, jump to the middle
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0
1/3
2/3
1
0
1/3
2/3
1
0
1/3
2/3
1
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One-Layer 1/3-Agreement
Protocol
0
1
I
¥
0?
01
01
?1
P
0
1/3
2/3
1
O
δ
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No 1-Layer 1/5-Agreement
Protocol
0
1
I
¥
0?
δ
0
01
01
?1
P
(no map possible)
1/5
2/5
3/5
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4/5
1
O
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2-Layer 1/5-Agreement
0
1
I
¥
0?
01
layer 1
01
?1
layer 2
¥
δ
0
1/5
2/5
3/5
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4/5
O
1
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Fact
In the layered read-write model,
The 1/K-Agreement Task
Has a dlog3 Ke–layer protocol
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Road Map
Elementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Fact
The protocol graph for any L-layer protocol with
input graph I is a subdivision of I, where each
edge is subdivided 3L times.
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Main Theorem
The two-process task (I, O, ¢) is solvable in the
layered read-write model if and only if there
exists a connected carrier map ©: I ! 2O
carried by ¢.
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Corollary
The consensus task has no layered
read-write protocol
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Corollary
Any ²–agreement task has a layered
read-write protocol
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Distributed Computing through
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