Leader Election in Rings

1
Leader Election in Rings

2
A Ring Network
3
Sense of direction
left
2
1
1
right
2
2
1
1
2
2
1
1
2
2
1
1
2
4
Links are bi-directional for messages
At most one message in each direction
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
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Anonymous Ring
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
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Eponymous (non-anonymous) Ring
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Leader Election
Initial state
Final state
leader
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Leader election algorithms are affected by
Anonymous Ring Eponymous Ring
The size of the network is known The size of
the network is not known
Synchronous Algorithm Asynchronous Algorithm
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Synchronous Anonymous Rings
Every processor runs the same algorithm
Every processor does exactly the same execution
10
Initial state
Final state
leader
If one node is elected a leader, then every node
is elected a leader
11
Final state
Conclusion 1
Leader election cannot be solved in
synchronous anonymous rings
leader
12
Final state
Conclusion 2
Leader election cannot be solved in
asynchronous anonymous rings either
leader
Why?
The asynchronous ring may behave Like the
synchronous ring
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Asynchronous Eponymous Rings
The maximum id node is elected leader
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1
2
5
6
3
4
7
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Each node sends a message with its id to the left
neighbor
1
8
8
1
5
2
2
5
6
3
6
3
4
7
7
4
15
If message received id current node id
Then forward message
5
8
1
2
8
5
6
7
3
4
7
6
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If message received id current node id
Then forward message
8
1
7
2
5
6
8
3
4
7
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If message received id current node id
Then forward message
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8
1
2
5
6
3
4
7
8
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If message received id current node id
Then forward message
8
1
2
5
6
3
4
8
7
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If a node receives its own message
Then it elects itself a leader
8
8
1
2
5
6
3
4
7
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If a node receives its own message
Then it elects itself a leader
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1
leader
2
5
6
3
4
7
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The leader sends a message in the
network declaring itself as the leader of the
ring
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1
leader
2
5
6
3
4
7
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nodes
Time complexity
8
1
leader
2
5
6
3
4
7
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Message complexity
nodes
worst case scenario
n
1
n-1
2
n-2
n-3
24
Message complexity
nodes
n
1
messages
n-1
2
n-2
n-3
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Message complexity
nodes
n
1
messages
n-1
2
n-2
n-3
26
Message complexity
nodes
n
1
messages
n-1
2
n-2
n-3
27
Message complexity
nodes
Total messages
n
1
n-1
2
n-2
n-3
28
Notes
does not need to be known to the algorithm
The algorithm can be converted to asynchronous
29
An O(n log n) Mess. Algorithm
Again, the maximum id node is elected leader
8
1
2
5
6
3
4
7
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nodes
8
nodes
1
2
5
6
3
4
7
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Phase 1 send id to 1-neighborhood
1
8
8
1
8
5
2
2
1
2
5
6
6
5
3
4
6
3
3
7
4
7
7
4
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If received id current id Then send
a reply
8
1
2
5
6
3
4
7
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If a node receives both replies Then it
becomes a temporal leader
8
1
2
5
6
3
4
7
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Phase 2 send id to 2-neighborhood
8
8
1
8
2
5
5
6
6
5
6
3
7
4
7
7
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If received id current id Then
forward the message
8
6
1
8
5
2
5
8
6
7
3
7
6
4
5
7
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At second step
If received id current id Then send
a reply
8
1
2
5
6
3
4
7
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If a node receives both replies Then it
becomes a temporal leader
8
1
2
5
6
3
4
7
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Phase 3 send id to -neighborhood
8
1
8
2
5
8
7
7
6
3
4
7
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If received id current id Then
forward the message
8
1
8
2
5
8
7
7
6
3
4
7
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At the step
If received id current id Then send
a reply
8
1
2
5
6
3
4
7
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If a node receives both replies Then it
becomes the leader
8
1
2
5
6
3
4
7
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8
1
leader
2
5
6
3
4
7
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In general
nodes
phases
8
1
leader
2
5
6
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Phase i send id to -neighborhood
8
1
leader
2
5
6
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Time complexity
The leader spends time in
Phase 1 2 Phase 2 4 Phase i Phase log n
Total time
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Message complexity
Messages per leader
Max leaders
Phase 1 4 Phase 2 8 Phase i Phase log n
47
Messages per leader
Max leaders
Phase 1 4 Phase 2 8 Phase i Phase log n
Total messages
48
Notes
The algorithm does not need to know
It can be converted to an asynchronous algorithm
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An Mess. Synchronous Algorithm
is known
The node with smallest id is elected leader
There are rounds

• If in round there is a node with id
• this is the new leader
• the algorithm terminates

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Round 1 ( time steps) no message sent
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9
22
15
nodes
16
33
24
57
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Round 2 ( time steps) no message sent
48
9
22
15
nodes
16
33
24
57
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Round 9
new leader
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9
22
15
nodes
16
33
24
57
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Round 9 ( time steps) messages sent
new leader
48
9
22
15
nodes
16
33
24
57
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Round 9 ( time steps) messages sent
new leader
48
9
22
15
nodes
16
33
24
57
Algorithm Terminates
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Round 9 ( time steps) messages sent
new leader
48
9
22
15
nodes
16
33
24
57
Total number of messages
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Another Synchronous Algorithm
is not known
The node with smallest id is elected leader
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The algorithm

• Each node injects a message with its id

• Message with id is injected and
• transferred with rate

• Nodes which have seen smaller id
• absorb higher id messages

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Time 1
Transfer rate
0
0
4
2
5
6
3
1
7
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Time 2
0
4
2
rate
5
0
6
3
1
7
1
60
Time 3
0
4
2
5
6
3
0
1
7
1
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Time 4
0
4
2
5
6
3
1
1
7
0
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Time 5
0
4
2
5
6
3
1
0
7
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Time 6
0
4
2
5
6
0
3
1
7
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Time 8
0
0
4
2
5
6
3
1
7
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Time 8
New leader
0
0
4
2
5
6
3
1
7
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Message complexity
Assume leader has (smallest) id
Total time of algorithm
Note that if then algorithm is
exponentially slow
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Take the node with immediately higher id
Total number of messages
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Take the node with immediately higher id
Total number of messages
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messages
id
lower
higher
Total messages
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An Lower Bound
Assume we have algorithms in which

• the network is asynchronous

• the maximum identifier is elected leader

• all the nodes must know the leader

• the size of the network is not known

We will prove
at least messages are needed
to elect a leader
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Messages may be pending to cross open edge
open edge
no messages cross it
(i.e. a very slow edge)
There is a possible asynchronous execution with
an open edge
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Messages pending to cross open edge
open edge
Quiescent State
If the edge remains open, then the execution will
eventually reach a quiescent state where not more
messages cross the ring
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Messages pending to cross open edge
open edge
Quiescent State
In the quiescent state, a node can send a
message only after it receives a message (thus no
messages cross the ring)
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Time t
Suppose that the edge closes
This may cause a propagation of messages until
the algorithm terminates
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Time t1
This may cause a propagation of messages until
the algorithm terminates
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Time t2
This may cause a propagation of messages until
the algorithm terminates
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Time tk
k
k
After k time steps the affected radius is k
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Time tx
Eventually, the ring stabilizes with a leader
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We will show, that there is an execution with
nodes such that

• there is an open edge

• at least messages are received

where
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Proof by induction
basis case
x
y
If then is the leader
Therefore a message is sent
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open edge
x
y
The message can be sent on one edge
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The case
From induction hypothesis, we have an execution
open edge
nodes
Messages sent
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The case
From induction hypothesis, we have an execution
open edges
nodes
nodes
Messages sent
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If no message is sent along
Observation
then ring cannot distinguish the
two scenarios
(similarly for ring )
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Therefore, the same number of messages have to be
sent in sub-rings
True open edge
nodes
nodes
True open edge
Messages sent
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All nodes in should learn about
open edge
max id
open edge
(assume that max id is in )
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All nodes in should learn about
max id
(At least one of these is sent)
Thus, after the open edges close, at least
messages are sent
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Suppose edges close at time
max id
At least one message is sent
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time
max id
At least one message is sent
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time
max id
At least messages have been sent since
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time
radius
max id
At least messages have been sent since
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time
radius
max id
Independent areas
At least messages have been sent since
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Messages
time
radius
max id
Independent areas
At least messages have been sent in one area
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Messages
time
radius
max id
open edge
Since areas are independent, we could have closed
only one edge
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Messages
time
radius
max id
open edge
Total Messages
Induction Hypothesis