Transcript Leader Election

Leader Election
Ch. 3, 4.1, 15.1, 15.2
Chien-Liang Fok
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Leader Election
process
connection



Numerous processes are connected in a digraph
One process eventually declares itself leader
Motivation:
 token
ring networks
 efficient communication (e.g., lighthouse routing)
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Impossibility Result for Identical
processes

Cannot solve leader election problem with
identical processes
 Proof
by induction on the sequence of states
 Every process will declare self as leader
simultaneously

Solution: Assume identical processes
distinguished by a unique identifier (UID)
 UID
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is a positive integer
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Synchronous Network
All processing done in lock-step formation
 There are rounds of computation
 Message-passing is done between rounds

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Model 1: Synchronous Ring
1
5
2
4
3
n processes connected in a ring
 Processes can distinguish clockwise from
counterclockwise neighbor

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Variations on the Theme
Non-leaders may have to declare
themselves as such
 Ring may be unidirectional or bidirectional
 Each node may or may not know n, the
size of the ring

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1
LCR Algorithm
5



Developed by Le Lann, Chang,
and Roberts (1978)
Unknown n, unidirectional ring,
only leader outputs
General idea:
2
4
3
 Every
process passes its UID around the ring
 Upon reception, compare to own UID



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If larger: Pass to next process
If smaller: Discard
If equal: Declare self as leader
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LCR Algorithm Proof
1
5
2
4


3
r
UIDmax
r is the number of rounds
Proof by induction on r
 After
r rounds, the node r away from the Pmax will be
sending UIDmax
 When r = n, UIDmax will have traveled around the ring
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Halting LCR

LCR algorithm never halts
 Non-leaders
continuously listen for incoming
messages

Have leader pass a report message around the
ring
 This
is the general way to transform a non-halting
algorithm into a halting one
 Transformation Cost:


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n additional rounds
n additional messages
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LCR Complexity Analysis

Time Complexity:
Non-Halting
Halting
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Time
Communication
n
2n
O(n2)
O(n2)
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HS Algorithm
Developed by Hirshberg and Sinclair
 Achieves O(n log n) communication
complexity, O(n) time complexity
 Assumptions:
1

 Unknown
n
 Bidirectional ring
5
2
4
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1
HS Algorithm (Cont.)
5

2
General idea:
4
 Operate
3
in phases 0, 1, 2, . . .
 During phase p, send UID in both directions
for a distance of 2p
Processing of received UID same as LCR
 If its UID comes back, continue to next round
 If it receives its outbound UID, declare self as
leader
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Comparison-based Algorithms

LCS and HS are comparison-based
algorithms
 Only

The communication lower-bound is:
Ω(n log n)
 See

compare UIDs
Lynch section 3.6 for proof
Is this the lower bound in general?
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TimeSlice Algorithm
Non-comparison-based algorithm
 Achieves O(n) communication complexity
 Assumptions:

n
is known to all processes
 Unidirectional communication
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TimeSlice Algorithm (Cont.)

1
5
General idea:
2
 Have
phases 1, 2, 3, . . .
 Phase p


4
3
Contains n rounds, rounds (p-1)n + 1, …, pn
If a node with UID p exists, it is circulated around the ring
 If
we reach round (p-1)n + 1 and the node with UID =
p has not received any UID before, declare self as
leader

Process with minimum UID becomes the leader
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TimeSlice Algorithm (Cont.)

Proof foundation:
 No
messages are sent except between
rounds (UIDmin-1) ·n + 1 and UIDmin·n
Only n messages are sent
 Problem:

 Time
complexity is n · UIDmin
 Unbounded!
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VariableSpeeds Algorithm



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Another non-comparison-based algorithm
Does not require knowledge of n
Unidirectional links
General idea:
 Each
process sends its UID around the ring
 The UID travels one hop every 2UID rounds
 Each process keeps track of lowest UID seen and
discards anything higher
 If token returns, declare self as leader
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VariableSpeeds Algorithm (Cont.)

Complexity Analysis
 Message:
2·n
 Time: O(n · 2Umin)
Outrageous unbounded time complexity
 This algorithm serves as a counterexample
algorithm

Impractical
 Shows that impossibility result cannot be proved

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Non-Comparison-based algorithms
Can achieve message costs of less than
Ω(n log n), but at the cost of unbounded
time complexity
 With bounded time complexity, the
message lower bound for noncomparison-based algorithms is Ω(n log n)

 See
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Lynch section 3.7 for proof
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Model 2: General Synch. Network
2
3
1
4
6
5
n processes each with UID
 Random connectivity
 Strongly connected graph

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Variations on the theme
Non-leaders may have to declare
themselves as such
 n may or may not be known
 diam may or may not be known
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FloodMax Algorithm
Assume diam is known
 General idea:

 Each
process remembers the largest UID it
has seen
 For each round, propagate UIDmax on all
outgoing links
 After diam rounds, if UIDmax is own UID,
declare self as leader
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FloodMax Algorithm (Cont.)
Basically a generalization of LCR
 Communication complexity is diam · |E|

 |E|

is the number of outgoing edges
Also works if only the upper-bound of diam
is known
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FloodMax Algorithm Proof Outline
Show that after r rounds, UIDmax has
reached all nodes within diameter r of the
node with the maximum UID
 Claim that when r = diam, all nodes will
2
have received UIDmax
3

1
4
6
5
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OptFloodMax Algorithm
A trivial optimization of FloodMax
 Only broadcast the UIDmax if it changed in
the current round
 Correctness is obvious, but the proof is not
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OptFloodMax Algorithm Proof

Use the simulation proof method
 Formally

Append a new-info flag to each node
 Only

relate OptFloodMax to FloodMax
broadcast UIDmax if new-info is true
First prove:
UIDmax,i > UIDmax,j for any j out-nbrsi, newinfoj must be true
 Proof by induction on r
 If
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OptFloodMax Proof (Cont.)

To complete the proof, relate the states of
OptFloodMax to FloodMax
 Assuming
both are started simultaneously,
after round r, the states of the two algorithms
are the same
 Proof by induction on r

See Lynch 4.1 for proof details
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Asynchronous Networks


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n processes connected via reliable FIFO
channels
Each process distinguished by UID
Processes do not operate in lock-step formation
Use I/O Automata to model processes and
channels
 Action
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of Pi include sendi, receivei, and leaderi
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Model 3: Asynchronous Ring
1
n processes arranged in a ring
5
 May be bidirectional or
unidirectional
4
3
 n may or may not be known
 Clockwise neighbor can be distinguished
from counterclockwise neighbor

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Asynchronous LCR algorithm

Same as synchronous LCR algorithm
except:
 Send
buffer of each process must be able to
hold up to n messages

This algorithm is intuitively correct, but the
proof is complex
 Do
not know where UIDmax is
 Cannot relate location of UIDmax to the round
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Asynchronous LCR Proof

Proof outline
 Show
that for 0 ≤ r ≤ n-1, eventually UIDmax
appears in the buffer sendimax+r

Proof by induction on r
 Show
that UIDmax will eventually appear on
channel connecting to the input of imax
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Asynchronous HS Algorithm

The synchronous HS algorithm can also
be trivially modified to work in
asynchronous rings
 Just
make sure all channels can handle
message pileups
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Peterson Algorithm
Requires O(n log n) messages
 Unidirectional links
 Unknown n
 Relies on asynchronously determined
phases
 Elects an arbitrary process as the leader

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Peterson Algorithm (Cont.)


Process can be in active or relay mode
In the first phase, all nodes send their UID two hops
clockwise




If UID of immediate counter-clockwise neighbor is highest of the
three, adopt it as its new UID and remain active
Otherwise, enter relay mode
In the following phases, active node sends their new UID
to their two clock-wise active neighbors and repeat
If an active node sees that its immediate
counterclockwise active neighbor has the same UID,
declare self as leader
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Peterson Algorithm Examples
Leader
!
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2
4
3
7
9
8
5
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Leader
!
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9
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Peterson Algorithm Time Cost

In each phase, the number of active
processes is decreased by two
 O(log

n) phases
For each phase p, the first p phases
complete in O(pn(l+d))
l
= upperbound on time to process message
 d = upperbound on output queue delay

Thus, the time cost is O(n log n (l+d))
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Peterson Algorithm Message Cost
There are at most log n  + 1 phases
 During each phase, each process sends at
most two messages (2n messages)
 Message cost = O(n log n)

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Communication Lower Bound

The communication lower bound of
asynchronous comparison-based
algorithms is Ω(n log n)
 Synchronous
rings is a restriction on
asynchronous rings
 See Lynch section 15.1.4 for proof
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Model 4: Asynchronous General

2
Assumptions:
3
 Bidirectional
1
links
 Random links
6
 Strongly connected graph
 Identical processes except UID
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5
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Asynchronous FloodMax
Since there are no rounds, FloodMax
cannot be trivially extended to asynch.
Networks
 Workaround:

 Append
round number to each message
 Each node collects all incoming round
messages before processing them
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Alternative techniques

There are many other general asynch.
network protocols that use:
 Spanning
Tree, Convergecast, and Broadcast
 Synchronizers to simulate synchronous
networks
 Consistent snapshots for termination
detection
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Conclusions
There are many leader election protocols
 Special protocols exist for ring networks
 Algorithms for asynchronous networks are
more complicated

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