1. Leader Election Ch. 3, 4.1, 15.1, 15.2 Chien-Liang Fok 4/29/2019
CS 673 Fall 2003

2. Leader Election Numerous processes are connected in a digraph
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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3. 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 is a positive integer
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4. Synchronous Network All processing done in lock-step formation
There are rounds of computation
Message-passing is done between rounds
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5. Model 1: Synchronous Ring5 2 4 3
n processes connected in a ring
Processes can distinguish clockwise from counterclockwise neighbor
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6. 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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7. LCR Algorithm Developed by Le Lann, Chang, and Roberts (1978)5 2
Developed by Le Lann, Chang, and Roberts (1978)
Unknown n, unidirectional ring, only leader outputs
General idea:
Every process passes its UID around the ring
Upon reception, compare to own UID
If larger: Pass to next process
If smaller: Discard
If equal: Declare self as leader 4 3
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8. LCR Algorithm Proof r is the number of rounds Proof by induction on r1 r 5 2 4 3
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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9. 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:
n additional rounds
n additional messages
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10. LCR Complexity Analysis
Time Complexity:
Time
Communication
Non-Halting n
O(n2)
Halting 2n
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11. HS Algorithm Developed by Hirshberg and Sinclair
Achieves O(n log n) communication complexity, O(n) time complexity
Assumptions:
Unknown n
Bidirectional ring 1 5 2 4 3
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12. HS Algorithm (Cont.) General idea: Operate in phases 0, 1, 2, . . .5 2
General idea:
Operate 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 4 3
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13. Comparison-based Algorithms
LCS and HS are comparison-based algorithms
Only compare UIDs
The communication lower-bound is: Ω(n log n)
See Lynch section 3.6 for proof
Is this the lower bound in general?
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14. TimeSlice Algorithm Non-comparison-based algorithm
Achieves O(n) communication complexity
Assumptions:
n is known to all processes
Unidirectional communication
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15. TimeSlice Algorithm (Cont.)1 5 2
General idea:
Have phases 1, 2, 3, . . .
Phase p
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 4 3
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16. 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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17. VariableSpeeds Algorithm
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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18. 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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19. 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 non-comparison-based algorithms is Ω(n log n)
See Lynch section 3.7 for proof
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20. Model 2: General Synch. Network3 1 4 6 5
n processes each with UID
Random connectivity
Strongly connected graph
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21. 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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22. 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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23. 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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24. 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 have received UIDmax 2 3 1 4 6 5
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25. 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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26. OptFloodMax Algorithm Proof
Use the simulation proof method
Formally relate OptFloodMax to FloodMax
Append a new-info flag to each node
Only broadcast UIDmax if new-info is true
First prove:
If UIDmax,i > UIDmax,j for any j out-nbrsi, new-infoj must be true
Proof by induction on r
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27. 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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28. Asynchronous Networks
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 of Pi include sendi, receivei, and leaderi
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29. Model 3: Asynchronous Ring
n processes arranged in a ring
May be bidirectional or unidirectional
n may or may not be known
Clockwise neighbor can be distinguished from counterclockwise neighbor 1 5 2 4 3
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30. 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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31. 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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32. 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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33. 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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34. 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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35. Peterson Algorithm Examples
Leader! 1 2 4 4 3
Leader! 7 9 8 9 8 5 9
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36. 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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37. 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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38. 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 for proof
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39. Model 4: Asynchronous General2
Assumptions:
Bidirectional links
Random links
Strongly connected graph
Identical processes except UID 3 1 4 6 5
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40. 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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41. 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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42. Conclusions There are many leader election protocols
Special protocols exist for ring networks
Algorithms for asynchronous networks are more complicated
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