Election Algorithms

3
Assumptions


Processes and channels are reliable System is fully asynchronous Processes are distinguished by unique identities
4
Election Algorithm Properties
19
Cont’d


Hischberg and Sinclair gave an algorithm with worst-case complexity of O(NlogN) requiring bidirectional channels in 1980. Burns has a slightly better bidirectional O(NlogN) algorithm in 1980. Burns formally defines the model and the problem and gives an (NlogN) lower bound for the bidirectional case.
17
Remark



Sense of direction is the ability of processors to distinguish neighbor processors in an anonymous setting. Sense of direction does not help in anonymous leader election The theorem also hห้องสมุดไป่ตู้lds for other symmetric network topologies The algorithm is not allowed to use randomization

22
(2)
Var Listp: set of P init {p}; statep; begin if p is initiator then begin statep:=cand; send <tok,p> to Nextp; receive <tok,q>; while qp do begin Listp:=Listp {q}; send <tok,q> to Nextp; receive <tok,q>; end; if p=min(Listp) then statep:= leader else statep:=lost end else while true do begin receive <tok,q>; send <tok,q> to Nextp; if statep=sleep then statep:=lost end end

Anonymous leader election Asynchronous ring Lower bounds Synchronous ring
11
Anonymous

A system is anonymous if processors do not have unique identifiers.
if p is initiator then { wsp:= true; for all q Neighp do send <wakeup> to q} while wrp < #Neighp do {receive <wakeup>; wrp := wrp + 1; if not wsp then { wsp :=true; forall q Neighp do send <wakeup> to q}}
20
Cont’d


Hirschberg and Sinclair conjecture that any unidirectional solution must be (N2). Petersen /Dolve-Klawe-Rodeh independently proposed an O(NlogN) solution for the unidirectional ring in 1982.



Each process has the same local algorithm. The algorithm is decentralized, i.e., a computation can be initialized by an arbitrary non-empty subset of the processes. The algorithm reaches a terminal configuration in each computation. There is exactly one process in the state leader and all other processes are in the state lost.
2
Remark



Processors are in one of three states: undecided, leader, lost Initially every process is in the undecided state When leaving the undecided state, a processor goes into a terminated state (leader or lost)
5
Election with the tree algorithm
The network topology is a tree or a spanning tree of the network All leaves are initiators of the algorithm Data structures: -- ws: boolean variable, to make every process send <wakeup> messages at most once -- wr: to count the number of <wakeup> messages a process has received
14
Conclusion

Non-uniform anonymous leader election for synchronous rings is impossible.
15
Lemma
After round k of a deterministic algorithm A, each processor is in state Sk. Proof with induction. -- All processors start in the same state. -- A round in a synchronous algorithm consists of the three steps sending, receiving, local computation. -- All processors send the same message(s), receive the same message(s), do the same local computation, and therefore end up in the same state.
12
Uniform


If the number of processors (“n”) is not known to the algorithm. If n is known, the algorithm is called non-uniform.
13
Remark

Whether or not a leader can be elected in an anonymous system depends whether the network is symmetric (ring, complete graph, complete bipartite graph, etc.) or asymmetric (star, single node with highest degree, etc.).
18
The History



Election algorithms for unidirectional rings Lelann solved the problem with message complexity O(N2) in 1977 Chang and Roberts solved the problem with a worst case complexity of O(N2) and an average case complexity of O(NlogN) in 1979
9
Algorithm Complexity Analysis

The election algorithm for trees solves the election problem using O(N) messages.
10
Election algorithms for ring networks
Election Algorithms
1
Election Problem


Starting from a configuration where each process is in the same state, a configuration is reached where exactly one process is in a special state leader, while all other processes are in the state lost. The process in state leader at the end of the computation is called the leader.