Distributed Computing 3. Leader Election – lower bound for ring networks Shmuel Zaks ©

Theorem: For every algorithm A for maximum finding in unidirectional rings, and every set I of n identities, PKR

Unidirectional ring Same messages in every execution

ring (4, 1, 3, 5 ) (1, 3, 5, 4 ) (3, 5, 4, 1 ) (5, 4, 1, 3 )

messages (4 ) (4,1 ) (4, 1, 3 )

Sequence Prefix of a sequence Concatenation of sequences subsequence of if s.t. C(S) = all cyclic permutations of S |C(s)| = length (s)

{ (4, 1, 3, 5 ), (1, 3, 5, 4 ), (3, 5, 4, 1 ), (5, 4, 1, 3 ) } s = (4, 1, 3, 5 ) C(s) = length (s) = 4 (4,1,3) is a prefix of s (1,3) is a subsequence of s

In every execution of a maximum finding algorithm A, at least one processor must see its own value In a ring labeled s, at least one message in C(s) is sent by A (4 ) (4,1 ) (4,1,3 ) (4,1,3,5 )

– set of all finite sequences of distinct integers For

E does not have to be a finite set. Note: though we wrote Why?

s = (4, 1, 3, 5 ) E={(4,5),(3,5),(5,4,1),(3),(6,2),(5,4) } N(s,E) = 4 N 1 (s,E) = 1 N 2 (s,E) = 2 N 3 (s,E) = 1 N k (s,E) = 0 for k ≥ 4

Definition: A setis exhaustive if it has the following two properties:  Prefix property: if then for every prefix s of u.  Cyclic permutation property:

is exhaustive: Prefix property: if then for every prefix s of u.  Cyclic permutation property: Example: the set

E contains also the following: (4), (4,1),(4,1,3), (1), (3), (5),(5,4),(5,4,1),(5,4,1,3) E is the set of messages sent by the Chang & Roberts’ algorithm!

Lemma: Let such that is a prefix of and, and let A be a maximum finding algorithm. If in the execution of A on ring a message is sent, then in the execution of A on ring a message is sent. st u u

Theorem: For every maximum finding algorithm A for unidirectional rings, there exists an exhaustive set E(A), such that, for every ring s, A sends at least N(s,E(A)) messages on s. Proof: Let

1. E(A) is exhaustive 1a. Prefix property

1. E(A) is exhaustive 1b. Cyclic permutation property

2. At least N(s,E(A)) messages sent by A on s

Theorem: For every maximum finding algorithm A for unidirectional rings, and a set of n identities I, we have:

Theorem: In a unidirectional ring whose size n is unknown, the Chang & Roberts algorithm has an optimal message complexity of Exercise: What if n is known? What if synchronous?

References J. E. Burns, A formal model for message passing systems, TR-91, Indiana University, September 1980.

References E. Chang and R. Roberts, An improved algorithm for decentralized extrema-finding in circular configurations of processes, Communications of the ACM}, 22, 5, 1979, pp

References J. Pachl, E. Korach and D. Rotem Lower bounds for distributed maximum- finding algorithm. Journal of Association for Computing Machinery, Vol. 31, No. 4, Oct. 1984, pp