TY - GEN
T1 - The complexity of leader election
T2 - 19th International Conference on Distributed Computing and Networking, ICDCN 2018
AU - Chatterjee, Soumyottam
AU - Pandurangan, Gopal
AU - Robinson, Peter
N1 - Funding Information:
Supported, in part, by NSF grants CCF-1527867
Publisher Copyright:
© 2018 Association for Computing Machinery.
PY - 2018/1/4
Y1 - 2018/1/4
N2 - Leader election is one of the fundamental problems in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message complexity of leader election in synchronous distributed networks, in particular, in networks of diameter two. Kutten et al. [JACM 2015] showed a fundamental lower bound of ω(m) (m is the number of edges in the network) on the message complexity of (implicit) leader election that applied also to Monte Carlo randomized algorithms with constant success probability; this lower bound applies for graphs that have diameter at least three. On the other hand, for complete graphs (i.e., diameter 1), Kutten et al. [TCS 2015] established a tight bound of η (√ n)1 on the message complexity of randomized leader election (n is the number of nodes in the network). For graphs of diameter two, the complexity was not known. In this paper, we settle this complexity by showing a tight bound of η(n) on the message complexity of leader election in diametertwo networks.We first give a simple randomized Monte-Carlo leader election algorithm that with high probability (i.e., probability at least 1-n-c , for some positive constant c) succeeds and uses O(n log3 n) messages and runs in O(1) rounds; this algorithm works without knowledge of n (and hence needs no global knowledge). We then show that any algorithm (even Monte Carlo randomized algorithms with large enough constant success probability) needs ω(n) messages (even when n is known), regardless of the number of rounds. We also present an O(n logn) messages deterministic algorithm that takes O(logn) rounds (but needs knowledge of n); we show that this message complexity is tight for deterministic algorithms. Our results show that leader election can be solved in diametertwo graphs in (essentially) linear (in n) message complexity and thus the ω(m) lower bound does not apply to diameter-Two graphs. Together with the two previous results of Kutten et al., our results fully characterize the message complexity of leader election vis-à-vis the graph diameter.
AB - Leader election is one of the fundamental problems in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message complexity of leader election in synchronous distributed networks, in particular, in networks of diameter two. Kutten et al. [JACM 2015] showed a fundamental lower bound of ω(m) (m is the number of edges in the network) on the message complexity of (implicit) leader election that applied also to Monte Carlo randomized algorithms with constant success probability; this lower bound applies for graphs that have diameter at least three. On the other hand, for complete graphs (i.e., diameter 1), Kutten et al. [TCS 2015] established a tight bound of η (√ n)1 on the message complexity of randomized leader election (n is the number of nodes in the network). For graphs of diameter two, the complexity was not known. In this paper, we settle this complexity by showing a tight bound of η(n) on the message complexity of leader election in diametertwo networks.We first give a simple randomized Monte-Carlo leader election algorithm that with high probability (i.e., probability at least 1-n-c , for some positive constant c) succeeds and uses O(n log3 n) messages and runs in O(1) rounds; this algorithm works without knowledge of n (and hence needs no global knowledge). We then show that any algorithm (even Monte Carlo randomized algorithms with large enough constant success probability) needs ω(n) messages (even when n is known), regardless of the number of rounds. We also present an O(n logn) messages deterministic algorithm that takes O(logn) rounds (but needs knowledge of n); we show that this message complexity is tight for deterministic algorithms. Our results show that leader election can be solved in diametertwo graphs in (essentially) linear (in n) message complexity and thus the ω(m) lower bound does not apply to diameter-Two graphs. Together with the two previous results of Kutten et al., our results fully characterize the message complexity of leader election vis-à-vis the graph diameter.
KW - Distributed Algorithm
KW - Leader Election
KW - Lower Bounds
KW - Message Complexity
KW - Randomized Algorithm
KW - Time Complexity
UR - http://www.scopus.com/inward/record.url?scp=85041168407&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85041168407&partnerID=8YFLogxK
U2 - 10.1145/3154273.3154308
DO - 10.1145/3154273.3154308
M3 - Conference contribution
AN - SCOPUS:85041168407
T3 - ACM International Conference Proceeding Series
BT - Proceedings of the 19th International Conference on Distributed Computing and Networking, ICDCN 2018
PB - Association for Computing Machinery
Y2 - 4 January 2018 through 7 January 2018
ER -