On the Complexity of Universal Leader Election

Abstract

Electing a leader is a fundamental task in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message and time complexity of randomized implicit leader election in synchronous distributed networks. Surprisingly, the most "obvious" complexity bounds have not been proven for randomized algorithms. In particular, the seemingly obvious lower bounds of Ω(m) messages, where m is the number of edges in the network, and Ω(D) time, where D is the network diameter, are non-trivial to show for randomized (Monte Carlo) algorithms. (Recent results, showing that even Ω(n), where n is the number of nodes in the network, is not a lower bound on the messages in complete networks, make the above bounds somewhat less obvious). To the best of our knowledge, these basic lower bounds have not been established even for deterministic algorithms, except for the restricted case of comparison algorithms, where it was also required that nodes may not wake up spontaneously and that D and n were not known. We establish these fundamental lower bounds in this paper for the general case, even for randomized Monte Carlo algorithms. Our lower bounds are universal in the sense that they hold for all universal algorithms (namely, algorithms that work for all graphs), apply to every D, m, and n, and hold even if D, m, and n are known, all the nodes wake up simultaneously, and the algorithms can make any use of node's identities. An interesting fundamental problem is whether both upper bounds (messages and time) can be reached simultaneously in the randomized setting for all graphs. The answer is known to be negative in the deterministic setting. We answer this problem partially by presenting a randomized algorithm that matches both complexities in some cases. This already separates (for some cases) randomized algorithms from deterministic ones. As first steps towards the general case, we present several universal leader election algorithms with bounds that trade-off messages versus time. We view our results as a step towards understanding the complexity of universal leader election in distributed networks.

Multi-leader election in dynamic sensor networks

“…In [21][22][23], although they gave the deterministic lower bounds of time and message complexities, these lower bounds are not suitable for our model. When nodes leave or enter the network dynamically, the network topology changes over time.…”

“…The message complexity of the synchronous algorithm is O(n log n), and they also proved that any message-optimal synchronous algorithm requires (log n) time. In [23], Kutten et al focused on studying the message and time complexities of randomized implicit leader election in synchronous distributed networks. The lower bounds of message complexity and time complexity are (m) and (D), respectively, where m denotes the number of edges and D is the diameter of the network.…”

Leader Election in Well-Connected Graphs

“…The algorithm can also be used for solving the explicit variant of leader election by adding a broadcasting procedure, wherein the leader broadcasts its identity to all other nodes. For well connected graphs, this breaks the Ω(m) lower bound given in [24] (where m denotes the number of edges of the graph) and nearly matches the Ω( √ n) lower bound for clique graphs [25] (as cliques have constant conductance).We show that a dependence on the graph conductance is unavoidable, by presenting a message complexity lower bound of Ω( √ n/φ 3/4 ) messages that holds for any leader election algorithm that succeeds with probability at least 1 − o(1). This nearly matches the upper bound since we know that Θ(1/φ) t mix Θ(1/φ 2 ) from [37].By a similar analysis, we also provide lower bounds for other graph problems like broadcast and spanning tree construction in terms of the graph's conductance.…”

“…Our lower bounds also apply for the LOCAL model [32], where there are no restrictions on the message size. Other than the Ω(m) bound in [24], to the best of our knowledge, this is the first non-trivial lower bound for randomized leader election in general networks. Also, ours is one of the first results to show the dependence of the time and message complexity to solve leader election on the connectivity of the graph G, which is often characterized by the graph's conductance φ.Additionally, we show that the knowledge of the network size n is critical for our algorithm to succeed by giving a lower bound of Ω(m) for all graphs if n is not known.…”

“…To the best of our knowledge, this is the first work that shows a dependence between the time and message complexity to solve leader election and the connectivity of the graph G, which is often characterized by the graph's conductance φ. Apart from the Ω(m) bound in [24] (where m denotes the number of edges of the graph), this work also provides one of the first non-trivial lower bounds for leader election in general networks.Leader election is one of the most classical and fundamental problem in the field of distributed computing having applications in numerous problems relating to synchronization, resource allocation, reliable replication, load balancing, job scheduling (in master slave environment), crash recovery, membership maintenance etc. Computing a leader can be thought of as a form of symmetry breaking, where exactly one special node or process (denoted as leader) is chosen to take some critical decisions.Loosely speaking, the problem of leader election requires a set of nodes in a distributed network to elect a unique leader among themselves, i.e., exactly one node must output the decision that it is the leader.…”