Optimal lower bounds for some distributed algorithms for a complete network of processors

Korach, Ephraim; Moran, Shlomo; Zaks, Shmuel

doi:10.1016/0304-3975(89)90103-5

“…13. Consider a universal leader election algorithm R that succeeds with probability 1 − β, assuming that β < 1/16 in the anonymous setting and β < 1/16 − 15/16n 2 , if nodes have unique ids.…”

Section: Time Complexity Lower Boundmentioning

confidence: 99%

On the complexity of universal leader election

Kutten

¹

,

Pandurangan

²

,

Peleg

³

et al. 2013

Proceedings of the 2013 ACM Symposium on Principles of Distributed Computing

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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.

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“…There are several known lower bounds for deterministic leader election algorithms in cycles (e.g., [8]) and complete graphs (e.g., [13,2]), which also imply bounds for (deterministic) universal algorithms. These results alone do not, however, imply that a universal algorithm cannot do significantly better in other classes of networks.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Universal Leader Election

Kutten

¹

,

Pandurangan

²

,

Peleg

³

et al. 2015

J. ACM

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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.

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“…Thus we have the same assumptions that were made in [23]. Among the assumptions, the second one is crucial.…”

Section: Lower Boundmentioning

confidence: 99%

“…Our lower bound is based on a classical lower bound due to Korach et al [23], which shows that Ω(n log n) messages are needed by any distributed algorithm for constructing a spanning tree (or equivalently, leader election) in a complete network. Thus we have the same assumptions that were made in [23].…”

Section: Lower Boundmentioning

confidence: 99%

Energy-Optimal Distributed Algorithms for Minimum Spanning Trees

Choi

¹

,

Pandurangan

²

,

Khan

³

et al. 2009

IEEE J. Select. Areas Commun.

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Traditionally, the performance of distributed algorithms has been measured in terms of time and message complexity. Message complexity concerns the number of messages transmitted over all the edges during the course of the algorithm.However, in energy-constrained ad hoc wireless networks (e.g., sensor networks), energy is a critical factor in measuring the ef ciency of a distributed algorithm. Transmitting a message between two nodes has an associated cost (energy) and moreover this cost can depend on the two nodes (e.g., the distance between them among other things). Thus in addition to the time and message complexity, it is important to consider energy complexity that accounts for the total energy associated with the messages exchanged among the nodes in a distributed algorithm. This paper addresses the minimum spanning tree (MST) problem, a fundamental problem in distributed computing and communication networks. We study energy-ef cient distributed algorithms for the Euclidean MST problem assuming random distribution of nodes. We show a non-trivial lower bound of Ω(log n) on the energy complexity of any distributed MST algorithm. We then give an energy-optimal distributed algorithm that constructs an optimal MST with energy complexity O(log n) on average and O(log n log log n) with high probability. This is an improvement over the previous best known bound on the average energy complexity of Ω(log 2 n). Our energy-optimal algorithm exploits a novel property of the giant component of sparse random geometric graphs. All the above results assume that nodes do not know their geometric coordinates. If the nodes know their own coordinates, then we give an algorithm with O(1) energy complexity (which is the best possible) that gives an O(1) approximation to the MST.

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Optimal lower bounds for some distributed algorithms for a complete network of processors

Cited by 40 publications

References 5 publications

On the complexity of universal leader election

On the complexity of universal leader election

On the Complexity of Universal Leader Election

Energy-Optimal Distributed Algorithms for Minimum Spanning Trees

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