B. Awerbuch. Optimal Distributed Algorithms for Minimum Weight Spanning Tree, Counting, Leader Election, and Related Problems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), New York City, New York, May 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... does there exist a neighborhood optimal algorithm for MST . The algorithm presented in [GKP] has time complexity O(n Diam) hence it is neighborhood optimal for the case that Diam n . Previous distributed MST algorithms had running time O(n log n) GHS] O(n log n) CT, G] and O(n) [A2]. Using the fast k Dominating Set al..gorithm we manage to present an improved MST construction algorithm whose time complexity is O( n Diam(G) Thus the MST algorithm is now neighborhood optimal for all graphs with Diam n 1=2 n. Let us hint about how the improved MST algorithm was ....

....most one message over each edge at each time unit. We will also make the assumption that edge weights are polynomial in n, so an edge weight can be sent in a single message. This assumption is required for the time analysis of the previous algorithms that use edge weights or nodes identity, e.g. [GHS, A2, GKP]. Let us now de ne the Minimum Spanning Tree task solved later on as an application of the k Dominating Set al..gorithm. The goal is to have the nodes (processors) cooperate to construct a minimum weight spanning tree (MST) for G, namely, a tree covering the nodes in V whose total edge weight is ....

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B. Awerbuch, Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems, Proc. 19th Symp. on Theory of Computing, pp. 230-240, May 1987.

.... real systems lack such guarantees, these results have been valuable mostly in a theoretical rather than a practical sense. Non randomized leader election algorithms for a failure prone asynchronous network model broadly fall into the following flavors. 1) Gallager HumbletSpira type algorithms [1, 13, 21] that work by constructing several spanning trees in the network, with a prospective leader at the root of each of these, and recursively reduce the number of these spanning trees to one. The correctness guarantees of these algorithms are violated in the face of pathological process and message ....

....by a bit string A I . For example, A I could be the (source address, sequence number) pair of message I. Each group member M i that receives this message computes a hash of the concatenation of A I and M i s address, using a hash function H that deterministically maps bit strings to the interval [0, 1]. Next, M i calculates the filter value H(M i A I ) N i for the initiating message, where N i is the size of Failure Detection I Initiating Mcast I mcast elected leaders (among filter members) 1 2 3 Group 4 N 1 N Relay Phase Relay Phase members Phase Hash(I,self) N K ....

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B. Awerbuch, "Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems", Proc. 19th Symp. on Theory of Computing, 1987, pp. 230-240.

....For example, the leader can be the commander s mobile for a platoon of soldiers in a mission. If it is impossible to designate any leader, a distributed leader election algorithm can be applied to find out a leader. This adds message and time complexity. The best leader election algorithm (see [4]) takes time O(n) and message O(n log n) and these are the best achievable results. Assume host s is the leader. Phase 1. Host s first colors itself black and broadcasts message DOMINATOR. Any white host u receiving DOMINATOR message the first time from v colors itself gray and broadcasts message ....

B. Awerbuch, Optimal distributed algorithm for minimum weight spanning tree, counting, leader election and related problems, Proceedings of the 19th ACM Symposium on Theory of Computing, ACM, pp. 230-240, 1987.

....For example, the leader can be the commander s mobile for a platoon of soldiers in a mission. If it is impossible to designate any leader, a distributed leader election algorithm can be applied to find out a leader. This adds message and time complexity. The best leader election algorithm (see [3]) takes time O(n) and message O(n log n) and these are the best achievable results. Assume host s is the leader. Phase 1. Host s first colors itself black and broadcasts message DOMINATOR. Any white host u receiving DOMINATOR message the first time from v colors itself gray and broadcasts message ....

B. Awerbuch, Optimal distributed algorithm for minimum weight spanning tree, counting, leader election and related problems, Proceedings of the 19th ACM Symposium on Theory of Computing, ACM, pp. 230-240, 1987.

....or event driven hello messages. As mentioned earlier, we designate a host as the leader. If it is impossible to specify any leader, a distributed leader election algorithm can be applied. This increases message and time complexities. The best leader election algorithm in literature (see [4]) takes time O(n) and message O(n log n) and these are the best achievable results. Algorithm I computes a tree rooted at the leader. 5 Each host maintains the following parameters: dom, which is the dominator, or the parent of the host in the tree; rank, which defines a relative relationship ....

B. Awerbuch, Optimal distributed algorithm for minimum weight spanning tree, counting, leader election and related problems, Proceedings of the 19th ACM Symposium on Theory of Computing, ACM, pp. 230-240, 1987.

....whp 1 and terminates in O(log n log ) time whp, where n is the number of nodes in the network. Our algorithm consists of a simple local procedure in which each node repeatedly performs a small number of xed operations. For our analysis, we adopt a standard synchronous model of computation [2, 3, 21, 25] in which, in a communication step, each node can exchange a message with each of its neighbors; the time complexity of an algorithm is measured by the total number of communication steps. Our focus in this paper is on the time complexity and the approximation factor achieved; we do not attempt to ....

B. Awerbuch. Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pages 230-240, May 1987.

....node IDs; another standard assumption is that an edge weight can be contained in a single message. Our lower bounds do not rely on any of these assumptions; our upper bounds (algorithms) do. In this B model (with B = O(log n) the classical distributed MST construction algorithms of [GHS83] and [A87] were communication optimal and required O(n log n) and O(n) time, respectively. Subsequently, the distributed MST construction algorithms of [GKP98] and [KP98] had time complexity O(D n 0:613 log n) and O(D p n log n) respectively. Recently, a near tight lower bound has been ....

B. Awerbuch, Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems, Proc. 19th Symp. on Theory of Computing, pp. 230--240, May 1987.

....optimality will be defined. 1. 1 Basic Algorithm of Gallager, Humblet and Spira In their pioneering paper [GHS83] Gallager, Humblet and Spira introduced the distributed MST problem and presented an algorithm that has formed the basis of subsequent work in the area, for example [CT85] Gaf85] [Awe87] and [Fal95] In their algorithm, each node is initially the root of its own fragment (a trivial connected subgraph of the MST) and all the edges are Unlabeled. Thereafter, adjacent fragments join to form larger fragments by labeling their intermediate edge as a Branch of the MST. The new branch ....

....procedure. If F joined finally with the chain, then it would get a great level increase and it would be compensated for the delay. However, this is not guaranteed, since F could end up joining with some other fragment Fx and get a very small level increase. 1. 2 Awerbuch s Optimal Algorithm In [Awe87], Awerbuch proposed an innovative three phase distributed MST algorithm, which achieves optimal performance in terms of both message and time complexity. The different phases represent a tradeoff between the demands of the initial part of the problem (involving large numbers of small fragments, ....

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B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. Proc. 19th Symp. on Theory of Computing, pages 230--240, May 1987.

....none of them has any special status nor is aware of any network topology except for its adjacent edges. Thus, we cannot simply send all the information to one node and solve a centralized problem, because this would involve a Leader Election problem which turns out to be comparably difficult [Awe87]. Fortunately, finding distributed MST algorithms is straightforward because the centralized MST problem can be easily solved by greedy algorithms. A distributed algorithm was proposed by Gallager et al. [GHS83] requiring O(E N log(N ) messages and O(N log(N ) time units. We refer to this ....

....the basic algorithm is O( DMST d) Delta log(N ) units, where DMST is the diameter of the resulting MST and d is the maximum degree of the nodes. Following [GHS83] various authors have proposed enhancements to the basic algorithm to achieve efficient time and message complexity [CT85] Gaf85] [Awe87], FM95] and [Fal95] As it will become apparent later, there is a trade off between termination time and messages. The first contribution of this paper is the introduction of further refinements to the basic algorithm, which are based on a novel approach called Distributed Information. The ....

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B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. Proc. 19th Symp. on Theory of Computing, pages 230--240, May 1987.

.... = u; v) if u and v have same label, then e does not connect different clusters, and we do not use e in this stage otherwise, e has weight by triple fR; min(u; v) max(u; v)g, where R = 1 if u in S or v in S, and R = 2 if neither is in S Run minimum spanning tree algorithm on weighted edges [13] Lemma 1. Algorithm I approximates C with a performance ratio of 3H ( Delta) in O( n jCj) Delta) time, using O(njCj m n log n) messages. Proof The ratio of jSj=jC j is at most H ( Delta) 7] The MST algorithm finds jSj Gamma 1 edges and adds at most two nodes per edge to C. Therefore, ....

....over all nodes and rounds, we get the O(njCj) term in the number of messages and the O(jCj Delta) term in the time bound. For the MST algorithm, we could use a sub linear time algorithm [17,15] but these algorithms use a large number of messages. Instead, we use the MST algorithm of Awerbuch [13], with modifications by Faloutsos and Molle [14] This 10 algorithm takes O(n Delta) time in our model, using O(m n log n) time. 4. Optimal Spine Routing (OSR) In Optimal Spine Routing, the spine is used to provide optimal, up to date routes to sources in reply to route queries or update ....

B. Awerbuch, "Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems," in Proceedings of the 19th Annual ACM Symposium on Theory of Computing, (1987) 230--240.

....the model of computation, that is, can we use only a EREW PRAM to achieve the same time and processor bounds As mentioned in Section 2, Cheng et al. 16] present a distributed algorithm for multiple edge and node insertions and deletions. Unfortunately, the basic distributed MST algorithms [25, 26, 27, 28] do not appear to lend themselves to the same implicit simulation that FNR 24 and FNRP use. Designing an efficient distributed algorithm for ANR remains an open problem. 25 ....

B. Awerbuch. "Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems." In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (1987), 230--240.

....Gallager et al. ensured a lower bound on a fragments level. In a later work, Chin and Ting [CT85] improved Gallager s algorithm to O(n Delta lg (n) time, estimating the fragment s size and updating its level accordingly, thus making a fragment s level dependent upon its estimated size. In [Awe87], Awerbuch proposed an optimal O(n) time and O(n Delta lg(n) 2 Delta m) message complexity algorithm, constructed in three phases. In the first phase, the number of nodes in the graph is established. In the second phase, a MST is built according to Gallager s algorithm, until the fragments ....

B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. Proc. 19th Symp. on Theory of Computing, pages 230 -- 240, May 1987.

....halved. Gallager et al. ensured a lower bound on a fragments level. In a later work, Chin and Ting [4] improved Gallager s algorithm to O(n Delta lg (n) time, estimating the fragment s size and updating its level accordingly, thus making a fragment s level dependent upon its estimated size. In [1], Awerbuch proposed an optimal O(n) time and O(n Delta lg(n) 2 Delta m) message complexity algorithm, constructed in three phases. In the first phase, the number of nodes in the graph is established. In the second phase, a MST is built according to Gallager s algorithm, until the fragments ....

B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. Proc. 19th Symp. on Theory of Computing, pages 230 -- 240, May 1987.

....The sequence of solutions to the Leader Election problem (LE) exemplify the reasoning above. Following the O(n log n) running time, and a first improvement by Chin and Ting [CT] and Gafni [G] with an O(n log n) running time) Awerbuch gave an optimal O(n) time solution to the problem [A1]. Again, this solution is optimal in the sense that there exist networks for which this is the best possible. This type of optimality may be thought of as existential optimality. Namely, there are points in the class of input instances under consideration, for which the algorithm is optimal. A ....

.... before as a canonical example for a graph algorithmic problem whose communication efficient distributed solution poses some surprisingly nontrivial subtleties [GHS] The time complexity of the algorithm of [GHS] is O(n log n) which was later improved to the (existentially) optimal O(n) in [A1]. As with other problems, such as the above LE example, it is natural to ask whether O(n) is universally optimal, or it can be improved. Once again, the MST problem proves to be a worthy candidate for this type of study. In other tree constructions, such as the Breadth First Search (BFS) tree ....

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B. Awerbuch, Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems, Proc. 19th Symp. on Theory of Computing, pp. 230--240, May 1987.

....Asynchronous Network Let s consider a static network (no server or network failures) with asynchronous communication. We distinguish two steps in this situation. Initialization: Construct a spanning tree over the given network. This is a well studied problem with a wide range of solutions [GHS83, Awe87]. In the appendix we present a simple algorithm that could be used for the Initialization procedure . This procedure assumes a stable underlying network and builds a spanning tree over the connected nodes (Table 1) Synchronization: The algorithm emulates a synchronous environment, by ....

....topology change any number of failures that are detected simultaneously . The present algorithm has constant overhead in time and space, and log n overhead in number of messages sent per link per set of topological changes . The logarithmic factor is the price paid to compute a spanning tree [GHS83, Awe87]. Previous solutions based on group communication require linear (# n) where n is the number of nodes) overhead in communication and space, because of the inherent membership learning component, making them Note that this is a worst case analysis. If only one failure occurs at a given moment, ....

Baruch Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems (detailed summary). In ACM Symposium on Theory of Computing, pages 230240, 1987.

....as one topology change as any number of failures that are detected simultaneously . The present algorithm has constant overhead in time and space, and log n overhead in number of messages sent per link per topological change. The logarithmic factor is the price paid to compute a spanning tree [GHS83, Awe87]. Previous solutions based on group communication require linear( n) where n is the number of nodes) overhead in communication and space, because of the inherent membership learning component, making them unscalable. Membership learning means that all the nodes must know the ID of all other ....

....Network Solution Let s consider a static network (no server or network failures) with asynchronous communication. We distinguish two steps in this situation. 4 Initialization: Construct a spanning tree over the given network. This is a well studied problem with a wide range of solutions [GHS83, Awe87]. In the appendix we give a simple possible solution. Synchronization: The algorithm will simply emulate a synchronous environment, by maintaining a running virtual clock . Each pulse of the clock is generated by ooding a message over the xed tree constructed in the previous step and ....

Baruch Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems (detailed summary). In ACM Symposium on Theory of Computing, pages 230-240, 1987.

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B. Awerbuch. Optimal Distributed Algorithms for Minimum Weight Spanning Tree, Counting, Leader Election, and Related Problems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), New York City, New York, May 1987.

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B. Awerbuch, "Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems," in Proceedings of STOC, 1987, pp. 230--240.

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B. Awerbuch, Optimal distributed algorithms for minimum-weight spanning trees, counting, leader election and related problems, in Proc., 19th Symp. on Theory of Computing, pp. 230 - 240, 1987.

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B. Awerbuch, "Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems," in Proceedings of STOC, 1987, pp. 230--240.

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Baruch Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. In Proceedings of the 19 ACM Symposium on Theory of Computing , 230-240, 1987.

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B. Awerbuch. Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems. In Proc. 19th Symp. on Theory of Computing, pages 230-240, 1987.

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B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. In Proc. 19th Symp. on Theory of Computing, pages 230-240, May 1987.

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B. Awerbuch. Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (1987), pp. 230--240.

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: B. Awerbuch, "Optimal Distributed Algorithms for Minimum Weight Spanning Tree, Counting, Leader Election and related problems", Proc. 19th Symp. on Theory of Computing, May 1987, pp. 230-240.

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