This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in @math time and exchanges @math messages (both with high probability), where @math is the number of nodes of the network, @math is the diameter, and @math is the number of edges. This is the first distributed MST algorithm that matches the time lower bound of @math [Elkin, SIAM J. Comput. 2006] and the message lower bound of @math [, J.ACM 2015] (which both apply to randomized algorithms). The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires @math rounds and @math messages.
It is important to point out that this paper and all the prior results discussed above (including the prior MST results @cite_6 @cite_26 @cite_18 @cite_38 @cite_10 @cite_2 @cite_35 ) assume the so-called clean network model , a.k.a. @math @cite_44 (cf. Section ), where nodes do not have initial knowledge of the identity of their neighbors. However, one can assume a model where nodes do have such a knowledge. This model is called the @math model . Although the distinction between @math and @math has clearly no bearing on the asymptotic bounds for the time complexity, it is significant when considering message complexity. @cite_21 show that @math is a message lower bound for MST in the @math model, if one allows only (possibly randomized Monte Carlo) comparison-based algorithms, i.e., algorithms that can operate on IDs only by comparing them. (We note that all prior MST algorithms mentioned earlier are comparison-based, including ours.) Hence, the result of @cite_21 implies that our MST algorithm (which is comparison-based and randomized) is time- and message-optimal in the @math model if one considers comparison-based algorithms only.
@cite_38 @cite_18 @cite_26 @cite_35 @cite_21 @cite_6 @cite_44 @cite_2 @cite_10
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This paper considers the question of identifying the parameters governing the behavior of fundamental global network problems. Many papers on distributed network algorithms consider the task of optimizing the running time successful when an O(n) bound is achieved on an n-vertex network. We propose that a more sensitive parameter is the network's diameter @math . This is demonstrated in the paper by providing a distributed minimum-weight spanning tree algorithm whose time complexity is sublinear in n, but linear in @math (specifically, @math for @math ). Our result is achieved through the application of graph decomposition and edge-elimination-by-pipelining techniques that may be of independent interest. A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities. Initially each node knows only the weight of each of its adjacent edges. When the algorithm terminates, each node knows which of its adjacent edges are edges of the tree. For a graph with n nodes and e edges, the total number of messages required by our algorithm is at most 5nlogn+2e, and each message contains at most one edge weight or one node identity plus 3+logn bits. Although our algorithm has the same message complexity as the previously known algorithm by , the time complexity of our algorithm takes at most O(nG(n))+ time units, an improvement from Gallager's O(nlogn)+. A worst case O(nG(n)) is also possible. This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time O(Λ(G)+n⋅log∗n) due to Kutten and Peleg [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27], where Λ(G) denotes the diameter of the graph G. Peleg and Rubinovich [D. Peleg, V. Rubinovich, A near-tight lower bound on the time complexity of distributed MST construction, in: Proc. 40th IEEE Symp. on Foundations of Computer Science, 1999, pp. 253–261] have shown that Ω˜(n) time is required for constructing MST even on graphs of small diameter, and claimed that their result “establishes the asymptotic near-optimality” of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. In this paper we refine this claim, and devise a protocol that constructs the MST in Ω˜(μ(G,ω)+n) rounds, where μ(G,ω) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27] by a factor of Ω˜(n). Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest. This paper develops linear time distributed algorithms for a class of problems in an asynchronous communication network. Those problems include Minimum-Weight Spanning Tree (MST), Leader Election, counting the number of network nodes, and computing a sensitive decomposable function (e.g. majority, parity, maximum, OR, AND). The main problem considered is the problem of finding the MST. This problem, which has been known for at least 9 years, is one of the most fundamental and the most studied problems in the field of distributed network algorithms. Any algorithm for any one of the problems above requires at least O( E + V log V ) communication and O( V ) time in the general network. In this paper, we present new algorithms, which achieve those lower bounds. The best previous algorithm requires T( E + V log V ) in communication and T( V log V ) in time. Our result enables to improve algorithms for many other problems in distributed computing, achieving lower bounds on their communication and time complexities. This article presents a fast distributed algorithm to compute a smallk-dominating setD(for any fixedk) and to compute its induced graph partition (breaking the graph into radiuskclusters centered around the vertices ofD). The time complexity of the algorithm isO(klog*n). Smallk-dominating sets have applications in a number of areas, including routing with sparse routing tables, the design of distributed data structures, and center selection in a distributed network. The main application described in this article concerns a fast distributed algorithm for constructing a minimum-weight spanning tree (MST). On ann-vertex network of diameterd, the new algorithm constructs an MST in time, improving on previous results.