A faster distributed protocol for constructing a minimum spanning tree

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) + sqrt(n) ṡ 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 over(Ω, ̃) (sqrt(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 over(Ω, ̃) (μ (G, ω) + sqrt(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 over(Ω, ̃) (sqrt(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.