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.
From a practical perspective, given that MST construction can take as much as @math time even in low-diameter networks, it is worth investigating whether one can design distributed algorithms that run faster and output an approximate minimum spanning tree. The question of devising faster approximation algorithms for MST was raised in @cite_3 . Elkin @cite_5 later established a hardness result on distributed MST approximation, showing that approximating the MST problem on a certain family of graphs of small diameter (e.g., @math ) within a ratio @math requires essentially @math time. Khan and Pandurangan @cite_22 showed that there can be an exponential time gap between exact and approximate MST construction by showing that there exist graphs where any distributed (exact) MST algorithm takes @math rounds, whereas an @math -approximate MST can be computed in @math rounds. The distributed approximation algorithm of Khan and Pandurangan is message-optimal but not time-optimal.
@cite_5 @cite_22 @cite_3
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The design of distributed approximation protocols is a relatively new and rapidly developing area of research. However, so far, little progress has been made in the study of the hardness of distributed approximation. In this paper we initiate the systematic study of this subject and show strong unconditional lower bounds on the time-approximation trade-off of the distributed minimum spanning tree problem, and show some of its variants. We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in time O(D(G) + L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H-approximation to the MST for any (H , ,[1, ( log n)] ) . Our result also shows that there can be a significant time gap between exact and approximate MST computation: there exists graphs in which the running time of our approximation algorithm is exponentially faster than the time-optimal distributed algorithm that computes the MST. Finally, we show that our algorithm can be used to find an approximate MST in wireless networks and in random weighted networks in almost optimal O(D(G)) time. This paper presents a lower bound of @math on the time required for the distributed construction of a minimum-weight spanning tree (MST) in weighted n-vertex networks of diameter @math , in the bounded message model. This establishes the asymptotic near-optimality of existing time-efficient distributed algorithms for the problem, whose complexity is @math .