Distributed network optimization problems, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require Ω(√ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an O(D^2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.
Some early work that tried to circumvent the @math bound was done by Khan and Pandurangan @cite_8 who argued that their MST algorithm performed in an manner on some restricted classes of graphs. In particular, they gave a @math round MST algorithm for (1) unit disk networks where weights match the distances, and (2) networks with IID random weights. Their approach used a novel parameter called local shortest path diameter'' that takes the edge weights into account. This makes their approach unsuitable for arbitrary weights and topologically constrained networks such as planar graphs.
@cite_8
2058191123
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.