A distributed algorithm for directed minimum-weight spanning tree

In the directed minimum spanning tree problem (DMST, also called weight arborescence), we are given a weighted graph, and root node r. Our goal is to construct minimum-weight tree, rooted at r oriented outwards. We present first sub-quadratic DMST algorithm in distributed $${\mathsf {CONGEST}}$$ network model, where messages exchanged between nodes bounded size. consider three versions of model: communication links bidirectional but can have different weights two directions; unidirectional; Congested Clique all communicate directly with each other. based on variant Lovász’ for PRAM uses single-source shortest-path (SSSP) graphs as black box. our has roughly same running time SSSP that used box; using state-of-the-art due Chechik Mukhtar (in: Symposium principles computing (PODC), ACM, 2020, pp 464–473), obtain $${\widetilde{O}}(\sqrt{n}D^{1/4}+D))$$ rounds case. For unidirectional model give an $${\widetilde{O}}(n)$$ algorithm, show it nearly optimal. And finally, Clique, again matches best known algorithm: runs $${\widetilde{O}}(n^{1/3})$$ rounds. On negative side, adapt observation sequential setting models, least hard (s, t)-shortest path problem. Thus, terms round complexity, lies shortest path.