Abstract
In the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), the network is given a root node r, and needs to construct a minimumweight directed spanning tree, rooted at r and oriented outwards. In this paper we present the first subquadratic DMST algorithms in the distributed CONGEST network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions: a model where the communication links are bidirectional but can have different weights in the two directions; a model where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other.
Our algorithm is based on a variant of LovĂˇsz' DMST algorithm for the PRAM model, and uses a distributed singlesource shortestpath (SSSP) algorithm for directed graphs as a black box. In the bidirectional CONGEST model, our algorithm has roughly the same running time as the SSSP algorithm; using the stateoftheart SSSP algorithm, we obtain a running time of O~(min(sqrt{nD},sqrt{n}D^{1/4} + n^{3/5} +D)) rounds for the bidirectional communication case.
For the unidirectional communication model we give an O~(n) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in O~(n^{1/3}) rounds.
On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s,t)shortest path problem. Thus, in terms of round complexity, distributed DMST lies between singlesource shortest path and (s,t)shortest path.
BibTeX  Entry
@InProceedings{fischer_et_al:LIPIcs:2019:11323,
author = {Orr Fischer and Rotem Oshman},
title = {{A Distributed Algorithm for Directed MinimumWeight Spanning Tree}},
booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)},
pages = {16:116:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771269},
ISSN = {18688969},
year = {2019},
volume = {146},
editor = {Jukka Suomela},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11323},
URN = {urn:nbn:de:0030drops113236},
doi = {10.4230/LIPIcs.DISC.2019.16},
annote = {Keywords: Distributed Computing, Directed Minimum Spanning Tree, Minimum Arborescence, CONGEST}
}
Keywords: 

Distributed Computing, Directed Minimum Spanning Tree, Minimum Arborescence, CONGEST 
Collection: 

33rd International Symposium on Distributed Computing (DISC 2019) 
Issue Date: 

2019 
Date of publication: 

08.10.2019 