SIAM J. COMPUT. c  2005 Society for Industrial and Applied Mathematics Vol. 35, No. 1, pp. 120–131 MINIMUM-WEIGHT SPANNING TREE CONSTRUCTION IN O(log log n) COMMUNICATION ROUNDS * ZVI LOTKER † , BOAZ PATT-SHAMIR † , ELAN PAVLOV ‡ , AND DAVID PELEG § Abstract. We consider a simple model for overlay networks, where all n processes are connected to all other processes, and each message contains at most O(log n) bits. For this model, we present a distributed algorithm which constructs a minimum-weight spanning tree in O(log log n) communica- tion rounds, where in each round any process can send a message to every other process. If message size is Θ(n ǫ ) for some ǫ> 0, then the number of communication rounds is O(log 1 ǫ ). Key words. minimum-weight spanning tree, distributed algorithms AMS subject classifications. 05C05, 05C85, 68Q22, 68Q25, 68R10 DOI. 10.1137/S0097539704441848 1. Introduction. A minimum-weight spanning tree (MST) is one of the most useful distributed constructs, as it minimizes the cost associated with global opera- tions such as broadcasts and convergecasts. This paper presents an MST construction algorithm that works in O(log log n) communication rounds, where in each round each process can send O(log n) bits to every other process (intuitively allowing each mes- sage to contain the identity and weight of only a constant number of edges). Our result shows that an MST can be constructed with little communication: throughout the execution of the algorithm, each pair of processes exchanges at most O(log n log log n) bits; the overall number of bits sent is Θ(n 2 log n), which is optimal. The algorithm extends to larger message sizes, in the sense that the number of communication rounds is O(log 1 ǫ ) if each message can contain n ǫ bits for some ǫ> 0. Note that if messages are not restricted in size, then the MST can be trivially constructed in a single round of communication: each process sends all its information to all its neighbors, allowing each node to locally compute the MST. The number of communication rounds dominates the time complexity in situa- tions where latency is high and bandwidth is scarce. This may be the situation in some overlay networks. Briefly, the idea in overlay networks is to think of the un- derlying communication network (e.g., the Internet) as a “black box” that provides reliable point-to-point communication. On top of that network run distributed ap- plications. This approach (whose precursor is the Internet’s “end-to-end argument” [13]) is different from classical distributed models, where processes reside in networks nodes (i.e., switches or routers), and thus their implementation would require using low-level communication. Rather, the pragmatic view now is that distributed applica- ∗ Received by the editors February 11, 2004; accepted for publication (in revised form) April 27, 2005; published electronically September 8, 2005. A preliminary version of this work appeared in Proceedings of the 15th Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Diego, CA, 2003. http://www.siam.org/journals/sicomp/35-1/44184.html † Department of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (zvilo@eng.tau. ac.il, boaz@eng.tau.ac.il). ‡ Department of Computer Science, The Hebrew University, Jerusalem 91904, Israel (elan@cs.huji. ac.il). § Department of Computer Science and Applied Mathematics, The Weizmann Institute, Rehovot 76100, Israel (david.peleg@weizmann.ac.il). The work of this author was supported in part by a grant from the Israel Science Foundation. 120