# Fast distributed construction of k-dominating sets and applications

## Abstract

This paper presents a fast distributed algorithm to compute a small k-dominating set D (for any fixed k) and its induced graph partition (breaking the graph into radius k clusters centered around the vertices of D). The time complexity of the algorithm is O(k log* n). Small k-dominating sets have applications in a number of areas, including routing with sparse routing tables via the scheme of [PU], the design of distributed data structures [P2], and center selection in a distributed network (cf. [BKP]). The main application described in this paper concerns a fast distributed algorithm for constructing a minimum weight spanning tree (MST). On an n-vertex network of diameter d, the new algorithm constructs an MST in time O(√n log* n + d), improving on the results of [GKP]. The new MST algorithm is conceptually simpler than the three-phase algorithm of [GKP]. In addition to exploiting small k-dominating sets, it uses a very simple convergecast protocol to inform a center about graph edges, that avoids forwarding messages about edges that close cycles. This convergecast protocol is similar to the one used in the third phase of the algorithm of [GKP], and most of the novelty lies in a new careful analysis proving that the convergecast process is fully pipelined, and no waiting occurs at intermediate nodes. This enables the new algorithm to skip the complicated second phase of the algorithm of [GKP].