Distributed Minimum Degree Spanning Trees

Abstract

The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree T for graph G, such that the maximum degree of T is the smallest among all spanning trees of G. Let d be this MDST degree for a given graph. In this paper, we present a randomized distributed approximation algorithm for the MDST problem that constructs a spanning tree with maximum degree in O(d+log n ). With high probability in n, the algorithm runs in O((D + √n) log4 n) rounds, in the broadcast-CONGEST model, where D is the graph diameter and n is the graph size. We then show how to derandomize this algorithm, obtaining the same asymptotic guarantees on degree and time complexity, but now requiring the standard CONGEST model. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990's (finding an exact solution is NP-hard), our algorithms are the first efficient distributed solutions. We conclude by proving a lower bound that establishes that any randomized MDST algorithm that guarantees a maximum degree in ∼Ω (d) requires &Ø#8764;Ω (n1/3) rounds, and any deterministic solution requires ∼Ω (n1/2) rounds. These bounds proves our deterministic algorithm to be asymptotically optimal, and eliminates the possibility of significantly more efficient randomized solutions.