摘要：This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT_1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., algorithms that have simultaneously optimal time and message complexity (up to polylogarithmic factors), for many fundamental problems in the standard KT_0 model (where nodes have only local knowledge of themselves and not their neighbors). The situation is less clear in the KT_1 model. In this paper, we present several new distributed algorithms in the KT_1 model that trade off between time and message complexity.
Our distributed algorithms are based on a uniform and general approach which involves constructing a sparsified spanning subgraph of the original graph - called a danner - that trades off the number of edges with the diameter of the sparsifier. In particular, a key ingredient of our approach is a distributed randomized algorithm that, given a graph G and any delta in [0,1], with high probability constructs a danner that has diameter O~(D + n^{1-delta}) and O~(min{m,n^{1+delta}}) edges in O~(n^{1-delta}) rounds while using O~(min{m,n^{1+delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. Using our danner construction, we present a family of distributed randomized algorithms for various fundamental problems that exhibit a trade-off between message and time complexity and that improve over previous results. Specifically, we show the following results (all hold with high probability) in the KT_1 model, which subsume and improve over prior bounds in the KT_1 model (King et al., PODC 2014 and Awerbuch et al., JACM 1990) and the KT_0 model (Kutten et al., JACM 2015, Pandurangan et al., STOC 2017 and Elkin, PODC 2017):
1) Leader Election, Broadcast, and ST. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,1]. 2) MST and Connectivity. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. In particular, for delta = 0.5 we obtain a distributed MST algorithm that runs in optimal O~(D+sqrt{n}) rounds and uses O~(min{m,n^{3/2}}) messages. We note that this improves over the singularly optimal algorithm in the KT_0 model that uses O~(D+sqrt{n}) rounds and O~(m) messages.
3) Minimum Cut. O(log n)-approximate minimum cut can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. 4) Graph Verification Problems such as Bipartiteness, Spanning Subgraph etc. These can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5].
