We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are $n$ players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex -- this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when $r$ rounds of interaction are allowed, at least one player needs to communicate $\Omega(n^{1/20^{r+1}})$ bits. In particular, with logarithmic bandwidth, finding an MIS requires $\Omega(\log\log{n})$ rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrovi\'c, and Rubinfeld [PODC 2018] that solves MIS in $O(\log\log{n})$ rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. To prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of edge-sharing between players. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.
翻译:我们考虑在共享黑板通信模式中找到一个最大独立的数据集(MIS) 的问题。 在共享的黑板通信模式中找到一个最大独立的数据集(MIS ), 并有顶端配置投入 。 虽然在其它分布式模式中已经很好地研究了MIS问题, 而共享的黑板或许是简单的运行周期模式, 而每个玩家看到其顶端的边缘事件。 这样, 每个边端都被其端点所知道, 并且因此由两个玩家共享。 玩家同时在共享的黑板上同时发布信息, 以计算所有玩家可见的黑板, 目的是计算图示的 MIS 。 至少在其它分布式模式中很好地研究了MIS问题, 也许简单最简单的广播模式, 我们的问题的下边框, 我们的圆环交易交易在这种模式中显示一个较低的连接点。 具体地说, 当允许用美元进行互动的时候, 至少一个玩家需要用 将 $Omelga(n) IMIS(n\20+Q), liver+1+%) 来将我们的新运行的游戏中, 。