基于代数流言的有序最优信息传播
Order Optimal Information Spreading Using Algebraic Gossip
摘要：
在本文中，我们研究了基于流言的信息传播与有限的消息大小。我们使用代数流言传播$k$不同的消息到网络中的所有$n$节点。对于任意网络，我们给出了$O((k+\log n + D)Δ)$轮的高概率一致代数流言的一个新的上界，其中$D$和$Δ$分别是网络的直径和最大度.对于许多拓扑和$k$的选择，这个界改进了以前的结果，特别是，对于一个恒定的最大度的图，它意味着一致的八卦是\emph{order optimal}和停止时间是$Θ(k + D)$。为了消除$Δ$的因素从上限，我们提出了一个非均匀的八卦协议，TAG，它是基于代数八卦和任意生成树协议$§$。TAG的停止时间是$O(k+\log n +d(§)+t(§))$，其中$t(§)$是生成树协议的停止时间，$d(§)$是生成树的直径。我们提供了两个一般情况下，这个界限导致一个顺序最优协议。首先是$k=Ω(n)$，在那里，使用一个简单的八卦广播协议，创建一个生成树在最线性的时间，我们表明，TAG完成后，$Θ(n)$轮的任何图。第二个使用一个复杂的，最近的八卦协议，建立一个快速生成树的图与大弱电导。反过来，这导致了$k=Ω(\mathrm{polylog}(n))$在这些图上的TAG的最优性。在我们的证明中使用的技术依赖于排队论，这是一种有趣的方法，可以在未来的八卦分析中使用。
英文摘要：
In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate $k$ distinct messages to all $n$ nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of $O((k+\log n + D)Δ)$ rounds with high probability, where $D$ and $Δ$ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of $k$ this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is \emph{order optimal} and the stopping time is $Θ(k + D)$. To eliminate the factor of $Δ$ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol $§$. The stopping time of TAG is $O(k+\log n +d(§)+t(§))$, where $t(§)$ is the stopping time of the spanning tree protocol, and $d(§)$ is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for $k=Ω(n)$, where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after $Θ(n)$ rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for $k=Ω(\mathrm{polylog}(n))$. The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis.
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