来自自然的反馈：最大独立集选择和贪婪着色的简单随机分布算法
Feedback from Nature: Simple Randomised Distributed Algorithms for Maximal Independent Set Selection and Greedy Colouring
摘要：
我们提出了两个行之有效的问题，在极其恶劣的条件下有效地运作的分布式算法。我们的算法以一种简单而新颖的方式实现了最先进的性能。我们的最大独立集选择算法在相同的匿名处理器网络上运行。每个节点处的处理器没有关于网络的先验信息。在每个时间步，每个节点只能向所有邻居广播一个比特，或者保持沉默。每个节点可以检测是否有一个或多个邻居广播，但不能告诉它有多少邻居广播，或哪些邻居广播。我们建立在Afek等人最近的工作基础上，该工作受到果蝇~ \cite{Afek2011a}中细胞网络发育研究的启发。然而，我们第一次纳入了生物系统的另一个重要特征：根据相邻节点的局部反馈，改变每个节点使用的概率值。给定任意$n$节点网络，我们的算法实现了$O(\log n)$轮的最优期望时间复杂度和每个节点广播的$O(1)$单比特消息的最优期望消息复杂度，我们还表明，以前的方法，没有反馈，不能实现优于$Ω(\log^2 n)$的期望时间复杂度，无论使用全局方案来选择概率。我们的分布式贪婪着色算法在类似的苛刻条件下工作：每个相同的节点没有关于网络的先验信息，只能在每个时间步向所有邻居广播一条消息，代表所需的颜色，并且只能检测是否至少有一个邻居广播了每个颜色值。我们表明，我们的算法有一个预期的时间复杂度为$O(Δ+\log n)$，其中$Δ$是网络的最大程度，和预期的消息复杂度由每个节点广播的$O(1)$消息。
英文摘要：
We propose distributed algorithms for two well-established problems that operate efficiently under extremely harsh conditions. Our algorithms achieve state-of-the-art performance in a simple and novel way. Our algorithm for maximal independent set selection operates on a network of identical anonymous processors. The processor at each node has no prior information about the network. At each time step, each node can only broadcast a single bit to all its neighbours, or remain silent. Each node can detect whether one or more neighbours have broadcast, but cannot tell how many of its neighbours have broadcast, or which ones. We build on recent work of Afek et al. which was inspired by studying the development of a network of cells in the fruit fly~\cite{Afek2011a}. However we incorporate for the first time another important feature of the biological system: varying the probability value used at each node based on local feedback from neighbouring nodes. Given any $n$-node network, our algorithm achieves the optimal expected time complexity of $O(\log n)$ rounds and the optimal expected message complexity of $O(1)$ single-bit messages broadcast by each node.We also show that the previous approach, without feedback, cannot achieve better than $Ω(\log^2 n)$ expected time complexity, whatever global scheme is used to choose the probabilities. Our algorithm for distributed greedy colouring works under similar harsh conditions: each identical node has no prior information about the network, can only broadcast a single message to all neighbours at each time step representing a desired colour, and can only detect whether at least one neighbour has broadcast each colour value. We show that our algorithm has an expected time complexity of $O(Δ+\log n)$, where $Δ$ is the maximum degree of the network, and expected message complexity of $O(1)$ messages broadcast by each node.
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