论文标题
分布式MIS $ O（\ log \ log {n}）$醒着复杂性
Distributed MIS in $O(\log\log{n} )$ Awake Complexity
论文作者
论文摘要
最大独立集（MIS）是分布式图算法中的基本和最深思熟虑的问题之一。即使经过四十年的深入研究，最著名的（随机）错误算法在一般图上具有$ O（\ log {n}）$圆形复杂性[Luby，STOC，1986]（其中$ n $是节点的数量），而最不知名的下限是$ω（\ sqrt {\ sqrt {\ sqrt { Moscibroda，Wattenhofer，JACM 2016]。超越$ o（\ log {n}）$圆形复杂性上限或显示更强的下限是长期存在的开放问题。 我们的主要贡献是表明，与最著名的$ o（\ log n）$的圆形复杂度相比，可以计算出MIS的计算，并且绕过其基本的$ω（\ sqrt {\ sqrt {\ log {n}/\ log log \ log \ log \ \ \ \ {n}}）$ round Complactity bounder bundertity boundententy Exparnential exponenentallyly expeNentiallylysentiallyly exponalentiallylylylylyly。具体而言，我们表明MIS可以通过$ O（\ log \ log \ log {n}）$具有高概率（即，概率至少$ 1- n^{ - 1} $）中的随机分布式（Monte Carlo）算法计算。该算法的复杂性为$ o（（\ log^7 n）\ log \ log n）$。我们还表明，我们可以通过呈现一种随机分布的（Monte Carlo）算法来提高圆形复杂性，以减少清醒复杂性的成本，因为MIS的可能性很高，该算法很高的可能性可以计算出$ O（（（\ log \ log \ log \ log {n} n}）\ log^*n）\ log^*n）$（我们的算法在Conlest模型中起作用，其中大小$ o（\ log n）$位的消息每回合每回合都可以发送。
Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best-known (randomized) MIS algorithms have $O(\log{n})$ round complexity on general graphs [Luby, STOC 1986] (where $n$ is the number of nodes), while the best-known lower bound is $Ω(\sqrt{\log{n}/\log\log{n}})$ [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the $O(\log{n})$ round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of $O(\log n)$ and also bypassing its fundamental $Ω(\sqrt{\log{n}/\log\log{n}})$ round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in $O(\log\log{n} )$ awake complexity with high probability (i.e., with probability at least $1 - n^{-1}$). This algorithm has a round complexity of $O((\log^7 n) \log \log n)$. We also show that we can improve the round complexity at the cost of a slight increase in awake complexity, by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability, computes an MIS in $O((\log\log{n})\log^*n)$ awake complexity and $O((\log^3 n) (\log \log n) \log^*n)$ round complexity. Our algorithms work in the CONGEST model where messages of size $O(\log n)$ bits can be sent per edge per round.
