Random selection, leader election, and collective coin flipping are fundamental tasks in fault-tolerant distributed computing. We study these problems in the full-information model where despite decades of study, key gaps remain in our understanding of the trade-offs between round complexity, communication per player in each round, and adversarial resilience. We make progress by proving improved bounds for these problems. We first show that any $k$-round coin flipping protocol over $\ell$ players, each player sending one bit per round, can be biased by $O(\ell/\log^{(k)}(\ell))$ bad players. We obtain a similar lower bound for leader election. This strengthens prior best bounds [RSZ, SICOMP 2002] of $O(\ell/\log^{(2k-1)}(\ell))$ for coin flipping protocols and $O(\ell/\log^{(2k+1)}(\ell))$ for leader election protocols. Our result implies that any (1-bit per player) protocol tolerating linear fraction of bad players requires at least $\log^* \ell$ rounds, showing existing protocols [RZ, JCSS 2001; F, FOCS 1999] are near-optimal. We next initiate the study of one-round, (1-bit per player) random selection. For all $m\ge (\log(\ell))^2$, we obtain an optimal protocol (a first in the full information model for any task): We construct a protocol resilient to $O(\ell / m)$ bad players that outputs $m$ uniform random bits. And, we show that any protocol that outputs $m$ uniform random bits can be corrupted using $O(\ell / m)$ bad players. This also implies a one-round leader election protocol resilient to $\ell / (\log \ell)^2$ bad players, improving the prior best protocol [RZ, JCSS 2001] which was resilient to $\ell / (\log \ell)^3$ bad players. Our resilience matches that of the best one-round coin flipping protocol by Ajtai & Linial. To obtain our lower bound, we introduce multi-output influence, an extension of influence of boolean functions to the multi-output setting.

翻译：随机选择、领导者选举和集体抛硬币是容错分布式计算中的基本任务。我们在全信息模型下研究这些问题，尽管已有数十年的研究，但在轮复杂度、每轮每玩家通信量以及对抗性鲁棒性之间的权衡方面仍存在关键空白。我们通过证明这些问题的改进界取得了进展。首先，我们证明任何基于ℓ个玩家、每轮每位玩家发送一比特的k轮抛硬币协议，可被O(ℓ/log^(k)(ℓ))个恶意玩家偏置。我们获得了领导者选举的类似下界。这加强了先前的最佳结果[RSZ, SICOMP 2002]：抛硬币协议为O(ℓ/log^(2k-1)(ℓ))，领导者选举协议为O(ℓ/log^(2k+1)(ℓ))。我们的结果表明，任何容忍线性比例恶意玩家的（每位玩家一比特）协议至少需要log^* ℓ轮，这表明现有协议[RZ, JCSS 2001; F, FOCS 1999]已接近最优。接下来，我们开创性地研究了单轮、每位玩家一比特的随机选择问题。对于所有m ≥ (log(ℓ))^2，我们获得了一个最优协议（在全信息模型中针对任何任务的首次）：我们构造了一个能容忍O(ℓ/m)个恶意玩家并输出m个均匀随机比特的协议。并且，我们证明任何输出m个均匀随机比特的协议可被O(ℓ/m)个恶意玩家破坏。这也蕴含一个能容忍ℓ/(log ℓ)^2个恶意玩家的单轮领导者选举协议，改进了先前最佳协议[RZ, JCSS 2001]中容忍ℓ/(log ℓ)^3个恶意玩家的结果。我们的鲁棒性与Ajtai & Linial提出的最优单轮抛硬币协议匹配。为获得下界，我们引入了多输出影响度，这是布尔函数影响度在多输出设置中的扩展。