We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost ($\textsf{LIC}$) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that $\Omega(\textsf{LIC}_\gamma(P)/ \log\tau \log n)$ bits are required for solving a graph problem $P$ with a $\tau$-round algorithm that errs with probability at most $\gamma$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a $(2t-1)$-spanner that consists of at most $O(n^{1+1/t + \epsilon})$ edges, where $\epsilon = \Theta(1/t^2)$. Our main result is that any $O(\textsf{poly}(n))$-time algorithm must send at least $\tilde\Omega((1/t^2) n^{1+1/2t})$ bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for this problem. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with $\tilde O(n)$ communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.

中文翻译：

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快速意味着健谈：Graph Spanner 的本地信息成本

我们引入了一种新的度量方法来量化网络中的节点需要学习以共同解决图问题的信息量。我们表明本地信息成本（$\textsf{LIC}$）呈现了分布式算法通信复杂度的自然下限。对于同步 CONGEST-KT1 模型，其中每个节点都有其邻居 ID 的初始知识，我们证明 $\Omega(\textsf{LIC}_\gamma(P)/ \log\tau \log n)$ 位是使用 $\tau$-round 算法解决图问题 $P$ 所需，该算法错误的概率至多为 $\gamma$。我们的结果是第一个下限，它在 CONGEST-KT1 模型中的图问题的通信和时间之间产生一般权衡。我们演示了如何通过计算最多包含 $O(n^{1+1/t + \epsilon} 的 $(2t-1)$-spanner 的通信复杂度的下限来应用本地信息成本)$ 边，其中 $\epsilon = \Theta(1/t^2)$。我们的主要结果是任何 $O(\textsf{poly}(n))$-time 算法必须至少发送 $\tilde\Omega((1/t^2) n^{1+1/2t})$ KT1 假设下的 CONGEST 模型中的位。以前，对于这个问题，只有 $\tilde \Omega(n)$ 位的微不足道的下界是已知的。我们下界的结果是，在设计分布式生成器算法时，不可能同时实现时间和通信最优。根据 King、Kutten 和 Thorup 的工作（PODC 2015），这表明在考虑具有 $\tilde O(n)$ 通信复杂度的算法时，计算最小生成树可以比找到生成器快得多。我们的结果还暗示了在 Augustine 等人的节点拥塞团中构造扳手的时间复杂度下限。(2019) 和有限带宽的推挽八卦模型。