We introduce the \emph{local information cost} (LIC), which quantifies the amount of information that nodes in a network need to learn when solving a graph problem. We show that the local information cost 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 $\Omega(\frac{\text{LIC}_\gamma(P)}{\log\tau \log n})$ 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 spanner with multiplicative stretch $2t-1$ that consists of at most $O(n^{1+\frac{1}{t} + \epsilon})$ edges, where $\epsilon = O( {1}/{t^2} )$. More concretely, we show that any $O(\text{poly}(n))$-time spanner algorithm must send at least $\tilde\Omega(\tfrac{1}{t^2} n^{1+{1}/{2t}})$ bits. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for this problem. (See PDF for the full abstract.)

翻译：我们引入了局部信息代价（LIC），它量化了网络节点在解决图问题时需要学习的信息量。我们证明局部信息代价为分布式算法的通信复杂度提供了自然下界。在同步CONGEST KT1模型中（每个节点预先知道其邻居的ID），我们证明：对于任意以概率至多γ出错的τ轮算法，解决图问题P至少需要Ω( LIC_γ(P)/(logτ * logn) )比特。该结果首次为CONGEST KT1模型中的图问题建立了通信与时间的一般性权衡。通过推导乘法拉伸因子为2t-1、边数至多为O(n^{1+1/t+ε})（其中ε=O(1/t^2)）的图子图计算通信复杂度下界，我们展示了局部信息代价的应用方法。具体而言，任何O(poly(n))时间的子图算法至少需要发送Ω̃( (1/t^2) * n^{1+1/(2t)} )比特。此前该问题仅已知平凡下界Ω̃(n)比特。（完整摘要见PDF文件。）