We introduce the 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 $KT_1$ 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 $KT_1$ model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing routing tables for all-pairs-shortest-paths (APSP) routing, as well as for 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 derive the following lower bounds in the CONGEST model under the $KT_1$ assumption: For constructing routing tables, we show that any $O(\text{poly}(n))$-time algorithm has a communication complexity of $\Omega( {n^2}/{\log^2 n} )$ bits. Our main result is for constructing graph spanners: We show that any $O(\text{poly}(n))$-time 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 these problems.

翻译：我们引入局部信息代价（local information cost, LIC），该量度量化了网络中的节点在解决图问题时需要学习的信息量。我们证明，局部信息代价构成了分布式算法通信复杂度的一个自然下界。针对同步CONGEST $KT_1$模型（其中每个节点预先知道其邻居的ID），我们证明：对于以概率最多$\gamma$出错、运行$\tau$轮的算法，解决图问题$P$需要至少$\Omega(\frac{\text{LIC}_\gamma(P)}{\log\tau \log n})$比特。这是首个在CONGEST $KT_1$模型中为图问题建立通信与时间之间普遍权衡的下界。我们通过导出计算全源最短路径（APSP）路由表以及计算乘法伸展度$2t-1$且边数不超过$O(n^{1+\frac{1}{t} + \epsilon})$（其中$\epsilon = O( {1}/{t^2} )$）的图拉伸子（spanner）的通信复杂度下界，展示了如何应用局部信息代价。具体而言，我们在CONGEST $KT_1$假设下导出以下下界：对于构建路由表，我们证明任何$O(\text{poly}(n))$时间算法的通信复杂度为$\Omega( {n^2}/{\log^2 n} )$比特。我们的主要结果针对图拉伸子构建：我们证明任何$O(\text{poly}(n))$时间算法必须发送至少$\tilde\Omega(\tfrac{1}{t^2} n^{1+{1}/{2t}})$比特。此前，对于这些问题仅已知一个平凡的$\tilde \Omega(n)$比特下界。