AI 中文总结
针对分布式图草图模型，我们证明了在微小误差下解决连通性和生成树构造问题需要多项式长度的消息，显著缩小了现有下界与上界之间的差距，并扩展至k边连通性。
AI 中文摘要
我们首次在微小误差机制下，为分布式图草图模型中的几个基本问题给出了多项式下界，该机制将确定性算法作为特例。在图草图模型中，每个节点向裁判发送一条消息，裁判对图没有任何先验知识，必须输出答案。虽然 Nelson 和 Yu (SODA 2019) 以及 Yu (SODA 2021) 的工作表明，对于构造生成森林或判断图是否连通且误差至多为 $\frac{1}{\text{poly}(n)}$，$\Theta( \log^3n )$ 是最优的，但他们的方法无法对显著更小的误差概率给出更强的界。我们的主要结果是证明：以误差至多 $\delta$ 解决连通性或生成树构造问题，需要长度为 $\Omega( \min\{n, \log_2 \frac{1}{\delta}\}^{1/3} )$ 的消息，这意味着具有指数级小误差的算法在最坏情况下必须发送 $\Omega( n^{1/3} )$ 比特的消息。我们的结果显著缩小了 Jelani-Yu 阈值 $\Theta( \log^3n )$ 与确定性图草图中每个节点发送 $O(n)$ 比特的平凡上界之间的现有差距。我们还将结果扩展到 $k$ 边连通性。对于任何 $k=O(n^{1/7})$，我们恢复了 Robinson 和 Tan (PODS 2026) 仅针对确定性算法所证明的、对于具有指数级小误差的算法消息长度的相同下界 $\Omega( k )$。最后，对于 $k=n^{o(1)}$，我们的结果意味着对于任何常数 $\epsilon<\tfrac{1}{3}$，更强的下界 $\Omega_\epsilon( n^{\epsilon} )$ 比特。
英文摘要
We present the first polynomial lower bounds for several fundamental problems in the distributed graph sketching model in the tiny-error regime, which includes deterministic algorithms as a special case. In the graph sketching model, every node sends a single message to the referee who does not have any prior knowledge of the graph and must output the answer. While the work of Nelson and Yu (SODA 2019) and Yu (SODA 2021) showed that $Θ( \log^3n )$ is optimal for constructing a spanning forest or deciding whether the graph is connected with error at most $\frac{1}{\text{poly}(n)}$ , their approach does not yield any stronger bounds for significantly smaller error probabilities. Our main result is to show that solving either connectivity or spanning tree construction with error at most $δ$ requires messages of length $Ω( \min\{n, \log_2 \frac{1}δ\}^{1/3} )$, which implies that algorithms with exponentially small error must send messages of $Ω( n^{1/3} )$ bits in the worst case. Our results significantly narrows the current gap between the Jelani-Yu threshold of $Θ( \log^3n )$ and the trivial upper bound of sending $O(n)$ bits per node for deterministic graph sketching. We also extend our results to $k$-edge connectivity. For any $k=O(n^{1/7})$, we recover the same bound of $Ω( k )$ on the message length for algorithms with exponentially small error that was shown by Robinson and Tan (PODS 2026) only for deterministic algorithms. Finally, for $k=n^{o(1)}$, our result implies a stronger lower bound of $Ω_ε( n^ε )$ bits, for any constant $ε<\tfrac{1}{3}$.
