发表机构
Technische Universität Berlin（柏林工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对动态网络多维渐近共识问题，提出非分裂网络模型下的新收缩上界，给出首个高维无坐标匿名确定性算法的收缩下界，证明BallMidpoint策略单轮渐近最优，改进了此前的收缩率结果。
AI 中文摘要
本研究探讨动态网络中的多维渐近一致性问题。假设节点拥有d维输入，需在保持向量处于所有输入的凸包内的同时，相互收敛至任意接近的状态。针对非分裂网络模型，我们给出渐近一致性收缩率的新上界√(2d/(3d+1))，其中d为输入维度，当d→∞时，该值趋近于√(2/3)≈0.816。为此，我们采用了近期为全对通信模型提出的BallMidpoint算法（Melnyk, 2026），该算法使节点选择所接收向量的最小包围球的中点。由于节点的凸包无公共交集，仅两两相交，该上界严格差于全对通信网络中容错近似一致性的收缩率√(1/2)≈0.707。我们的界改进了此前动态网络中通过MidExtremes算法（Függer和Nowak, 2018）得到的最优收缩率√(7/8)≈0.935。对于d≥8的非分裂网络中无坐标、无记忆、匿名的确定性算法的收缩，我们给出首个多维下界2/3，该结果表明，此前已知的n≥3且d=1时收缩率下界1/2在更高维度并不紧。若n为无穷大，我们将该下界扩展至√(1/2·d/(d+1))，当d→∞时，其趋近于√(1/2)。我们进一步证明，单轮收缩至少为√((2d-2)/(3d+7))，当d→∞时，该值趋近于√(2/3)，表明BallMidpoint策略在单轮中是渐近最优的。
英文摘要
This work studies the multidimensional asymptotic agreement problem in dynamic networks. We assume that nodes have $d$-dimensional inputs and need to converge arbitrarily close to each other's inputs while their vectors stay inside the convex hull of all inputs. We present a new upper bound of $\sqrt{2d/(3d+1)}$ on the contraction rate of asymptotic agreement in the non-split network model, where $d$ is the dimension of the input and $\sqrt{2d/(3d+1)}\rightarrow \sqrt{2/3}\approx 0.816$ for $d\rightarrow\infty$. To this end, we adapt the BallMidpoint algorithm (Melnyk, 2026) recently introduced for the all-to-all communication model. This algorithm lets the nodes choose the midpoint of the smallest enclosing ball of the received vectors. This bound is strictly worse than the $\sqrt{1/2}\approx 0.707$ contraction rate for fault-tolerant approximate agreement in all-to-all communication networks because the convex hulls of the nodes do not have a common intersection, and only intersect pairwise. Our bound improves over the previously best-known contraction rate of $\sqrt{7/8}\approx 0.935$ for dynamic networks via the MidExtremes algorithm (Függer and Nowak, 2018). We present the first multi-dimensional lower bound of $2/3$ for the contraction of coordinate-free memoryless anonymous deterministic algorithms in non-split networks, for $d\ge 8$. This result shows that the previously best-known lower bound of $1/2$ on the contraction rate for $n\ge 3$ and $d=1$ is not tight in higher dimensions. If $n$ is infinite, we extend this lower bound to $\sqrt{1/2\cdot d/(d+1)}$, which converges to $\sqrt{1/2}$ for $d\rightarrow \infty$. We further show that for a single round, the one-round contraction is at least $\sqrt{\frac{2d-2}{3d+7}}$, which converges to $\sqrt{2/3}$ for $d\rightarrow\infty$, showing that the BallMidpoint strategy is asymptotically optimal for one round.
Comments21 pages, 2 figures
