平面图中面距离模式的紧界
A Tight Bound for Facial Distance Patterns in Planar Graphs
AI总结：
该研究将平面图面距离模式的唯一模式数上界改进为$O(k^2)$，解决了相关猜想，带来度量压缩、距离预言机、分布式直径算法等改进，还得到了更优的集中式无权重平面图直径计算算法。
AI中文摘要：
设$G$为无向无权重平面图，$S=(s_0,\dots,s_{k-1})$是某指定面的顶点，按循环顺序排列。考虑存储任意顶点$v$到$S$所有顶点距离的向量，$v$的模式由该向量中每对连续值的差值构成。Li与Parter在STOC'19中证明，$G$所有顶点的唯一模式数的上界为$O(k^3)$。我们将该上界改进为$O(k^2)$，与已知下界匹配，解决了ISAAC'22中的一个猜想。该简洁证明由OpenAI的GPT 5.6-Sol模型得出。将此新上界代入已知结果，对无向无权重平面图有三个直接推论：(1) 改进Okamura-Seymour度量的压缩效果；(2) 降低常数时间精确距离预言机所需的空间；(3) 优化计算直径的最快分布式算法。我们还提出了一个此前未知且非平凡的推论：得到了一个（集中式）$\tilde{O}(n^{8/5})$时间的直径计算算法，优于SODA'18中针对带权有向平面图的$\tilde{O}(n^{5/3})$算法。因此，当前带权与无权重平面图的直径计算时间之间存在差距。
英文摘要：
Let $G$ be an undirected unweighted planar graph and let $S=(s_0,\dots,s_{k-1})$ be the vertices of a designated face, listed in cyclic order. Consider a vector that stores the distances from an arbitrary vertex $v$ to all vertices of $S$. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values in this vector. Li and Parter [STOC'19] proved an upper bound of $O(k^3)$ on the number of unique patterns over all vertices of $G$. We improve this to $O(k^2)$, matching a known lower bound and settling a conjecture in [ISAAC'22]. The simple proof was found by OpenAI's GPT 5.6-Sol model. Plugging this new bound into known results has the following three immediate implications for undirected unweighted planar graphs: (1) it gives an improved compression of the Okamura-Seymour metric (2) it improves the space required by constant-time exact distance oracles, and (3) it improves the fastest distributed algorithm for computing the diameter. We further present a previously unknown and nontrivial implication: a (centralized) $\tilde{O}(n^{8/5})$-time algorithm for computing the diameter, improving over the $\tilde{O}(n^{5/3})$ algorithm of [SODA'18] which works for weighted directed planar graphs. Thus, there is currently a gap between the time for computing the diameter between weighted and unweighted planar graphs.
