AI 中文总结
研究有限有向图中顶点间基于桥接距离的度量，定义了\(a\)到\(b\)的距离\(\mu(a,b)\)，可多项式时间计算且满足度量不等式，证明其是某不等式在\(r = s \rightarrow 0\)时的特殊极限情况，还涉及其他三种极限情况。
AI 中文摘要
设\(G=(V,E)\)为有限有向图，给每条有向边\(e\in E\)赋予非负实长度\(\mu_e\)，非边的\(\mu_e = +\infty\)。固定\(V\)中任意两个不同顶点\(a,b\)。从\(a\)到\(b\)的有向路径称为\((a,b)\)-路径，属于所有\((a,b)\)-路径的边\(e\)称为\((a,b)\)-桥。所有\((a,b)\)-路径按相同顺序经过所有\((a,b)\)-桥。定义\(a\)到\(b\)的距离\(\mu(a,b)\)为所有\((a,b)\)-桥长度之和，无\((a,b)\)-路径时\(\mu(a,b)=\infty\)，有路径但无桥时\(\mu(a,b)=0\)。\(\mu(a,b)\)可在多项式时间计算，且满足度量不等式\(\mu(a,b) \leq \mu(a,c) + \mu(c,b)\)。本文将证明这是论文“有向图的度量和超度量不等式”中不等式\(\mu(a,b)^{s/r} \leq \mu(a,c)^{s/r} + \mu(c,b)^{s/r}\)在\(r = s \rightarrow 0\)时的特殊极限情况，还提及了其他三种极限情况。
英文摘要
Let $G = (V,E)$ be a finite directed graph with a non-negative real length $μ_e$ assigned to every directed edge $e \in E$. We assume that $μ_e = +\infty$ for every non-edge $e \not\in E$. Fix any two distinct vertices $a, b \in V$. A directed path from $a$ to $b$ is called an $(a,b)$-path. An edge $e$ is called an $(a,b)$-bridge if it belongs to all $(a,b)$-paths. Furthermore, it is not difficult to show that all $(a,b)$-paths pass all $(a,b)$-bridges in the same order. Define the distance $μ(a,b)$ from $a$ to $b$ as the sum of lengths of all $(a,b)$-bridges. Furthermore, $μ(a,b) = \infty$ if there are no $(a,b)$-paths and $μ(a,b) = 0$ if $(a,b)$-paths exist but there are no $(a,b)$-bridges. It is easily seen that $μ(a,b)$ can be computed in polynomial time and the metric inequality $μ(a,b) \leq μ(a,c) + μ(c,b)$ holds for every $a,b,c \in V$. Furthermore, equality holds if and only if each $(a,b)$-bridge is either an $(a,c)$- or a $(c,b)$-bridge. \newline We will show that this is a special limit case $r=s \rightarrow 0$ of the inequality $μ(a,b)^{s/r} \leq μ(a,c)^{s/r} + μ(c,b)^{s/r}$ obtained for all positive real parameters $r$ and $s$ in the paper ``Metric and ultrametric inequalities for directed graphs'', Discrete Appl. Math. 314 (2022) 93--104, along with 3 other limit cases $r=s \rightarrow \infty$, $r=1, s \rightarrow \infty$, and $s = 1, r \rightarrow 0$, considered in that paper.
