We introduce a generalized family of $\left( 2\cdot \left\lfloor \frac{k}{2} \right\rfloor-1, 2\cdot \left\lceil \frac{k}{2} \right\rceil \cdot W_{1} +\max\left\{0,2\cdot\left(\left\lceil\frac{k}{2}\right\rceil-2\right)\right\}\cdot W_{2} \right)$-emulators with $\tilde O \left(n^{1+\frac{1}{k}}\right)$ edges, for any $k\in\mathbb{N}$, where $W_{i}$ is the $i$th heaviest edge on a shortest path between two vertices. Our construction generalizes the $+2W_{1}$-spanner of size $\tilde O\left(n^{\frac{3}{2}}\right)$ and the $+4W_{1}$-emulator of size $\tilde O \left(n^{\frac{4}{3}}\right)$, both by Elkin, Gitlitz and Neiman [DISC'21 and DICO'23]. When $k$ is even, these are $\left(k-1,k\cdot W_{1} + \left(k-4\right)\cdot W_{2}\right)$-emulators and when $k$ is odd, these are $\left(k-2,\left(k+1\right)\cdot W_{1} + \left(k-3\right) \cdot W_{2}\right)$-emulators. Our framework not only expands known constructions for weighted graphs but also yields an improved stretch over state of the art emulators and spanners for unweighted graphs within a specific distance regime. In particular, for all vertex pairs separated by a distance of $δ\leq O\left(3^{k^{2}}\right)$, our construction improves upon the seminal additive $+\tilde O\left(δ^{1-\frac{1}{k}}\right)$-emulator of size $\tilde O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$ by Thorup and Zwick [SODA'06].

翻译：我们引入一族广义的$\left( 2\cdot \left\lfloor \frac{k}{2} \right\rfloor-1, 2\cdot \left\lceil \frac{k}{2} \right\rceil \cdot W_{1} +\max\left\{0,2\cdot\left(\left\lceil\frac{k}{2}\right\rceil-2\right)\right\}\cdot W_{2} \right)$-仿真器，其边数为$\tilde O \left(n^{1+\frac{1}{k}}\right)$，对任意$k\in\mathbb{N}$成立，其中$W_{i}$是两顶点间最短路径上的第$i$重边。我们的构造推广了Elkin、Gitlitz和Neiman [DISC'21和DICO'23] 的$+2W_{1}$-展缩图（边数为$\tilde O\left(n^{\frac{3}{2}}\right)$）和$+4W_{1}$-仿真器（边数为$\tilde O \left(n^{\frac{4}{3}}\right)$）。当$k$为偶数时，这些是$\left(k-1,k\cdot W_{1} + \left(k-4\right)\cdot W_{2}\right)$-仿真器；当$k$为奇数时，这些是$\left(k-2,\left(k+1\right)\cdot W_{1} + \left(k-3\right) \cdot W_{2}\right)$-仿真器。我们的框架不仅扩展了加权图的已知构造，还在特定距离范围内对无权图的现有最优仿真器和展缩图实现了改进的拉伸。特别地，对于所有距离为$δ\leq O\left(3^{k^{2}}\right)$的顶点对，我们的构造改进了Thorup和Zwick [SODA'06] 开创性的加法$+\tilde O\left(δ^{1-\frac{1}{k}}\right)$-仿真器（边数为$\tilde O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$）。