Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs

Abstract: 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 $\tilde O \left(n<sup>{1+\frac{1}{k}}\right)$ 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 $\tilde O\left(n<sup>{\frac{3}{2}}\right)$ and the $+4W_{1}$-emulator of size $\tilde O \left(n<sup>{\frac{4}{3}}\right)$, 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 $δ\leq O\left(3<sup>{k<sup>{2}}\right)$, our construction improves upon the seminal additive $+\tilde O\left(δ<sup>{1-\frac{1}{k}}\right)$-emulator of size $$\tilde O\left(n<sup>{1+\frac{1}{2<sup>{k+1}-1}}\right)$$ by Thorup and Zwick [SODA'06].