Given an input graph $G = (V, E)$, an additive emulator $H = (V, E', w)$ is a sparse weighted graph that preserves all distances in $G$ with small additive error. A recent line of inquiry has sought to determine the best additive error achievable in the sparsest setting, when $H$ has a linear number of edges. In particular, the work of [Kogan and Parter, ICALP 2023], following [Pettie, ICALP 2007], constructed linear size emulators with $+O(n^{0.222})$ additive error. It is known that the worst-case additive error must be at least $+\Omega(n^{2/29})$ due to [Lu, Vassilevska Williams, Wein, and Xu, SODA 2022]. We present a simple linear-size emulator construction that achieves additive error $+O(n^{0.191})$. Our approach extends the path-buying framework developed by [Baswana, Kavitha, Mehlhorn, and Pettie, SODA 2005] and [Vassilevska Williams and Bodwin, SODA 2016] to the setting of sparse additive emulators.

翻译：给定输入图 $G = (V, E)$，加性仿真器 $H = (V, E', w)$ 是一个稀疏加权图，它以较小的加性误差保留 $G$ 中的所有距离。最近的一系列研究试图确定在 $H$ 具有线性边数的最稀疏设置中可实现的最佳加性误差。具体来说，继 [Pettie, ICALP 2007] 之后，[Kogan 和 Parter, ICALP 2023] 的工作构建了具有 $+O(n^{0.222})$ 加性误差的线性规模仿真器。根据 [Lu, Vassilevska Williams, Wein 和 Xu, SODA 2022] 的研究，已知最坏情况下的加性误差必须至少为 $+\Omega(n^{2/29})$。我们提出了一种简单的线性规模仿真器构建方法，实现了 $+O(n^{0.191})$ 的加性误差。我们的方法将 [Baswana, Kavitha, Mehlhorn 和 Pettie, SODA 2005] 以及 [Vassilevska Williams 和 Bodwin, SODA 2016] 开发的路径购买框架扩展到了稀疏加性仿真器的设置。