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 $G = (V, E)$，可加仿真器 $H = (V, E', w)$ 是一类稀疏加权图，能够以较小的可加误差保留 $G$ 中的所有距离。近期的研究趋势旨在确定当 $H$ 具有线性数量边时，在最稀疏设定下所能达到的最佳可加误差。具体而言，继 [Pettie, ICALP 2007] 之后，[Kogan and Parter, ICALP 2023] 的工作构造了可加误差为 $+O(n^{0.222})$ 的线性大小仿真器。已知最坏情况下的可加误差至少为 $+\Omega(n^{2/29})$（参考 [Lu, Vassilevska Williams, Wein, and Xu, SODA 2022]）。我们提出一种简洁的线性大小仿真器构造方法，实现了可加误差 $+O(n^{0.191})$。我们的方法将 [Baswana, Kavitha, Mehlhorn, and Pettie, SODA 2005] 和 [Vassilevska Williams and Bodwin, SODA 2016] 发展的路径购买框架扩展至稀疏可加仿真器的设定中。