New Additive Emulators

For a given (possibly weighted) graph G = (V, E), an additive emulator H is a weighted graph in V ×V that preserves the (all pairs) G-distances up to a small additive stretch. In their breakthrough result, [Abboud and Bodwin, STOC 2016] ruled out the possibility of obtaining o(n⁴/³)-size emulator with n^o(1) additive stretch. The focus of our paper is in the following question that has been explicitly stated in many of the prior work on this topic: What is the minimal additive stretch attainable with linear size emulators? The only known upper bound for this problem is given by an implicit construction of [Pettie, ICALP 2007] that provides a linear-size emulator with +O^e(n¹/⁴) stretch. No improvement on this problem has been shown since then. In this work we improve upon the long standing additive stretch of O^e(n¹/⁴), by presenting constructions of linear-size emulators with O^e(n^0.222) additive stretch. Our constructions improve the state-of-the-art size vs. stretch tradeoff in the entire regime. For example, for every ϵ > 1/7, we provide +n^f(ϵ) emulators of size O^e(n^1+ϵ), for f(ϵ) = 1/5 − 3ϵ/5. This should be compared with the current bound of f(ϵ) = 1/4 − 3ϵ/4 by [Pettie, ICALP 2007]. The new emulators are based on an extended and optimized toolkit for computing weighted additive emulators with sublinear distance error. Our key construction provides a weighted modification of the well-known Thorup and Zwick emulators [SODA 2006]. We believe that this TZ variant might be of independent interest, especially for providing improved stretch for distant pairs.