Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs

2026-04-29

AI Summary

This work addresses the problem of constructing graph emulators for undirected weighted graphs that achieve improved additive stretch while preserving sparsity. The authors propose a general construction framework that, for the first time, incorporates both the heaviest edge $W_1$ and the second-heaviest edge $W_2$ along shortest paths into the stretch analysis, thereby unifying and extending prior results on $+2W_1$ and $+4W_1$ emulators. By integrating graph sparsification, hierarchical clustering, and inter-cluster connection strategies, the method yields emulators with $\tilde{O}(n^{1+1/k})$ edges and additive stretch guarantees of $(k-1, kW_1 + (k-4)W_2)$ and $(k-2, (k+1)W_1 + (k-3)W_2)$. These bounds significantly outperform the classical Thorup–Zwick emulators for distances up to $\delta \leq O(3^{k^2})$.

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 $\tilde O \left(n^{1+\frac{1}{k}}\right)$ 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].

Problem

• emulators
• spanners
• weighted graphs
• stretch
• shortest paths

Innovation

• weighted emulators
• graph spanners
• additive stretch
• shortest paths
• sparse graph representations