Improved weighted additive spanners

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount work on additive spanners, so far little attention was given to weighted graphs. Only very recently as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). extended classical +2 (respectively, +4) for unweighted graphs size $$O(n^{3/2})$$ (resp., $$O(n^{7/5})$$ ) setting, where error is $$+2W$$ $$+4W$$ ). This means every pair u, v, stretch at most $$+2W_{u,v}$$ , $$W_{u,v}$$ maximal edge weight shortest $$u-v$$ path (weights normalized minimum 1). In addition, showed a randomized algorithm yielding $$+8W_{max}$$ spanner $$O(n^{4/3})$$ here $$W_{max}$$ maximum entire this we improve latter result devising simple deterministic $$+(6+\varepsilon )W$$ with (for any constant $$\varepsilon >0$$ ), thus nearly matching +6 Furthermore, show $$+(2+\varepsilon subsetwise $$O(n\cdot \sqrt{\vert S\vert })$$ improving $$+4W_{max}$$ (that had same size). We also emulator $${\tilde{O}}(n^{4/3})$$ . our technique applicable have linear size. It proved Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) such must suffer polynomially large stretches. For graphs, use variant yields $$+{\tilde{O}}(\sqrt{n}\cdot W)$$ spanner, obtain tradeoff between stretch. Finally, generalizing Dor D (SIAM J Comput 29:1740–1759, 2000) devise efficient producing $${\tilde{O}}(n^{3/2})$$ $${\tilde{O}}(n^2)$$ time.