We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)Δ$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $Δ= \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2Δ- 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

翻译：我们在LOCAL模型中获得了计算更平衡的定向和度分裂的更优算法。我们的关键结果与超图无汇定向问题相关联[BMNSU, SODA'25]。我们设计了一个复杂度为$\mathcal{O}(\varepsilon^{-1} \cdot \log n)$的算法，用于计算一个平衡定向，使得每个顶点$v \in V$的偏差不超过$\varepsilon \cdot \mathrm{deg}(v)$。这改进了[GHKMSU, Distrib. Comput. 2020]的先前结果，其复杂度为$\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$。此外，我们证明这一结果也能扩展到计算具有相同偏差和运行时间的无向度分裂。作为应用，我们证明在LOCAL模型中，$(3/2 + \varepsilon)Δ$边着色现在可以在$\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$轮内解决。注意，对于常数$\varepsilon$和$Δ= \mathcal{O}(2^{\log^{1/3} n})$，此运行时间与[Ghaffari & Kuhn, FOCS'21]中针对$(2Δ- 1)$边着色的当前最优结果相匹配。