Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths. We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to $ 1 + \frac{1}{\sqrt{2}} $. This bound is tight in the following sense. We also prove the gap $ 1 + \frac{1}{\sqrt{2}} $ between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use. Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis.

翻译：均匀代价-距离斯坦纳树旨在最小化总长度与从专用根节点到其他终端节点的加权路径长度之和。当该树用于信号传输（例如芯片设计或通信网络）时，此类问题具有重要应用。它是广义代价-距离斯坦纳树问题的一个特例，后者对总长度和路径长度采用不同的距离函数。我们将均匀代价-距离斯坦纳树问题的最佳公开近似因子从2.39改进至2.05。若我们能以任意精度近似最小长度斯坦纳树问题，则本算法实现的近似因子可任意接近$ 1 + \frac{1}{\sqrt{2}} $。该上界在以下意义上具有紧致性：我们还证明了最优解与所有现有近似算法及本文共同采用的下界之间存在间隙$ 1 + \frac{1}{\sqrt{2}} $。与先前方法类似，我们以近似最小长度斯坦纳树为初始解，将其分割为子树后重新连接。为改进近似因子，我们充分考虑代价结构，对分割组件的方法进行精细化处理，并显著增强了理论分析。