$\renewcommand{\Re}{\mathbb{R}}$We present a new optimal construction of a semi-separated pair decomposition (i.e., SSPD) for a set of $n$ points in $\Re^d$. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties. As an application of the new construction, for a fixed $t>1$, we present a new construction of a $t$-spanner with $O(n)$ edges and maximum degree $O(\log^2 n)$ that has a separator of size $O\pth{n^{1-1/d}}$.

翻译：$\\renewcommand{\\Re}{\\mathbb{R}}$我们提出了一种新的最优半分离对分解（即SSPD）构造方法，适用于$\\Re^d$空间中包含$n$个点的点集。在新构造中，每个点仅参与少量对，且该方法易于推广至低倍增维度的空间。这是首个具备这些性质的最优构造。作为新构造的应用，针对固定参数$t>1$，我们提出了一种新的$t$-生成子构造，其具有$O(n)$条边和$O(\\log^2 n)$的最大度，且包含一个大小为$O\\pth{n^{1-1/d}}$的分隔器。