A geometric $t$-spanner $\mathcal{G}$ on a set $S$ of $n$ point sites in a metric space $P$ is a subgraph of the complete graph on $S$ such that for every pair of sites $p,q$ the distance in $\mathcal{G}$ is a most $t$ times the distance $d(p,q)$ in $P$. We call a connection between two sites in the spanner a link. In some settings, such as when $P$ is a simple polygon with $m$ vertices and a link is a shortest path in $P$, links can consist of $\Theta (m)$ segments and thus have non-constant complexity. The total spanner complexity is a recently-introduced measure of how compact a spanner is. In this paper, we study what happens if we are allowed to introduce $k$ Steiner points to reduce the spanner complexity. We study such Steiner spanners in simple polygons, polygonal domains, and edge-weighted trees. Surprisingly, we show that Steiner points have only limited utility. For a spanner that uses $k$ Steiner points, we provide an $\Omega(nm/k)$ lower bound on the worst-case complexity of any $(3-\varepsilon)$-spanner, and an $\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the worst-case complexity of any $(t-\varepsilon)$-spanner, for any constant $\varepsilon\in (0,1)$ and integer constant $t \geq 2$. These lower bounds hold in all settings. Additionally, we show NP-hardness for the problem of deciding whether a set of sites in a polygonal domain admits a $3$-spanner with a given maximum complexity using $k$ Steiner points. On the positive side, for trees we show how to build a $2t$-spanner that uses $k$ Steiner points and of complexity $O(mn^{1/t}/k^{1/t} + n \log (n/k))$, for any integer $t \geq 1$. We generalize this result to forests, and apply it to obtain a $2\sqrt{2}t$-spanner in a simple polygon or a $6t$-spanner in a polygonal domain, with total complexity $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} + n\log^2 n)$.

翻译：设 $P$ 为度量空间，$S$ 为 $P$ 中 $n$ 个点位的集合，几何 $t$-生成图 $\mathcal{G}$ 是 $S$ 上完全图的子图，满足对于任意两个点位 $p,q$，$\mathcal{G}$ 中的距离不超过 $P$ 中距离 $d(p,q)$ 的 $t$ 倍。我们将生成图中两个点位之间的连接称为一条链路。在某些设置中（例如 $P$ 为具有 $m$ 个顶点的简单多边形，且链路为 $P$ 中的最短路径），链路可能由 $\Theta(m)$ 个线段组成，因而具有非常数复杂度。总生成图复杂度是近期提出的衡量生成图紧致性的指标。本文研究允许引入 $k$ 个 Steiner 点以降低生成图复杂度的情况。我们在简单多边形、多边形区域和边权树上研究此类 Steiner 生成图。令人惊讶的是，我们证明 Steiner 点的效用有限。对于使用 $k$ 个 Steiner 点的生成图，我们证明了任意 $(3-\varepsilon)$-生成图的最坏情况复杂度下界为 $\Omega(nm/k)$，且对任意常数 $\varepsilon\in (0,1)$ 和整数常数 $t \geq 2$，任意 $(t-\varepsilon)$-生成图的最坏情况复杂度下界为 $\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$。这些下界在所有设置中均成立。此外，我们证明判断多边形区域中点位集合是否允许使用 $k$ 个 Steiner 点构造最大复杂度为某值的 $3$-生成图的问题是 NP-难的。在正面结果方面，对于树，我们展示了如何构造一个使用 $k$ 个 Steiner 点且复杂度为 $O(mn^{1/t}/k^{1/t} + n \log (n/k))$ 的 $2t$-生成图（对任意整数 $t \geq 1$）。我们将该结果推广到森林，并将其应用于简单多边形中构造 $2\sqrt{2}t$-生成图或多边形区域中构造 $6t$-生成图，总复杂度为 $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} + n\log^2 n)$。