A geometric $t$-spanner for a set $S$ of $n$ point sites is an edge-weighted graph for which the (weighted) distance between any two sites $p,q \in S$ is at most $t$ times the original distance between $p$ and $q$. We study geometric $t$-spanners for point sets in a constrained two-dimensional environment $P$. In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let $S$ be a set of $n$ point sites in a simple polygon $P$ with $m$ vertices. We present an algorithm to construct, for any constant $\varepsilon>0$ and fixed integer $k \geq 1$, a $(2k + \varepsilon)$-spanner with complexity $O(mn^{1/k} + n\log^2 n)$ in $O(n\log^2n + m\log n + K)$ time, where $K$ denotes the output complexity. When we consider sites in a polygonal domain $P$ with holes, we can construct such a $(2k+\varepsilon)$-spanner of similar complexity in $O(n^2 \log m + nm\log m + K)$ time. Additionally, for any constant $\varepsilon \in (0,1)$ and integer constant $t \geq 2$, we show a lower bound for the complexity of any $(t-\varepsilon)$-spanner of $\Omega(mn^{1/(t-1)} + n)$.

翻译：设$S$为平面上$n$个点位置的集合，几何$t$-扳手是边赋权图，其中任意两个位置$p,q \in S$间的（赋权）距离至多为$p$与$q$原始距离的$t$倍。我们研究受限二维环境$P$中点集的几何$t$-扳手。在此类情形中，扳手边可能具有非常数复杂度。为此，我们引入一项新性质：扳手复杂度，即扳手中所有边的总复杂度。设$S$为简单多边形$P$内$n$个点位置构成的集合，$P$有$m$个顶点。我们提出一种算法，对任意常数$\varepsilon>0$及固定整数$k \geq 1$，能在$O(n\log^2n + m\log n + K)$时间内构造复杂度为$O(mn^{1/k} + n\log^2 n)$的$(2k + \varepsilon)$-扳手，其中$K$表示输出复杂度。当考虑带孔洞的多边形域$P$中的位置时，我们可在$O(n^2 \log m + nm\log m + K)$时间内构造复杂度类似的$(2k+\varepsilon)$-扳手。此外，对任意常数$\varepsilon \in (0,1)$及整数常数$t \geq 2$，我们证明任意$(t-\varepsilon)$-扳手的复杂度下界为$\Omega(mn^{1/(t-1)} + n)$。