The Complexity of Geodesic Spanners using Steiner Points

TL;DR

This paper studies Steiner spanners in simple polygons, polygonal domains, and edge-weighted trees, and shows 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 Steiner points.

Abstract

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 a \emph{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 $Θ(m)$ segments and thus have non-constant complexity. The spanner complexity is a measure of how compact a spanner is, which is equal to the sum of the complexities of all links in the spanner. 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. We show that Steiner points have only limited utility. For a spanner that uses $k$ Steiner points, we provide an $Ω(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the worst-case complexity of any $(t-\varepsilon)$-spanner, for any constant $\varepsilon \in (0,1)$ and integer constant $t \geq 2$. 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 of complexity $O(mn^{1/t}/k^{1/t} + n \log (n/k))$, for any integer $t \geq 1$. We generalize this to forests, and use it to obtain a $2\sqrt{2}t$-spanner in a simple polygon with complexity $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} + n\log^2 n)$. When a link can be any path between two sites, we show how to improve the spanning ratio to $(2k+\varepsilon)$, for any constant $\varepsilon \in (0,2k)$, and how to build a $6t$-spanner in a polygonal domain with the same complexity.

Figures (4)

• Figure 1: A spanner in a simple polygon that uses two Steiner points (red squares). By adding the two Steiner points, the spanner has a small spanning ratio and low complexity, as we no longer need multiple links that pass through the middle section of $P$.
• Figure 2: (a) Our construction for an $\Omega(mn^2/k^2)$ lower bound on the complexity of any $(2-\varepsilon\xspace)$-spanner. (b) A more detailed version of the comb of a pitchfork highlighted in the orange disk, which is also used for our $\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the complexity of any $(t-\varepsilon\xspace)$-spanner.
• Figure 5: The tree $T_i$ is the subtree whose edges and vertices have color $i$. A Steiner point (square) is placed at the root of $T_i$. The shaded areas show the trees $T_i'$. The examples show the case when the Steiner points are (a) at different vertices or (b) share a vertex.
• Figure 7: The shortest path tree of $\lambda$ in $P'$ and its $\mathit{SPT}_{i,j}$. The grey nodes and edges are not included in $\mathit{SPT}_{i,j}$, but can be assigned to a $T_i'$ as indicated by the colored backgrounds. The squares show the Steiner points in $\mathit{SPT}_{i,j}$ and $P'$. The sites in $P'$ are colored as the trees $T_i'$.

Theorems & Definitions (18)

• Lemma 1
• Corollary 2
• Lemma 2
• Lemma 2
• Lemma 3
• Lemma 4: Dinitz et al. shallow_low_light_trees
• Lemma 5: de Berg complexity_spanners
• Lemma 6
• Lemma 6
• Lemma 6