The Complexity of Geodesic Spanners

TL;DR

This work studies geodesic spanners for point sets in polygonal environments with a novel measure of compactness: spanner complexity, the total edge complexity under geodesic paths. It develops a hierarchical framework that reduces edge complexity while controlling the spanning ratio, starting from a 1D additively weighted spanner and lifting it to simple polygons, yielding a geodesic $2\sqrt{2}$-spanner with $O(mn^{1/k}+n\log^2n)$ complexity for any fixed $k$ in simple polygons. The results extend to polygonal domains with holes via balanced shortest-path separators, producing a relaxed geodesic $6k$-spanner of similar asymptotics, and provide lower bounds showing near-optimality gaps, e.g., $\Omega(mn^{1/(t-1)}+n)$ for $(t-\\varepsilon)$-spanners. A key technique is the recasting of subdivision and projection steps through separators and 1D projections, enabling low-complexity spanners despite potentially high per-edge path complexity. Overall, the paper advances both constructive upper bounds and fundamental limits for geodesic spanners with bounded complexity in complex polygonal environments, with implications for routing, network design, and geometric data structures.

Abstract

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 fixed integer $k \geq 1$, a $2\sqrt{2}k$-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 relax the restriction that the edges in the spanner are shortest paths, such that an edge in the spanner can be any path between two sites, we obtain for any constant $\varepsilon \in (0,2k)$ a relaxed geodesic $(2k + \varepsilon)$-spanner of the same complexity, where the constant is dependent on $\varepsilon$. When we consider sites in a polygonal domain $P$ with holes, we can construct a relaxed geodesic $6k$-spanner of complexity $O(mn^{1/k} + n\log^2 n)$ in $O((n+m)\log^2n\log m+ K)$ 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 $Ω(mn^{1/(t-1)} + n)$.

Figures (15)

• Figure 1: A spanner on a set of point sites in a polygonal domain. Because of the orange edges, the spanner has a relatively high complexity.
• Figure 2: Construction of the additively weighted 1-dimensional spanner. The green triangle represents all points that are at distance at most $d_w(c,O)$ from $O$.
• Figure 4: A simple polygon with a set of sites for which any $(3-\varepsilon)$-spanner has complexity $\Omega(nm)$.
• Figure 5: The shortest path tree of $c$. The polygon vertices are grey in the tree. Each group $S_i$ has an associated polygonal region $R_i$ in $P$.
• Figure 9: No shortest path between two points on the boundary of $P$ can separate the sites into two groups. The sites can be separated by three shortest paths, for example using the orange triangle.

Theorems & Definitions (36)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 6
• Lemma 6
• Lemma 6
• Lemma 7