Emanation Graph: A Plane Geometric Spanner with Steiner Points

TL;DR

This paper introduces emanation graphs as plane geometric spanners with Steiner points, improves their spanning ratio bounds, and demonstrates their advantages over traditional triangulations in network visualization.

Contribution

It provides a tighter bound on the spanning ratio of emanation graphs, introduces a heuristic for simplification, and empirically compares their performance to constrained Delaunay triangulations.

Findings

Improved upper bound on spanning ratio to √10, which is tight.

Emanation graphs of fixed grade are constant spanners.

Simplified emanation graphs outperform constrained Delaunay triangulations in key quality metrics.

Abstract

An emanation graph of grade $k$ on a set of points is a plane spanner made by shooting $2^{k+1}$ equally spaced rays from each point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. Emanation graphs of grade one were studied by Mondal and Nachmanson in the context of network visualization. They proved that the spanning ratio of such a graph is bounded by $(2+\sqrt{2})\approx 3.414$. We improve this upper bound to $\sqrt{10} \approx 3.162$ and show this to be tight, i.e., there exist emanation graphs with spanning ratio $\sqrt{10}$. We show that for every fixed $k$, the emanation graphs of grade $k$ are constant spanners, where the constant factor depends on $k$. An emanation graph of grade two may have twice the number of edges compared to grade one graphs. Hence we introduce a heuristic method for simplifying them. In particular, we compare simplified emanation graphs against Shewchuk's constrained Delaunay triangulations on both synthetic and real-life datasets. Our experimental results reveal that the simplified emanation graphs outperform constrained Delaunay triangulations in common quality measures (e.g., edge count, angular resolution, average degree, total edge length) while maintaining a comparable spanning ratio and Steiner point count.