Colored Spanning Graphs for Set Visualization

TL;DR

This paper investigates the problem of constructing minimal ink-efficient spanning graphs for colored point sets, providing polynomial algorithms, special case solutions, and approximation methods for set visualization.

Contribution

It introduces polynomial-time algorithms for minimum red-blue-purple spanning graphs and develops efficient solutions for linear and circular point arrangements, along with a new approximation algorithm.

Findings

Polynomial-time algorithm for general RBP spanning graphs.

Efficient algorithms for points on a line or circle.

A fast approximation algorithm with ratio based on Steiner ratio.

Abstract

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(\frac 12\rho+1)$-approximation algorithm, where $\rho$ is the Steiner ratio.