### 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 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 ([Formula presented]ρ+1)-approximation algorithm, where ρ is the Steiner ratio.

Original language | English |
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Pages (from-to) | 262-276 |

Number of pages | 15 |

Journal | Computational Geometry: Theory and Applications |

Volume | 68 |

DOIs | |

Publication status | Published - 2018 Mar |

### Keywords

- Approximation
- Colored point set
- Matroid intersection
- Minimum spanning tree
- Set visualization

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Computational Geometry: Theory and Applications*,

*68*, 262-276. https://doi.org/10.1016/j.comgeo.2017.06.006