TY - JOUR

T1 - Colored spanning graphs for set visualization

AU - Hurtado, Ferran

AU - Korman, Matias

AU - van Kreveld, Marc

AU - Löffler, Maarten

AU - Sacristán, Vera

AU - Shioura, Akiyoshi

AU - Silveira, Rodrigo I.

AU - Speckmann, Bettina

AU - Tokuyama, Takeshi

N1 - Funding Information:
M. K. was supported in part by MEXT KAKENHI Nos. 17K12635, and 15H02665. M. L. and B. S. were supported by the Netherlands Organisation for Scientific Research (NWO) under project numbers 639.021.123 and 639.023.208, respectively. V. S., and R.I. S. were supported by project Gen. Cat. DGR 2014SGR46 and MTM2015-63791-R (MINECO/FEDER). A. S. is supported by JSPS/MEXT KAKENHI Grant Numbers 24500002, 25106503. R.I. S. was funded by Portuguese funds through CIDMA and FCT, within project PEst-OE/MAT/UI4106/2014, by FCT grant SFRH/BPD/88455/2012, and by Spanish MINECO through the Ram?n y Cajal program. T. T. is supported by ELC project of Grant-in-Aid for Scientific Research on Innovative Areas 24106007, Grant-in-Aid for Scientific Research (B) 40312631, Grant-in-Aid for Exploratory Research 24650001, JSPS Grant Scientific Research (B) 15H02665, MEXT Japan, and Kawarabayashi Big Graph ERATO project, Japan Science and Technology Agency. The authors would like to thank Hugo Alves Akitaya for his input in the analysis of the algorithm in Section 4.
Funding Information:
M. K. was supported in part by MEXT KAKENHI Nos. 17K12635 , and 15H02665 . M. L. and B. S. were supported by the Netherlands Organisation for Scientific Research (NWO) under project numbers 639.021.123 and 639.023.208 , respectively. V. S., and R.I. S. were supported by project Gen. Cat. DGR 2014SGR46 and MTM2015-63791-R (MINECO/FEDER). A. S. is supported by JSPS/MEXT KAKENHI Grant Numbers 24500002 , 25106503 . R.I. S. was funded by Portuguese funds through CIDMA and FCT , within project PEst-OE/MAT/UI4106/2014 , by FCT grant SFRH/BPD/88455/2012 , and by Spanish MINECO through the Ramón y Cajal program. T. T. is supported by ELC project of Grant-in-Aid for Scientific Research on Innovative Areas 24106007 , Grant-in-Aid for Scientific Research (B) 40312631 , Grant-in-Aid for Exploratory Research 24650001 , JSPS Grant Scientific Research (B) 15H02665 , MEXT Japan , and Kawarabayashi Big Graph ERATO project, Japan Science and Technology Agency . The authors would like to thank Hugo Alves Akitaya for his input in the analysis of the algorithm in Section 4.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/3

Y1 - 2018/3

N2 - 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.

AB - 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.

KW - Approximation

KW - Colored point set

KW - Matroid intersection

KW - Minimum spanning tree

KW - Set visualization

UR - http://www.scopus.com/inward/record.url?scp=85026774013&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85026774013&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2017.06.006

DO - 10.1016/j.comgeo.2017.06.006

M3 - Article

AN - SCOPUS:85026774013

VL - 68

SP - 262

EP - 276

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -