TY - JOUR

T1 - Bottleneck bichromatic full Steiner trees

AU - Abu-Affash, A. Karim

AU - Bhore, Sujoy

AU - Carmi, Paz

AU - Chakraborty, Dibyayan

N1 - Funding Information:
Work by P. Carmi was partially supported by the Lynn and William Frankel Center for Computer Science and by Grant 2016116 from the United States-Israel Binational Science Foundation.
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/2/1

Y1 - 2019/2/1

N2 - Given two sets of points in the plane, Q of n (terminal) points and S of m (Steiner) points, where each of Q and S contains bichromatic points (red and blue points), a full bichromatic Steiner tree is a Steiner tree in which all points of Q are leaves and each edge of the tree is bichromatic, i.e., connects a red and a blue point. In the bottleneck bichromatic full Steiner tree (BBFST) problem, the goal is to compute a bichromatic full Steiner tree T, such that the length of the longest edge in T is minimized. In the k-BBFST problem, the goal is to find a bichromatic full Steiner tree T with at most k≤m Steiner points from S, such that the length of the longest edge in T is minimized. In this paper, we first present an O((n+m)logm) time algorithm that solves the BBFST problem. Then, we show that k-BBFST problem is NP-hard and cannot be approximated within a factor of 5 in polynomial time, unless P=NP. Finally, we give a polynomial-time 9-approximation algorithm for the k-BBFST problem.

AB - Given two sets of points in the plane, Q of n (terminal) points and S of m (Steiner) points, where each of Q and S contains bichromatic points (red and blue points), a full bichromatic Steiner tree is a Steiner tree in which all points of Q are leaves and each edge of the tree is bichromatic, i.e., connects a red and a blue point. In the bottleneck bichromatic full Steiner tree (BBFST) problem, the goal is to compute a bichromatic full Steiner tree T, such that the length of the longest edge in T is minimized. In the k-BBFST problem, the goal is to find a bichromatic full Steiner tree T with at most k≤m Steiner points from S, such that the length of the longest edge in T is minimized. In this paper, we first present an O((n+m)logm) time algorithm that solves the BBFST problem. Then, we show that k-BBFST problem is NP-hard and cannot be approximated within a factor of 5 in polynomial time, unless P=NP. Finally, we give a polynomial-time 9-approximation algorithm for the k-BBFST problem.

KW - Approximation algorithms

KW - Bichromatic full Steiner trees

KW - Geometric optimization

KW - Steiner trees

UR - http://www.scopus.com/inward/record.url?scp=85054831889&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2018.10.003

DO - 10.1016/j.ipl.2018.10.003

M3 - Article

AN - SCOPUS:85054831889

SN - 0020-0190

VL - 142

SP - 14

EP - 19

JO - Information Processing Letters

JF - Information Processing Letters

ER -