Abstract
Given the scheduling model of bike-sharing, we consider the problem of hitting a set of n axis-parallel line segments in \(\mathbb {R}^2\) by a square or an \(\ell _\infty \)-circle (and two squares, or two \(\ell _\infty \)-circles) whose center(s) must lie on some line segment(s) such that the (maximum) edge length of the square(s) is minimized. Under a different tree model, we consider (virtual) hitting circles whose centers must lie on some tree edges with similar minmax-objectives (with the distance between a center to a target segment being the shortest path length between them). To be more specific, we consider the cases when one needs to compute one (and two) centers on some edge(s) of a tree with m edges, where n target segments must be hit, and the objective is to minimize the maximum path length from the target segments to the nearer center(s). We give three linear-time algorithms and an \(O(n^2\log n)\) algorithm for the four problems in consideration.
Similar content being viewed by others
References
Agarwal PK, Sharir M, Welzl E (1998) The discrete 2-center problem. Discrete Comput Geom 20(3):287–305
Chan TM (1999) More planar two-center algorithms. Comput Geom 13(3):189–198
Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 2nd edn. MIT Press
Drezner Z (1987) On the rectangular p-center problem. Naval Res Log (NRL) 34(2):229–234
Du H, Xu Y (2014) An approximation algorithm for k-center problem on a convex polygon. J Comb Optim 27(3):504–518
Eppstein D (1997) Faster construction of planar two-centers. SODA 97:131–138
Hoffmann M (2005) A simple linear algorithm for computing rectilinear 3-centers. Comput Geom 31(3):150–165
Katz MJ, Kedem K, Segal M (2000) Discrete rectilinear 2-center problems. Comput Geom 15(4):203–214
Megiddo N (1983) Linear-time algorithms for linear programming in \(\text{ r }^{3}\) and related problems. SIAM J Comput 12(4):759–776
Sadhu S, Roy S, Nandy SC, Roy S (2017) Optimal covering and hitting of line segments by two axis-parallel squares. In: International computing and combinatorics conference. Springer, pp 457–468
Sharir M (1997) A near-linear algorithm for the planar 2-center problem. Discrete Comput Geom 18(2):125–134
Welzl E (1991) Smallest enclosing disks (balls and ellipsoids). In: New results and new trends in computer science. LNCS 555, Springer, pp 359–370
Acknowledgements
This research is partially supported by NNSF of China under Project 61628207. XH is supported by China Scholarship Council under program 201706240214 and by the Fundamental Research Funds for the Central Universities under Project 2012017yjsy219. ZL is supported by a Shandong Government Scholarship. YX is supported by the National Natural Science Foundation of China under Grant 71371129.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, X., Liu, Z., Su, B. et al. Efficient algorithms for computing one or two discrete centers hitting a set of line segments. J Comb Optim 37, 1408–1423 (2019). https://doi.org/10.1007/s10878-018-0359-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-018-0359-6