Abstract
In this paper we consider a special case of the Maximum Weighted Independent Set problem for graphs: given a vertex- and edge- weighted tree T = (V,E) where |V| = n, and a real number b, determine the largest weighted subset P of V such that the distance between the two closest elements of P is at least b. We present an O(n log3 n) algorithm for this problem when the vertices have unequal weights. The space requirement is O(n log n). This is the first known subquadratic algorithm for the problem. This solution leads to an O(n log4 n) algorithm to the previously-studied Weighted Max-Min Dispersion Problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
B. K. Bhattacharya and M. E. Houle. Generalized maximum independent sets for trees. Proceedings of Computing: the Australasian Theory Symposium (CATS97), Sydney, Feb. 1997.
P. Eades, T. Lin and X. Lin. Minimum size h-v drawings. Advanced Visual Interfaces (Proc. AVI’ 92), World Scientific Series in Computer Science, 36:386–394, 1992.
E. Erkut. The discrete p-dispersion problem. Research Paper No. 87-5, Dept. of Finance and Management Science, University of Calgary, Apr. 1989.
E. Erkut and S. Neuman. Analytical models for locating undesirable facilities. European Journal of Operations Research, 40:275–291, 1989.
E. Erkut, T. Baptie and B. von Hohenbalken. The discrete p-maxian location problem. Computers in OR, 17(1):51–61, 1990.
U. Friege, G. Kortsarz, D. Peleg. The dense k-subgraph problem. Manuscript, June 1998.
G. N. Frederickson and D. B. Johnson. Finding kth paths and p-centers by generating and searching good data structures. Journal of Algorithms, 4:61–80, 1983.
R. Hassin, S. Rubinstein and A. Tamir. Approximation algorithms for maximum facility dispersion. Operations Research Letters, 21:133–137, 1997.
M. M. Halldórsson, K. Iwano, N. Katoh and T. Tokuyama. Finding subsets maximizing minimum structures. In Proc. 6th ACM-SIAM Symposium on Discrete Algorithms (SODA), San Francisco, pp. 150–157, 1995.
G. Kortsarz and D. Peleg. On choosing a dense subgraph. In Proc. 34th IEEE FOCS, Palo Alto, USA, pp. 692–701, 1993.
M. J. Kuby. Programming models for facility dispersion: the p-dispersion and maximum dispersion problems. Geographical Analysis, 19(4):315–329, 1987.
F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
S. S. Ravi, D. J. Rosenkrantz and G. K. Tayi. Heuristic and special case algorithms for dispersion problems. Operations Research, 42(2):299–310, 1994.
D. J. Rosenkrantz, G. K. Tayi and S. S. Ravi. Capacitated facility dispersion problems. Manuscript, 1997.
L. Stockmeyer. Optimal orientation of cells in slicing floorplan design. Information and Control, 57:91–101, 1983.
A. Tamir. Obnoxious facility location on graphs. SIAM J. Disc. Math., 4:550–567, 1991.
A. Tamir. Comments on the paper `Heuristic and special case algorithms for dispersion problems’ by S. S. Ravi. D. J. Rosenkrantz and G. K. Tayi. Operations Research, 46:157–158, 1998.
D. J. White. The maximal dispersion problem and the `first point outside the neighbourhood’ heuristic. Computers and Operations Research, 18:43–50, 1991.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bhattacharya, B.K., Houle, M.E. (1999). Generalized Maximum Independent Sets for Trees in Subquadratic Time. In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_44
Download citation
DOI: https://doi.org/10.1007/3-540-46632-0_44
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66916-6
Online ISBN: 978-3-540-46632-1
eBook Packages: Springer Book Archive