Abstract
Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are integers such that 0 ≤ l ≤ u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an “almost uniform” partition is called an (l, u)-partition. We deal with three problems to find an (l, u)-partition of a given graph: the minimum partition problem is to find an (l, u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l, u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable for paths in linear time and for trees in polynomial time. In this paper, we give polynomial-time algorithms to solve the three problems for trees, which are much simpler and faster than the known algorithms.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Becker, R., Simeone, B., Chiang, Y.-I.: A shifting algorithm for continuous tree partitioning. Theoretical Computer Science 282, 353–380 (2002)
Bozkaya, B., Erkut, E., Laporte, G.: A tabu search heuristic and adaptive memory procedure for political districting. European J. Operational Research 144, 12–26 (2003)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Gonzalez, R.C., Wintz, P.: Digital Image Processing. Addison-Wesley, Reading (1977)
Ito, T., Zhou, X., Nishizeki, T.: Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size. J. Discrete Algorithms 4, 142–154 (2006)
Lucertini, M., Perl, Y., Simeone, B.: Most uniform path partitioning and its use in image processing. Discrete Applied Math. 42, 227–256 (1993)
Schröder, M.: Balanced Tree Partitioning, Ph. D. Thesis, University of Karlsruhe, Germany (2001)
Simone, C.D., Lucertini, M., Pallottino, S., Simeone, B.: Fair dissections of spiders, worms, and caterpillars. Networks 20(3), 323–344 (1990)
Tsichritzis, D.C., Bernstein, P.A.: Operating Systems. Academic Press, New York (1974)
Williams Jr., J.C.: Political redistricting: a review. Regional Science 74, 13–40 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ito, T., Uno, T., Zhou, X., Nishizeki, T. (2008). Partitioning a Weighted Tree to Subtrees of Almost Uniform Size. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-92182-0_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92181-3
Online ISBN: 978-3-540-92182-0
eBook Packages: Computer ScienceComputer Science (R0)