Abstract
A union-find data structure maintains a collection of disjoint sets under makeset, union and find operations. Kaplan, Shafrir and Tarjan [SODA 2002] designed data structures for an extension of the union-find problem in which elements of the sets maintained may be deleted. The cost of a delete operation in their implementations is the same as the cost of a find operation. They left open the question whether delete operations can be implemented more efficiently than find operations. We resolve this open problem by presenting a relatively simple modification of the classical union-find data structure that supports delete, as well as makeset and union, operations in constant time, while still supporting find operations in O(log n) worst-case time and O(α(n)) amortized time, where n is the number of elements in the set returned by the find operation, and α(n) is a functional inverse of Ackermann’s function.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)
Alstrup, S., Ben-Amram, A.M., Rauhe, T.: Worst-case and amortised optimality in union-find. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC 1999), May 1999, pp. 499–506 (1999)
Ben-Amram, A.M., Galil, Z.: A generalization of a lower bound technique due to Fredman and Saks. Algorithmica 30(1), 34–66 (2001)
Blum, N.: On the single-operation worst-case time complexity of the disjoint set union problem. SIAM J. Comput. 15(4), 1021–1024 (1986)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Fredman, M., Saks, M.: The cell probe complexity of dynamic data structures. In: Proceedings of the 21st Annual Symposium on Theory of Computing (STOC 1989), May 1989, pp. 345–354. ACM Association for Computing Machinery, New York (1989)
Galil, Z., Italiano, G.F.: Data structures and algorithms for disjoint set union problems. ACM Computing Surveys 23(3), 319 (1991)
Kaplan, H., Shafrir, N., Tarjan, R.E.: Union-find with deletions. In: Proc. of the 13th ACM-SIAM Symp. on Discrete Mathematics (SODA), pp. 19–28
Kozen, D.L.: The Design and Analysis of Algorithms. Springer, Berlin (1992)
Seidel, R., Sharir, M.: Top-down analysis of path compression. SIAM J. Comput. 34(3), 515–525 (2005)
Smid, M.: A data structure for the union-find problem having good single-operation complexity. ALCOM: Algorithms Review, Newsletter of the ESPRIT II Basic Research Actions Program Project no. 3075 (ALCOM), 1 (1990)
Tarjan, R.E.: Efficiency of a good but not linear disjoint set union algorithm. Journal of the ACM 22, 215–225 (1975)
Tarjan, R.E., van Leeuwen, J.: Worst-case analysis of set union algorithms. Journal of the ACM 31(2), 245–281 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alstrup, S., Li Gørtz, I., Rauhe, T., Thorup, M., Zwick, U. (2005). Union-Find with Constant Time Deletions. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_7
Download citation
DOI: https://doi.org/10.1007/11523468_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27580-0
Online ISBN: 978-3-540-31691-6
eBook Packages: Computer ScienceComputer Science (R0)