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.
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
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)