# Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

# Dynamic Trees

2005; Tarjan, Werneck
• Renato F. Werneck
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_121

Link-cut trees

## Problem Definition

The dynamic tree problem is that of maintaining an arbitrary n-vertex forest that changes over time through edge insertions (links) and deletions (cuts). Depending on the application, one associates information with vertices, edges, or both. Queries and updates can deal with individual vertices or edges, but more commonly they refer to entire paths or trees. Typical operations include finding the minimum-cost edge along a path, determining the minimum-cost vertex in a tree, or adding a constant value to the cost of each edge on a path (or of each vertex of a tree). Each of these operations, as well as links and cuts, can be performed in $${ O(\log n) }$$

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## Recommended Reading

1. 1.
Acar, U.A., Blelloch, G.E., Harper, R., Vittes, J.L., Woo, S.L.M.: Dynamizing static algorithms, with applications to dynamic trees and history independence. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 524–533. SIAM (2004)Google Scholar
2. 2.
Acar, U.A., Blelloch, G.E., Vittes, J.L.: An experimental analysis of change propagation in dynamic trees. In: Proceedings of the 7th Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 41–54 (2005)Google Scholar
3. 3.
Alstrup, S., Holm, J., de Lichtenberg, K., Thorup, M.: Minimizing diameters of dynamic trees. In: Proceedings of the 24th International Colloquium on Automata, Languages and Programming (ICALP), Bologna, Italy, 7–11 July 1997. Lecture Notes in Computer Science, vol. 1256, pp. 270–280. Springer (1997)Google Scholar
4. 4.
Alstrup, S., Holm, J., Thorup, M., de Lichtenberg, K.: Maintaining information in fully dynamic trees with top trees. ACM Trans. Algorithms 1(2), 243–264 (2005)
5. 5.
Bent, S.W., Sleator, D.D., Tarjan, R.E.: Biased search trees. SIAM J. Comput. 14(3), 545–568 (1985)
6. 6.
Frederickson, G.N.: Data structures for on-line update of minimum spanning trees, with applications. SIAM J. Comput. 14(4), 781–798 (1985)
7. 7.
Frederickson, G.N.: Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM J. Comput. 26(2), 484–538 (1997)
8. 8.
Frederickson, G.N.: A data structure for dynamically maintaining rooted trees. J. Algorithms 24(1), 37–65 (1997)
9. 9.
Goldberg, A.V., Grigoriadis, M.D., Tarjan, R.E.: Use of dynamic trees in a network simplex algorithm for the maximum flow problem. Math. Progr. 50, 277–290 (1991)
10. 10.
Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarihmic time per operation. In: Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), pp. 519–527 (1997)Google Scholar
11. 11.
Miller, G.L., Reif, J.H.: Parallel tree contraction and its applications. In: Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 478–489 (1985)Google Scholar
12. 12.
P˘atraşcu, M., Demaine, E.D.: Lower bounds for dynamic connectivity. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 546–553 (2004)Google Scholar
13. 13.
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
14. 14.
Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)
15. 15.
Tarjan, R.E.: Dynamic trees as search trees via Euler tours, applied to the network simplex algorithm. Math. Prog. 78, 169–177 (1997)Google Scholar
16. 16.
Tarjan, R.E., Werneck, R.F.: Self-adjusting top trees. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 813–822 (2005)Google Scholar
17. 17.
Tarjan, R.E., Werneck, R.F.: Dynamic trees in practice. In: Proceedings of the 6th Workshop on Experimental Algorithms (WEA). Lecture Notes in Computer Science, vol. 4525, pp. 80–93 (2007)Google Scholar
18. 18.
Werneck, R.F.: Design and Analysis of Data Structures for Dynamic Trees. Ph. D. thesis, Princeton University (2006)Google Scholar

## Copyright information

© Springer-Verlag 2008

## Authors and Affiliations

• Renato F. Werneck
• 1
1. 1.Microsoft Research Silicon ValleyLa AvenidaUSA