Abstract
Given a rooted tree with n nodes, the tree shortcut problem is to add a set of shortcut edges to the tree such that the shortest path from each node to any of its ancestors is of length \(O(\log n)\) and the degree increment of each node is constant. We consider in this paper the dynamic version of the problem, which supports node insertion and deletion. For insertion, a node can be inserted as a leaf node or an internal node by sub-dividing an existing edge. For deletion, a leaf node can be deleted, or an internal node can be merged with its single child. We propose an algorithm that maintains a set of shortcut edges in \(O(\log n)\) time for an insertion or deletion.
This research is partially funded by a grant from Hong Kong RGC under the contract HKU17200214E.
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References
Alon, N., Schieber, B.: Optimal preprocessing for answering on-line product queries. Technical report (1987)
Bagchi, A., Buchsbaum, A.L., Goodrich, M.T.: Biased skip lists. Algorithmica 42(1), 31–48 (2005)
Samuel, W.: Bent, Daniel D Sleator, and Robert E Tarjan. Biased search trees. SIAM Journal on Computing 14(3), 545–568 (1985)
Bodlaender, H.L., Tel, G., Santoro, N.: Trade-offs in non-reversing diameter. Nord. J. Comput. 1(1), 111–134 (1994)
Chazelle, B.: Computing on a free tree via complexity-preserving mappings. Algorithmica 2, 337–361 (1987)
Elkin, M., Solomon, S.: Optimal euclidean spanners: really short, thin and lanky. In: STOC, pp. 645–654 (2013)
Hesse, W.: Directed graphs requiring large numbers of shortcuts. In: SODA, pp. 665–669 (2003)
Raskhodnikova, S.: Transitive-Closure Spanners: A Survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 167–196. Springer, Heidelberg (2010)
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
Solomon, S.: From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics. In: STOC, pp. 363–372 (2014)
Solomon, S., Elkin, M.: Balancing Degree, Diameter and Weight in Euclidean Spanners. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 48–59. Springer, Heidelberg (2010)
Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 215–225 (1975)
Thorup, M.: On shortcutting digraphs. In: Proceedings of the 18th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1992, London, UK, pp. 205–211. Springer (1993)
Thorup, M.: Parallel shortcutting of rooted trees. J. Algorithms 23(1), 139–159 (1997)
Yao, A.C.-C.: Space-time tradeoff for answering range queries (extended abstract). In: STOC, pp. 128–136 (1982)
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Chan, TH.H., Wu, X., Zhang, C., Zhao, Z. (2015). Dynamic Tree Shortcut with Constant Degree. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_34
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DOI: https://doi.org/10.1007/978-3-319-21398-9_34
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