Randomized fully dynamic graph algorithms with polylogarithmic time per operation

Henzinger MH, King V. 1999. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM. 46(4), 502–516.

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Journal Article | Published | English

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Author
Henzinger, MonikaISTA ; King, Valerie

Corresponding author has ISTA affiliation

Abstract
This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of Ω(m0) operations, where m0 is the number of edges in the initial graph, the expected time for p updates is O(p log3 n) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. The worst-case time for one query is O(log n/log log n). For the k-edge witness problem (“Does the removal of k given edges disconnect the graph?”) the expected time for p updates is O(p log3 n) and the expected time for q queries is O(qk log3 n). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O(pk log3 n). This implies an algorithm to maintain a 1 + ε-approximation of the minimum spanning tree in expected time O((p log3 n logU)/ε) for p updates, where the weights of the edges are between 1 and U.
Publishing Year
Date Published
1999-07-01
Journal Title
Journal of the ACM
Publisher
Association for Computing Machinery
Volume
46
Issue
4
Page
502-516
ISSN
eISSN
IST-REx-ID

Cite this

Henzinger MH, King V. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM. 1999;46(4):502-516. doi:10.1145/320211.320215
Henzinger, M. H., & King, V. (1999). Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM. Association for Computing Machinery. https://doi.org/10.1145/320211.320215
Henzinger, Monika H, and Valerie King. “Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation.” Journal of the ACM. Association for Computing Machinery, 1999. https://doi.org/10.1145/320211.320215.
M. H. Henzinger and V. King, “Randomized fully dynamic graph algorithms with polylogarithmic time per operation,” Journal of the ACM, vol. 46, no. 4. Association for Computing Machinery, pp. 502–516, 1999.
Henzinger MH, King V. 1999. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM. 46(4), 502–516.
Henzinger, Monika H., and Valerie King. “Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation.” Journal of the ACM, vol. 46, no. 4, Association for Computing Machinery, 1999, pp. 502–16, doi:10.1145/320211.320215.

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