# Dynamic graph connectivity in polylogarithmic worst case time

@inproceedings{Kapron2013DynamicGC, title={Dynamic graph connectivity in polylogarithmic worst case time}, author={Bruce M. Kapron and Valerie King and Ben Mountjoy}, booktitle={SODA}, year={2013} }

The dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes which is undergoing a sequence of edge insertions and deletions, answer queries of the form q(a, b): "Is there a path between nodes a and b?" While data structures for this problem with polylogarithmic amortized time per operation have been known since the mid-1990's, these data structures have Θ(n) worst case time. In fact, no previously known solution has worst case time per operation which is o…

## 171 Citations

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This paper considers fully dynamic graph algorithms with both faster worst case update time and sublinear space, and shows that 2-edge connectivity can be maintained using O(n log^2 n) words with an amortized update time of O(log^6 n).

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- Computer Science2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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This is the first polylog update time for maximal matching that implies an exponential improvement from the previous results and can maintain a factor two approximate maximum matching in $O(\log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme.

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Fully Dynamic Maximal Matching in O(log n) Update Time

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An algorithm for maintaining maximal matching in a graph under addition and deletion of edges that can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme.

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This thesis presents several new algorithms for dynamic graph problems, including an algorithm for decremental connectivity in planar graphs, the maintenance of connected components in a dynamic graph, and the Steiner tree problem over a dynamic set of terminals.

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- Computer Science, MathematicsIADIS INTERNATIONAL JOURNAL ON COMPUTER SCIENCE AND INFORMATION SYSTEMS
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This paper shows that instead of solving the dynamic connectivity problem on a general graph G, it suffices to solve it on a graph the authors name aligned double-forest that has only 2n-1 edges where n is the number of vertices, and presents an algorithm that achieves all the operations in logarithmic worst-case time.

Fully Dynamic Spanners with Worst-Case Update Time

- Computer Science, MathematicsESA
- 2016

This paper gives fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update, and are the first dynamic spanner algorithms with sublinear worst- case update time guarantees.

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

- Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

This paper develops an efficient (1 + ε) -approximation algorithm for this query using fast matrix multiplication and obtains the first dynamic APSP algorithm with subquadratic update time and sublinear query time.

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- Computer ScienceSTOC
- 2017

A Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted undirected dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions, achieving an improvement over the O(√n) deterministic worst-case update time of Eppstein et al.

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