Fully Dynamic Maximal Matching in $O(\log n)$ Update Time
TLDR
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.Abstract:
We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivkovicź and Lloyd [Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we 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. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update.read more
Citations
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Proceedings ArticleDOI
Dynamic Matching Algorithms Under Vertex Updates
TL;DR: This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges, and focuses on Maximal Matching (MM) which is a 2-approximation to the MCM.
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Dynamic Rank Maximal Matchings
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Proceedings ArticleDOI
Decremental Matching in General Graphs
TL;DR: The gap between bipartite and general graphs is bridged, by giving an O ε (poly(log n )) update time algorithm that maintains a (1 + ε )-approximate maximum integral matching under adversarial deletions under partially dynamic matching.
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Dynamic geometric set cover and hitting set
TL;DR: This paper investigates dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and develops two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in $\mathbb{R}^1$ and $\ mathbb{ R}^2$.
Journal ArticleDOI
Fully Dynamic (Δ +1)-Coloring in O(1) Update Time
TL;DR: An improved randomized algorithm for (Δ +1)-coloring that achieves O(1) amortized update time and it is shown that this bound holds not only in expectation but also with high probability.
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