Fully Dynamic Connectivity in O(log n(log log n)2) Amortized Expected Time

Huang, Shaowu; Huang, Dawei; Kopelowitz, Tsvi; Pettie, Seth

doi:10.1137/1.9781611974782.32

Cited by 24 publications

(18 citation statements)

References 19 publications

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“…The extra O((log n) 2 ) factor in the running time of the algorithm comes from the (amortized) computational cost of checking whether the chosen edge is a cut-edge in each step of the FK-dynamics. This is equivalent to the fully dynamic connectivity problem which has been thoroughly studied (see, e.g., [37,51]).…”

Section: Log(1/δ))mentioning

confidence: 99%

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Blanca

Gheissari

2021

Commun. Math. Phys.

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We establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$ Δ -regular graphs for all $$q\ge 1$$ q ≥ 1 and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , where the threshold $$p_u(q,\varDelta )$$ p u ( q , Δ ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$ Δ -regular tree. It is expected that this threshold is sharp, and for $$q>2$$ q > 2 the Glauber dynamics on random $$\varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$q\ge 1$$ q ≥ 1 , $$\varDelta \ge 3$$ Δ ≥ 3 , and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , with probability $$1-o(1)$$ 1 - o ( 1 ) over the choice of a random $$\varDelta $$ Δ -regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$ Θ ( n log n ) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$ Δ -regular graphs for every $$q\ge 2$$ q ≥ 2 , in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$ O ( log n ) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

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Section: Log(1/δ))mentioning

confidence: 99%

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Blanca

Gheissari

2021

Commun. Math. Phys.

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“…The undirected counterpart of the dynamic strongly connected components problem, the dynamic connectivity problem, is very well-studied. Near-optimal deterministic amortized update bounds [51,52,83] and randomized worst-case update bounds [57,81] (see also [71]) are known for fully dynamic general graphs. An almost optimal deterministic worst-case update bound was very recently achieved in [28].…”

Section: :5mentioning

confidence: 99%

Single-source shortest paths and strong connectivity in dynamic planar graphs

Charalampopoulos

Karczmarz

2022

Journal of Computer and System Sciences

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Efficient algorithms for computing and processing additively weighted Voronoi diagrams on planar graphs have been instrumental in obtaining several recent breakthrough results, most notably the almost-optimal exact distance oracle for planar graphs [Charalampopoulos et al., STOC'19], and subquadratic algorithms for planar diameter [Cabello, SODA'17, Gawrychowski et al., SODA'18]. In this paper, we show how Voronoi diagrams can be useful in obtaining dynamic planar graph algorithms and apply them to classical problems such as dynamic single-source shortest paths and dynamic strongly connected components.First, we give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with O(n 4/5 ) worst-case update time and O(log 2 n) query time. Here, a single update can either change the graph by inserting or deleting an edge, or reset the source s of interest. All known non-trivial planarity-exploiting exact dynamic single-source shortest paths algorithms to date had polynomial query time. Further, note that a data structure with strongly sublinear update time capable of answering distance queries between all pairs of vertices in polylogarithmic time would refute the APSP conjecture [Abboud and Dahlgaard, FOCS'16].Somewhat surprisingly, the Voronoi diagram based approach we take for single-source shortest paths can also be used in the fully dynamic strongly connected components problem. In particular, we obtain a data structure maintaining a planar digraph under edge insertions and deletions, capable of returning the identifier of the strongly connected component of any query vertex. The worst-case update and query time bounds are the same as for our single-source distance oracle. To the best of our knowledge, this is the first fully dynamic strong-connectivity algorithm achieving both sublinear update time and polylogarithmic query time for an important class of digraphs.

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“…One, therefore, asks which static problems solvable in time f(m) can be fully "dynamized", in the sense of having dynamic algorithms that support updates in O(f(m) / m) time. This question has been answered affirmatively for many fundamental graph problems including connectivity (e.g., [30,33,34,52]), reachability [32], shortest paths (e.g., [8,18,31]), and maximum matching [9,27,49].…”

Section: Facing Velocity: Algorithms For Dynamic Big Datamentioning

confidence: 99%

Big data algorithms beyond machine learning

Mnich

2017

Künstl Intell

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Fully Dynamic Connectivity in O(log n(log log n)²) Amortized Expected Time

Cited by 24 publications

References 19 publications

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Single-source shortest paths and strong connectivity in dynamic planar graphs

Big data algorithms beyond machine learning

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