Dynamic Graph Algorithms

Demetrescu, Camil; Eppstein, David; Galil, Zvi; Italiano, Giuseppe F.

doi:10.1201/9781584888239-c9

Cited by 23 publications

(20 citation statements)

References 37 publications

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“…Several techniques have been used for the maintenance of other structures in dynamic graphs, such as spanning trees, transitive closure, and shortest paths. Surveys by Demetrescu et al [7,8] give a good overview of those. Recent progress on dynamic connectivity [14] and approximate single-source shortest paths [12] are witnesses of the current activity in this field.…”

Section: Related Resultsmentioning

confidence: 99%

Dynamic Graph Coloring

et al. 2018

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In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d > 0, the first algorithm maintains a proper O(CdN 1/d )-coloring while recoloring at most O(d) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(Cd)-coloring with O(dN 1/d ) recolorings per update. The two converge when d = log N , maintaining an O(C log N )-coloring with O(log N ) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on N vertices must recolor at least Ω(N 2 c(c−1) ) vertices per update, for any constant c ≥ 2.

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Section: Related Resultsmentioning

confidence: 99%

Dynamic Graph Coloring

et al. 2018

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“…A major open question is whether our techniques can be adapted to sequential dynamic graph algorithms, which constitutes a major area of research in the sequential setting [12, 13, 16, 22-24, 27-30, 33, 48, 49, 51, 52]. A formal definition and description of typical problems can be found in, e.g., [15]. Notice that our algorithms are fully dynamic, which means that they handle both insertions and deletions (of edges and nodes).…”

Section: Discussionmentioning

confidence: 99%

“…This makes solving tasks in a dynamic distributed setting of fundamental interest: Indeed, it is a widely studied area of research, especially in the context of asynchronous self-stabilization [17,18,26,50], and also in the context of severe graph changes, whether arbitrary [36] or evolving randomly [3]. In this paper, we consider a dynamic distributed setting which is synchronous and assumes topology changes with sufficient time for recovery in between, as is typically assumed in the literature on sequential dynamic algorithms [15].…”

Section: Introductionmentioning

confidence: 99%

Optimal Dynamic Distributed MIS

Censor-Hillel

Haramaty

Karnin³

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees [Luby 1986, Ghaffari 2015. By running a (static) distributed MIS algorithm after a topology change occurs, one can easily obtain a solution with the same complexity also for the dynamic distributed model, in which edges or nodes may be inserted or deleted.In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either synchronous or asynchronous, with only a single adjustment, meaning that a single node changes its output, and in a single round, in expectation. These strong guarantees hold for the complete fully dynamic setting: we handle all cases of insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former.We prove that for any deterministic algorithm, there is a topology change that requires n adjustments, thus we also strongly separate deterministic and randomized solutions.Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy sequential MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a 3-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing.Finally, our algorithm enjoys a useful history-independence property, which means that the distribution of the output structure depends only on the current graph, and does not depend on the history of topology changes that constructed that graph. This means that the output cannot be chosen, or even biased, by the adversary, in case its goal is to prevent us from optimizing some objective function. Moreover, history independent algorithms compose nicely, which allows us to obtain history independent coloring and matching algorithms, using standard reductions.

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“…The dynamic graph problem has a rich history and is one of the major topics for algorithms and data structures. See [4] for a survey.…”

Section: Related Workmentioning

confidence: 99%

“…We first show a contraction behavior of the Local-Resample (Algorithm 2) under condition (4). Fixed a graphical model I, the resampling procedure Local-Resample takes a pair (X, R) ∈ [q] V ×2 V as input and returns a pair (X ′ ,…”

Section: Proof Of Fast Convergencementioning

confidence: 99%

Dynamic sampling from graphical models

Feng

Vishnoi

Yin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

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Dynamic Graph Algorithms

Cited by 23 publications

References 37 publications

Dynamic Graph Coloring

Dynamic Graph Coloring

Optimal Dynamic Distributed MIS

Dynamic sampling from graphical models

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