Maintaining Minimum Spanning Forests in Dynamic Graphs

Henzinger, Monika; King, Valerie

doi:10.1137/s0097539797327209

Cited by 60 publications

(36 citation statements)

References 18 publications

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“…The decremental setting is often very important from a theoretical perspective, as decremental shortest paths (and decremental single source shortest paths especially) are used as a building block in a large variety of fully dynamic shortest paths algorithms; see e.g. [14,3,1,2]. In fact, as we discuss in Section 1.2, using our new decremental single source shortest paths result as a black box immediately yields new results for deterministic fully dynamic all pairs shortest paths.…”

Section: Introductionmentioning

confidence: 94%

Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

Bernstein¹,

Chechik²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. (Our algorithm can also be modified to work in the incremental setting, where the graph is initially empty and subject to a sequence of online adversarial edge insertions.)In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Later papers presented conditional lower bounds showing that O(mn) is optimal up to log factors.In SODA 2011, Bernstein and Roditty showed how to bypass these lower bounds and improve upon the Even and Shiloach O(mn) total update time bound by allowing a (1 + ) approximation. This triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a (1 + )-approximate algorithm whose total update time is near linear: O(m 1+O(1/ √ log n) ). However, every single one of these improvements over the Even-Shiloach algorithm was randomized and assumed a non-adaptive adversary. This additional assumption meant that the algorithms were not suitable for certain settings and could not be used as a black box data structure. Very recently Bernstein and Chechik presented in STOC 2016 the first deterministic improvement over Even and Shiloach, that did not rely on randomization or assumptions about the adversary: in an undirected unweighted graph the algorithm maintains (1+ )-approximate distances and has total update timeÕ(n 2 ). In this paper, we present a new deterministic algorithm for the problem with total update timeÕ(n 1.25 √ m) = O(mn 3/4 ): it returns a (1 + ) approximation, and is limited to undirected unweighted graphs. Although this result is still far from matching the randomized near-linear total update time, it presents important progress towards that direction, because unlike the STOC 2016Õ(n 2 ) algorithm it beats the Even and Shiloach O(mn) bound for all graphs, not just sufficiently dense ones. In particular, theÕ(n 2 ) algorithm relied entirely on a new sparsification technique, and so could not hope to yield an improvement for sparse graphs. We present the first deterministic improvement for sparse graphs

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Section: Introductionmentioning

confidence: 94%

Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

Bernstein¹,

Chechik²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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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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“…A number of algorithms in computer networks [22,8] also solve the problem of recomputing shortest path trees when edges are added to or removed from the graph. Some related work [6,3,19,4] for this problem proposes methods for exact computation of dynamic shortest paths in a variety of graph settings and in parallel or distributed scenarios. Our problem is however that of finding the maximum shortest path change between pairs of nodes, rather than that of designing incremental algorithms for maintaining shortest paths.…”

Section: Related Workmentioning

confidence: 99%

Finding Top-k Shortest Path Distance Changes in an Evolutionary Network

Gupta

Aggarwal²

2011

Advances in Spatial and Temporal Databases

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Abstract. Networks can be represented as evolutionary graphs in a variety of spatio-temporal applications. Changes in the nodes and edges over time may also result in corresponding changes in structural garph properties such as shortest path distances. In this paper, we study the problem of detecting the top-k most significant shortest-path distance changes between two snapshots of an evolving graph. While the problem is solvable with two applications of the all-pairs shortest path algorithm, such a solution would be extremely slow and impractical for very large graphs. This is because when a graph may contain millions of nodes, even the storage of distances between all node pairs can become inefficient in practice. Therefore, it is desirable to design algorithms which can directly determine the significant changes in shortest path distances, without materializing the distances in individual snapshots. We present algorithms that are up to two orders of magnitude faster than such a solution, while retaining comparable accuracy.

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Maintaining Minimum Spanning Forests in Dynamic Graphs

Cited by 60 publications

References 18 publications

Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

Big data algorithms beyond machine learning

Finding Top-k Shortest Path Distance Changes in an Evolutionary Network

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