Deterministic decremental single source shortest paths: beyond the o(mn) bound

Bernstein, Aaron; Chechik, Shiri

doi:10.1145/2897518.2897521

Cited by 47 publications

(87 citation statements)

References 39 publications

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“…We believe that this is an important question, since it may help in understanding how to develop deterministic dynamic algorithms in general. It is very challenging and interesting to design deterministic dynamic algorithms with performances similar to the randomized ones for many dynamic graph problems such as maximal matching [5,10,9,11,12,13], connectivity [24,28,32,27], and shortest paths [6,8,7,21,22].…”

Section: Open Problemsmentioning

confidence: 99%

Dynamic Algorithms for Graph Coloring

Bhattacharya¹,

Chakrabarty²,

Henzinger³

et al. 2018

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

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We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (∆ + 1)-vertex coloring and (2∆ − 1)-edge coloring in a graph with maximum degree ∆. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results.(1) We present a randomized algorithm which maintains a (∆ + 1)-vertex coloring with O(log ∆) expected amortized update time. (2) We present a deterministic algorithm which maintains a (1 + o (1)

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Section: Open Problemsmentioning

confidence: 99%

Dynamic Algorithms for Graph Coloring

Bhattacharya¹,

Chakrabarty²,

Henzinger³

et al. 2018

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

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“…2 Related work includes various approximation algorithms for undirected graphs [5,30,11,22,1,21,3], in particular also for the single-source shortest paths problem [16,11,19,21,20,10].…”

Section: Related Workmentioning

confidence: 99%

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

Abraham¹,

Chechik²,

Krinninger³

2017

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

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We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worstcase guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates).Our main result is a simple randomized algorithm that for any parameter c > 1 has a worst-case update time of O(cn 2+ 2 /3 log 4 /3 n) and answers distance queries correctly with probability 1 − 1/n c , against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time ofÕ(n 2+ 3 /4 ) and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.

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“…Very recently, Bernstein and Chechik [4] presented the first deterministic algorithm to go beyond O(mn) total update time: their algorithm achieves total update timeÕ(n 2 ) in undirected unweighted graphs, again with a necessary (1 + ) approximation. See Section 1.2 of their paper for a detailed discussion of why it is especially important to develop deterministic algorithms for this problem (in short: to avoid the non-adaptivity assumption), but also of why an entirely new set of techniques seems to be required.…”

Section: Related Workmentioning

confidence: 99%

“…In their deterministicÕ(n 2 ) total update time algorithm [4], Bernstein and Chechik partitioned the graph into light and heavy vertices according to their degree. Because light vertices had low degree they were easier to work with.…”

Section: Overview Of Techniquesmentioning

confidence: 99%

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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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Deterministic decremental single source shortest paths: beyond the o(mn) bound

Cited by 47 publications

References 39 publications

Dynamic Algorithms for Graph Coloring

Dynamic Algorithms for Graph Coloring

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

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