Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Gutenberg, Maximilian Probst; Wulff-Nilsen, Christian

doi:10.1137/1.9781611975994.155

Cited by 22 publications

(18 citation statements)

References 17 publications

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“…Very recently, Probst and Wulff-Nilsen [GW20a] improved upon this result and presented a randomized data structure for decremental directed SSSP against an oblivious adversary with total update time Õ(min{mn 3/4 log W, m 3/4 n 5/4 log W }). They also give a Las Vegas algorithm with total update time Õ(m 3/4 n 5/4 log W ) that works against an adaptive adversary.…”

Section: Prior Workmentioning

confidence: 99%

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

Gutenberg¹,

Williams²,

Wein³

2020

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In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph G = (V, E) subject to edge insertions and deletions and a source vertex s ∈ V , and the goal is to maintain the distance d(s, t) for all t ∈ V .Fine-grained complexity has provided strong lower bounds for exact partially dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15]. Thus much focus has been directed towards finding efficient partially dynamic (1+ ε)-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14, FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20]. Despite this rich literature, for directed graphs there are no known deterministic algorithms for (1 + ε)-approximate dynamic SSSP that perform better than the classic ES-tree [JACM'81]. We present the first such algorithm.We present a deterministic data structure for incremental SSSP in weighted directed graphs with total update time Õ(n 2 log W ) which is near-optimal for very dense graphs; here W is the ratio of the largest weight in the graph to the smallest. Our algorithm also improves over the best known partially dynamic randomized algorithm for directed SSSP by Henzinger et al. [STOC'14, ICALP'15] if m = ω(n 1.1 ).Complementing our algorithm, we provide improved conditional lower bounds. Henzinger et al. [STOC'15] showed that under the OMv Hypothesis, the partially dynamic exact s-t Shortest Path problem in undirected graphs requires amortized update or query time m 1/2−o(1) , given polynomial preprocessing time. Under a hypothesis about finding Cliques, we improve the update and query lower bound for algorithms with polynomial preprocessing time to m 0.626−o(1) . Further, under the k-Cycle hypothesis, we show that any partially dynamic SSSP algorithm with O(m 2−ε ) preprocessing time requires amortized update or query time m 1−o(1) , which is essentially optimal. All previous conditional lower bounds that come close to our bound [ESA'04,FOCS'14] only held for "combinatorial" algorithms, while our new lower bound does not make such restrictions.

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Section: Prior Workmentioning

confidence: 99%

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

Gutenberg¹,

Williams²,

Wein³

2020

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“…In particular, for none of the problems listed above there is a dynamic algorithm known that works against an adaptive adversary. Considering the recent progress on designing fast dynamic graph algorithms against an adaptive adversary [30,5,21,35,17,12], it is an obvious questions whether such fast algorithms are also possible for dynamic clustering. Furthermore, it is not known whether their quadratic dependence on k in the running time against an oblivious adversary is necessary, or whether we could improve it to, e.g., k or log k. Moreover, for k-sum-of-radii and k-sumof-diameters there is even no non-trivial dynamic algorithm known at all (for either type of adversary).…”

Section: Problemmentioning

confidence: 99%

On fully dynamic constant-factor approximation algorithms for clustering problems

Fichtenberger¹,

Henzinger²,

Wiese³

2021

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Clustering is an important task with applications in many fields of computer science. We study the fully dynamic setting in which we want to maintain good clusters efficiently when input points (from a metric space) can be inserted and deleted. Many clustering problems are APX-hard but admit polynomial time O(1)-approximation algorithms. Thus, it is a natural question whether we can maintain O(1)approximate solutions for them in subpolynomial update time, against adaptive and oblivious adversaries. Only a few results are known that give partial answers to this question. There are dynamic algorithms for k-center, k-means, and k-median that maintain constant factor approximations in expected Õ(k 2 ) update time against an oblivious adversary. However, for these problems there are no algorithms known with an update time that is subpolynomial in k, and against an adaptive adversary there are even no (non-trivial) dynamic algorithms known at all. Also, for the k-sum-of-radii and the k-sum-of-diameters problems it is open whether there is any dynamic algorithm, against either type of adversary.In this paper, we complete the picture of the question above for all these clustering problems.• We show that there is no fully dynamic O(1)-approximation algorithm for any of the classic clustering problems above with an update time in n o(1) h(k) against an adaptive adversary, for an arbitrary function h. So in particular, there are no such deterministic algorithms. • We give a lower bound of Ω(k) on the update time for each of the above problems, even against an oblivious adversary. This rules out update times with subpolynomial dependence on k. • We give the first O(1)-approximate fully dynamic algorithms for k-sum-of-radii and for k-sum-ofdiameters with expected update time of Õ(k O(1) ) against an oblivious adversary. • Finally, for k-center we present a fully dynamic (6 + ǫ)-approximation algorithm with an expected update time of Õ(k) against an oblivious adversary. This is the first dynamic O(1)-approximation algorithm for any of the clustering problems above whose update time is Õ(k) and in particular the first whose update time is asymptotically optimal in k.

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“…The O(mn) barrier was first surpassed in the decremental setting. Henzinger, Krinninger and Nanongkai [HKN14,HKN15] provided a randomized, oblivious Õ(mn 0.9 log W ) algorithm, which was improved on by Bernstein, Probst Gutenberg and Wulff-Nilsen [GWN20a,BGWN20] culminating in a randomized Õ(min{n 2 log 4 W, mn 2/3 log 3 W }) algorithm against an oblivious adversary.…”

Section: Prior Workmentioning

confidence: 99%

Incremental SSSP for Sparse Digraphs Beyond the Hopset Barrier

Kyng¹,

Meierhans²,

Gutenberg³

2021

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Given a directed, weighted graph G = (V, E) undergoing edge insertions, the incremental single-source shortest paths (SSSP) problem asks for the maintenance of approximate distances from a dedicated source s while optimizing the total time required to process the insertion sequence of m edges.Recently, Gutenberg, Williams and Wein [STOC'20] introduced a deterministic Õ(n 2 ) algorithm for this problem, achieving near linear time for very dense graphs. For sparse graphs, Chechik and Zhang [SODA'21] recently presented a deterministic Õ(m 5/3 ) algorithm, and an adaptive randomized algorithm with run-time Õ(m √ n + m 7/5 ). This algorithm is remarkable for two reasons: 1) in very spare graphs it reaches the directed hopset barrier of Ω(n 3/2 ) that applied to all previous approaches for partially-dynamic SSSP [STOC'14, SODA'20, FOCS'20] and 2) it does not resort to a directed hopset technique itself.In this article we introduce propagation synchronization, a new technique for controlling the error build-up on paths throughout batches of insertions. This leads us to a significant improvement of the approach in [SODA'21] yielding a deterministic Õ(m 3/2 ) algorithm for the problem. By a very careful combination of our new technique with the sampling approach from [SODA'21], we further obtain an adaptive randomized algorithm with total update time Õ(m 4/3 ). This is the first partially-dynamic SSSP algorithm in sparse graphs to bypass the notorious directed hopset barrier which is often seen as the fundamental challenge towards achieving truly near-linear time algorithms.

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Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Cited by 22 publications

References 17 publications

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

On fully dynamic constant-factor approximation algorithms for clustering problems

Incremental SSSP for Sparse Digraphs Beyond the Hopset Barrier

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