Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds

Gutenberg, Maximilian Probst; Wulff-Nilsen, Christian

doi:10.1137/1.9781611975994.156

Cited by 20 publications

(7 citation statements)

References 28 publications

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“…These modifications typically mean changing the weights of edges in a graph. Efficient algorithms for this problem have been proposed [9,17]. These algorithms could be used to solve the single edge deletion problem.…”

Section: Related Problemsmentioning

confidence: 99%

Efficient Network Analysis Under Single Link Deletion

Ward¹,

Datta²,

Nguyen³

et al. 2021

Preprint

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The problem of worst case edge deletion from a network is considered. Suppose that you have a communication network and you can delete a single edge. Which edge deletion causes the largest disruption? More formally, given a graph, which edge after deletion disconnects the maximum number of pairs of vertices, where ties for number of pairs disconnected are broken by finding an edge that increases the average shortest path length the maximum amount. This problem is interesting both practically and theoretically. In practice, it is a good tool to measure network robustness and find vulnerable edges. In theory, it is related, though subtly different, to classical problems including the dynamic all-pairs shortest paths problem, and the decremental shortest paths problem.

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Section: Related Problemsmentioning

confidence: 99%

Efficient Network Analysis Under Single Link Deletion

Ward¹,

Datta²,

Nguyen³

et al. 2021

Preprint

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“…This bound was recently improved to Õ(n 2.66 ) by Abraham et al [ACK17] who presented an adaptive randomized algorithm. Recently, Probst Gutenberg and Wulff-Nilsen [GW20c] gave a deterministic algorithm that breaks the Õ(n 2.75 ) update time bound by Thorup.…”

Section: Running Timementioning

confidence: 99%

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Evald,

Fredslund-Hansen,

Gutenberg

et al. 2020

Preprint

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Given a directed graph G = (V, E), undergoing an online sequence of edge deletions with m edges in the initial version of G and n = |V |, we consider the problem of maintaining all-pairs shortest paths (APSP) in G.Whilst this problem has been studied in a long line of research [ACM'81, FOCS'99, FOCS'01, STOC'02, STOC'03, SWAT'04, STOC'13] and the problem of (1+ǫ)-approximate, weighted APSP was solved to near-optimal update time Õ(mn) by Bernstein [STOC'13], the problem has mainly been studied in the context of oblivious adversaries, which assumes that the adversary fixes the update sequence before the algorithm is started.In this paper, we make significant progress on the problem in the setting were the adversary is adaptive, i.e. can base the update sequence on the output of the data structure queries. We present three new data structures that fit different settings:• We first present a deterministic data structure that maintains the exact distances with total update time Õ(n 3 ) 1 .• We also present a deterministic data structure that maintains (1 + ǫ)-approximate distance estimates with total update time Õ( √ mn 2 /ǫ) which for sparse graphs is Õ(n 2+1/2 /ǫ).

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“…Note that recomputing the distance matrix from scratch takes O(nm) = O(n 3 ) time [55]. Fully dynamic all-pairs shortest paths data structures with subcubic worst-case update bounds are also known [5,46,78]. There exist faster algorithms if the input graph is unweighted and partially dynamic (i.e., incremental or decremental) [6,7].…”

Section: Introductionmentioning

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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Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds

Cited by 20 publications

References 28 publications

Efficient Network Analysis Under Single Link Deletion

Efficient Network Analysis Under Single Link Deletion

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

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

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