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

Abraham, Ittai; Chechik, Shiri; Krinninger, Sebastian

doi:10.1137/1.9781611974782.28

Cited by 35 publications

(99 citation statements)

References 39 publications

(58 reference statements)

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“…We point out that for unweighted graphs, we can replace the Bellman-Ford procedure by a simple Breath-First-Search procedure (see for example [Cor + 09]) which improves the running time from O(n 2 h) to O(n 2 ). This was also exploited before in [ACK17].…”

Section: Batch Deletion Data Structure For Unweighted Graphsmentioning

confidence: 86%

“…Lemma 2.1 (see [Zwi02,ACK17]). Given a collection Π of the h-hop-improving shortest paths for all pairs (s, t) ∈ V 2 in G \ D, then there exists a procedure DetExtDistances(Π, h) that returns improving shortest paths for all pairs (s, t) ∈ V 2 in time O(n 3 log n/h + n 2 log 2 n).…”

Section: Preliminariesmentioning

confidence: 99%

“…We start by stating a classic reduction that was used in all existing approaches. Lemma 3.1 (see [HK01,Tho05,ACK17]). Given a data structure on G that supports a batch deletion of up to 2∆ vertices D ⊆ V from G such that the data structure computes for each (s, t) ∈ V 2 a shortest path π s,t in G \ D, and can return the k first edges on π s,t in time O(k) time.…”

Section: The Frameworkmentioning

confidence: 99%

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

Gutenberg

Wulff-Nilsen

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a directed weighted graph G = (V, E) undergoing vertex insertions and deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of G. In two breakthrough results, Italiano and Demetrescu [STOC '03] presented an algorithm that requiresÕ(n 2 ) amortized update time, and Thorup showed in [STOC '05] that worst-case update timeÕ(n 2+3/4 ) can be achieved. In this article, we make substantial progress on the problem. We present the following new results:• We present the first deterministic data structure that breaks theÕ(n 2+3/4 ) worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time toÕ(n 2+5/7 ) =Õ(n 2.71.. ) and toÕ(n 2+3/5 ) = O(n 2.6 ) for unweighted graphs.• We present a simple deterministic algorithm withÕ(n 2+3/4 ) worst-case update time (Õ(n 2+2/3 ) for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update timeÕ(n 2+2/3 ) (Õ(n 2+1/2 ) for unweighted graphs) that works against a nonoblivious adversary. Both data structures require spaceÕ(n 2 ). These are the first exact dynamic algorithms with truly-subcubic update time and space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC '03, ICALP'04, Encyclopedia of Algorithms '08] and is critical to algorithms in practice [TALG '06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA '17].The All-Pairs Shortest Paths problem is one of the most fundamental algorithmic problems and is commonly taught in undergraduate courses to every Computer Science student. Whilst static algorithms for the problem are well-known for several decades, the dynamic versions of the problem have recently received intense attention by the research community. In the dynamic setting, the underlying graph G = (V, E) undergoes updates, most commonly edge insertions and/or deletions. Most dynamic All-Pairs Shortest Paths algorithms can further handle vertex insertions (with up to n incident edges) and/or deletions.The problem. In this article, we are only concerned with the fully-dynamic All-Pairs Shortest Path (APSP) problem with worst-case update time, i.e. given a fully-dynamic graph G = (V, E), undergoing vertex insertions and deletions, we want to guarantee minimal update time after each vertex update to recompute the distance matrix of the new graph. This is opposed to the version of the problem that allows for amortized update time. Moreover, we focus on space-efficient data structures and show that our data structures even improve over the most space-efficient APSP algorithms with amortized update time. We further point out, that for the fully-dynamic setting, vertex updates are more general than edge updates, sinc...

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Section: Batch Deletion Data Structure For Unweighted Graphsmentioning

confidence: 86%

Section: Preliminariesmentioning

confidence: 99%

Section: The Frameworkmentioning

confidence: 99%

See 1 more Smart Citation

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

Gutenberg

Wulff-Nilsen

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…APSP The algorithm withÕ(n 2.75 ) worst-case update time of Thorup [Tho05] was among the first that addressed the issue of worst-case update time (Question 1.4). Despite much recent effort and progress on this issue 9 , the only improvement over Thorup's bound was theÕ(n 2+2/3 ) worstcase update time by Abraham, Chechik, and Krinninger [ACK17]. This bound holds for directed weighted graphs and can be improved toÕ(n 2.5 ) on directed unweighted graphs.…”

Section: Consequencesmentioning

confidence: 99%

“…Note that while previous algorithms [Tho05,ACK17] can also return the shortest path connecting two nodes in time proportional to the length of the path, our algorithms only maintain the distances. Also, previous algorithms can handle a more general update where the weights of all edges incident to the same node are updated at once.…”

Section: Consequencesmentioning

confidence: 99%

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Brand

Nanongkai

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Consider the following distance query for an n-node graph G undergoing edge insertions and deletions: given two sets of nodes I and J, return the distances between every pair of nodes in I × J. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient (1 + ǫ)-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Below are some highlights. Note that all our algorithms guarantee worstcase update time and are randomized (Monte Carlo), but do not need the oblivious adversary assumption.Subquadratic update time for SSSP, Diameter, Centralities, ect.: When we want to maintain distances from a single node explicitly (without queries), a fundamental question is to beat trivially calling Dijkstra's static algorithm after each update, taking Θ(n 2 ) update time on dense graphs. A better time complexity was not known even with amortization. It was known to be improbable for exact algorithms and for combinatorial any-approximation algorithms to polynomially beat the Ω(n 2 ) bound (under some conjectures) [Roditty, Zwick, ESA'04; Abboud, V. Williams, FOCS'14]. 1 Our algorithm with I = {s} and J = V (G) implies a (1+ǫ)-approximation algorithm for this, guaranteeingÕ(n 1.823 /ǫ 2 ) worst-case update time for directed graphs with positive real weights in [1, W ]. 2 With ideas from [Roditty, V. Williams, STOC'13], we also obtain the first subquadratic worst-case update time for (5/3 + ǫ)-approximating the eccentricities and (1.5 + ǫ)-approximating the diameter and radius for unweighted graphs (with small additive errors). We also obtain the first subquadratic worst-case update time for (1 + ǫ)-approximating the closeness centralities for undirected unweighted graphs.Worst-case update time for APSP: When we want to maintain distances between all-pairs of nodes explicitly, theÕ(n 2 ) amortized update time by Demetrescu and Italiano [STOC'03] already matches the trivial Ω(n 2 ) lower bound. A fundamental question is whether it can be made worst-case. The state-of-the-art algorithm takesÕ(n 2+2/3 ) worst-case update time to maintain the distances exactly [Abraham, Chechik, Krinninger, SODA'17; Thorup STOC'05]. When it comes to (1 + ǫ) approximation, this bound is still higher than calling theÕ(n ω /ǫ)-time static algorithm of Zwick [FOCS'98], where ω ≈ 2.373. Our algorithm with I = J = V (G) implies nearly tight bounds for this, namelyÕ(n 2 /ǫ 1+ω ) for undirected unweighted graphs and O(n 2.045 /ǫ 2 ) for directed graphs with positive real weights. Besides this, we also obtain the first dynamic APSP algorithm with subquadratic update time and sublinear query time.

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Improved Distance Sensitivity Oracles via Tree Partitioning

Duan

Zhang

2017

Lecture Notes in Computer Science

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We introduce an improved structure of distance sensitivity oracle (DSO). The task is to pre-process a non-negatively weighted graph so that a data structure can quickly answer replacement path length for every triple of source, terminal and failed vertex. The previous best algorithm constructs in time 1Õ (mn) a distance sensitivity oracle of size O(n 2 log n) that processes queries in O(1) time. As an improvement, our oracle takes up O(n 2 ) space, while preserving O(1) query efficiency andÕ(mn) preprocessing time. One should notice that space complexity and query time of our novel data structure are asymptotically optimal.

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

Cited by 35 publications

References 39 publications

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

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

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Improved Distance Sensitivity Oracles via Tree Partitioning

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