Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Gutenberg, Maximilian Probst; Wulff-Nilsen, Christian

doi:10.1137/1.9781611975994.154

Cited by 25 publications

(25 citation statements)

References 59 publications

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“…Interestingly, it seems that the other known approaches to fully dynamic APSP in real-weighted graphs [19,24,47], if adjusted, cannot easily yield subquadratic (in n) update times for "sparse" instances of MPSP where m, k = O(n). This is because they all reconstruct shortest paths in a hierarchical manner, by inductively stitching [24] or extending [19,47] paths recomputed earlier in the process. Even though the number of input source-target pairs of interest may be small, these may require answers for Θ(n 2 ) distinct source-target pairs at lower levels of the hierarchy.…”

Section: Our Resultsmentioning

confidence: 99%

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

Karczmarz

2021

Preprint

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We consider the directed minimum weight cycle problem in the fully dynamic setting. To the best of our knowledge, so far no fully dynamic algorithms have been designed specifically for the minimum weight cycle problem in general digraphs. One can achieve O(n 2 ) amortized update time by simply invoking the fully dynamic APSP algorithm of Demetrescu and Italiano [J. ACM '04]. This bound, however, yields no improvement over the trivial recompute-from-scratch algorithm for sparse graphs.Our first contribution is a very simple deterministic (1+ǫ)-approximate algorithm supporting vertex updates (i.e., changing all edges incident to a specified vertex) in conditionally nearoptimal O(m log (W )/ǫ) amortized time for digraphs with real edge weights in [1, W ]. Using known techniques, the algorithm can be implemented on planar graphs and also gives some new sublinear fully dynamic algorithms maintaining approximate cuts and flows in planar digraphs.Additionally, we show a Monte Carlo randomized exact fully dynamic minimum weight cycle algorithm with O(mn 2/3 ) worst-case update that works for real edge weights. To this end, we generalize the exact fully dynamic APSP data structure of Abraham et al. [SODA'17] to solve the multiple-pairs shortest paths problem, where one is interested in computing distances for some k (instead of all n 2 ) fixed source-target pairs after each update. We show that in such a scenario, O((m + k)n 2/3 ) worst-case update time is possible.

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Section: Our Resultsmentioning

confidence: 99%

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

Karczmarz

2021

Preprint

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“…The decremental shortest path problem was originally studied in the context of efficiently handling single-source shortest path queries while edges are deleted [20,3,16]. All-pairs shortest path variants of the problem have also been studied [5,2].…”

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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“…Even more recently, Chuzoy and Khanna [CK19] extended their framework and showed that it can be used to improve the static problems of vertex-capacitated max-flow and sparsest vertex cut. Probst Gutenberg and Wulff-Nilsen [GW20b] recently presented a deterministic algorithm that improves on the former bounds for sparse graphs with total update time mn 0.5+o(1) . The existing data structures where futher extended in [Ber+20] to be path-reporting.…”

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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Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Cited by 25 publications

References 59 publications

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

Efficient Network Analysis Under Single Link Deletion

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

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