Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

Karczmarz, Adam

doi:10.1137/1.9781611975031.5

Cited by 4 publications

(6 citation statements)

References 53 publications

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“…Since G ⊆ G * and the edges in G * correspond to paths in G, it is clear that δ G (s, t) = δ G * (s, t) for all s, t ∈ V (G). However, as proven in [17,32], the graph G * , although no longer planar, has hop-diameter O(log n). In other words, we in fact have δ (1) computing a feasible flow 11 for real-weighted capacities,…”

Section: A Faster Nearly Work-efficient Parallel Sssp Algorithmmentioning

confidence: 95%

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A Deterministic Parallel APSP Algorithm and its Applications

Karczmarz¹,

Sankowski²

2021

Preprint

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In this paper we show a deterministic parallel all-pairs shortest paths algorithm for realweighted directed graphs. The algorithm has O(nm + (n/d) 3 ) work and O(d) depth for any depth parameter d ∈ [1, n]. To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP'17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has O(nm + n 3 /d 2 ) work and O(d) depth [Ullman and Yannakakis, SIAM J. Comput. '91].Our APSP algorithm turns out to be a powerful tool for designing efficient planar graph algorithms in both parallel and sequential regimes. By suitably adjusting the depth parameter d and applying known techniques, we obtain: *

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Section: A Faster Nearly Work-efficient Parallel Sssp Algorithmmentioning

confidence: 95%

“…In order to solve the single-source shortest paths problem one can use a trick first described by Cohen [17] (and also used in [32]). Denote by H * a complete graph on ∂H 1 ∪ ∂H 2 whose edge weights represent distances in DDG H 1 ∪ DDG H 2 .…”

Section: A Faster Nearly Work-efficient Parallel Sssp Algorithmmentioning

confidence: 99%

A Deterministic Parallel APSP Algorithm and its Applications

Karczmarz¹,

Sankowski²

2021

Preprint

Self Cite

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“…The following lemmas establish the correctness and efficiency of the crucial parts of Algorithm The following lemma is essentially proved in [23,26]. However, as we use different notation, we give a complete proof below.…”

Section: A Path Removal Algorithmmentioning

confidence: 96%

“…To complete the construction, we show how to find a maximal set of edge-disjoint paths in O(n/ √ r) amortized time. To this end, we also exploit the properties of reachability in a dense distance graph, used previously in dynamic reachability algorithms for planar digraphs [23,26]. This way, we obtain O( √ mn/ √ r + mr) running time per scale.…”

Section: Introductionmentioning

confidence: 99%

Min-Cost Flow in Unit-Capacity Planar Graphs

Karczmarz,

Sankowski

2019

Preprint

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In this paper we give an O((nm) 2/3 log C) time algorithm for computing min-cost flow (or mincost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O(m 10/7 log C) time algorithm of Cohen et al. [SODA 2017], the O(m 3/2 log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O( √ nm log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Ω(m 3/2 ) time barrier for min-cost flow problem in planar graphs.To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O(m 3/2 log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs.

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“…Similarly, dynamic shortest paths problems have also been studied for important graph classes like planar graphs [3,4,38,56,63,66], or low treewidth-graphs [3,58]. The primary reason why faster dynamic shortest paths algorithms in these cases are possible is the existence of non-trivial distance oracles for these classes.…”

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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Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

Cited by 4 publications

References 53 publications

A Deterministic Parallel APSP Algorithm and its Applications

A Deterministic Parallel APSP Algorithm and its Applications

Min-Cost Flow in Unit-Capacity Planar Graphs

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

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