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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

In this paper we consider maintaining strongly connected components (SCCs) of a directed planar graph subject to edge insertions and deletions. We show a data structure maintaining an implicit representation of the SCCs within Õ(n^{6/7}) worst-case time per update. The data structure supports, in O(log²{n}) time, reporting vertices of any specified SCC (with constant overhead per reported vertex) and aggregating vertex information (e.g., computing the maximum label) over all the vertices of that SCC. Furthermore, it can maintain global information about the structure of SCCs, such as the number of SCCs, or the size of the largest SCC.
To the best of our knowledge, no fully dynamic SCCs data structures with sublinear update time have been previously known for any major subclass of digraphs. Our result should be contrasted with the n^{1-o(1)} amortized update time lower bound conditional on SETH, which holds even for dynamically maintaining whether a general digraph has more than two SCCs.

Adam Karczmarz and Marcin Smulewicz. Fully Dynamic Strongly Connected Components in Planar Digraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{karczmarz_et_al:LIPIcs.ICALP.2024.95, author = {Karczmarz, Adam and Smulewicz, Marcin}, title = {{Fully Dynamic Strongly Connected Components in Planar Digraphs}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {95:1--95:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.95}, URN = {urn:nbn:de:0030-drops-202388}, doi = {10.4230/LIPIcs.ICALP.2024.95}, annote = {Keywords: dynamic strongly connected components, dynamic strong connectivity, dynamic reachability, planar graphs} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We consider maintaining strongly connected components (SCCs) of a directed graph subject to edge insertions and deletions. For this problem, we show a randomized algebraic data structure with conditionally tight O(n^1.529) worst-case update time. The only previously described subquadratic update bound for this problem [Karczmarz, Mukherjee, and Sankowski, STOC'22] holds exclusively in the amortized sense.
For the less general dynamic strong connectivity problem, where one is only interested in maintaining whether the graph is strongly connected, we give an efficient deterministic black-box reduction to (arbitrary-pair) dynamic reachability. Consequently, for dynamic strong connectivity we match the best-known O(n^1.407) worst-case upper bound for dynamic reachability [van den Brand, Nanongkai, and Saranurak FOCS'19]. This is also conditionally optimal and improves upon the previous O(n^1.529) bound. Our reduction also yields the first fully dynamic algorithms for maintaining the minimum strong connectivity augmentation of a digraph.

Adam Karczmarz and Marcin Smulewicz. On Fully Dynamic Strongly Connected Components. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{karczmarz_et_al:LIPIcs.ESA.2023.68, author = {Karczmarz, Adam and Smulewicz, Marcin}, title = {{On Fully Dynamic Strongly Connected Components}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {68:1--68:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.68}, URN = {urn:nbn:de:0030-drops-187211}, doi = {10.4230/LIPIcs.ESA.2023.68}, annote = {Keywords: dynamic strongly connected components, dynamic strong connectivity, dynamic reachability} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in O(m + n poly log n) total time. Interspersed with the deletions, it can answer queries whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental c-edge-connectivity to a variant of fully-dynamic c-edge-connectivity on a sparse graph.
For non-sparse graphs with Ω(n poly log n) edges, our connectivity and 2-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an O(m log (n² / m) + n poly log n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n²) edges.
Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/log log n) worst-case time lower bound in the decremental setting [Alstrup, Husfeldt, and Rauhe FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

Anders Aamand, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal Decremental Connectivity in Non-Sparse Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aamand_et_al:LIPIcs.ICALP.2023.6, author = {Aamand, Anders and Karczmarz, Adam and {\L}\k{a}cki, Jakub and Parotsidis, Nikos and Rasmussen, Peter M. R. and Thorup, Mikkel}, title = {{Optimal Decremental Connectivity in Non-Sparse Graphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.6}, URN = {urn:nbn:de:0030-drops-180581}, doi = {10.4230/LIPIcs.ICALP.2023.6}, annote = {Keywords: decremental connectivity, dynamic connectivity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with Õ(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in Õ(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in Õ(n√m) worst-case time and queries in O(√m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008].

Adam Karczmarz and Piotr Sankowski. Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 84:1-84:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{karczmarz_et_al:LIPIcs.ICALP.2023.84, author = {Karczmarz, Adam and Sankowski, Piotr}, title = {{Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {84:1--84:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.84}, URN = {urn:nbn:de:0030-drops-181363}, doi = {10.4230/LIPIcs.ICALP.2023.84}, annote = {Keywords: dynamic shortest paths, dynamic reachability, dynamic transitive closure} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

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 Õ(n²) 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+ε)-approximate algorithm supporting vertex updates (i.e., changing all edges incident to a specified vertex) in conditionally near-optimal Õ(mlog{(W)}/ε) 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 Õ(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 n²) fixed source-target pairs after each update. We show that in such a scenario, Õ((m+k)n^{2/3}) worst-case update time is possible.

Adam Karczmarz. Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{karczmarz:LIPIcs.ICALP.2021.83, author = {Karczmarz, Adam}, title = {{Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {83:1--83:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.83}, URN = {urn:nbn:de:0030-drops-141521}, doi = {10.4230/LIPIcs.ICALP.2021.83}, annote = {Keywords: Dynamic graph algorithms, minimum weight cycle, dynamic shortest paths} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension d. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d = 2.
The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the A^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra’s algorithm. Specifically, if the n corresponding points of a weighted unit-disk graph G are picked from a unit square uniformly at random, and the connectivity radius is r ∈ (0,1), A^* finds a shortest path in G in O(n) expected time when r = Ω(√{log n/n}), even though G has Θ((nr)²) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges.
In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to A^*. Specifically, we show that, if we are able to report the set of all k points of G from an arbitrary rectangular region of the plane in O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r² + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r = Ω(√{log n/n}) and O(√n) expected bound for r = Ω(n^{-1/4}) after only near-linear preprocessing of the point set.
Our approach naturally generalizes to higher dimensions d ≥ 3 and yields sublinear expected bounds for all d = O(1) and sufficiently large r.

Adam Karczmarz, Jakub Pawlewicz, and Piotr Sankowski. Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{karczmarz_et_al:LIPIcs.SoCG.2021.46, author = {Karczmarz, Adam and Pawlewicz, Jakub and Sankowski, Piotr}, title = {{Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {46:1--46:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.46}, URN = {urn:nbn:de:0030-drops-138454}, doi = {10.4230/LIPIcs.SoCG.2021.46}, annote = {Keywords: unit-disk graphs, shortest paths, distance oracles} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

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 Õ(n^{4/5}) worst-case update time and O(log² 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.

Panagiotis Charalampopoulos and Adam Karczmarz. Single-Source Shortest Paths and Strong Connectivity in Dynamic Planar Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ESA.2020.31, author = {Charalampopoulos, Panagiotis and Karczmarz, Adam}, title = {{Single-Source Shortest Paths and Strong Connectivity in Dynamic Planar Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {31:1--31:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.31}, URN = {urn:nbn:de:0030-drops-128970}, doi = {10.4230/LIPIcs.ESA.2020.31}, annote = {Keywords: dynamic graph algorithms, planar graphs, single-source shortest paths, strong connectivity} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC’02, Bernstein STOC’13] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation).
Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions.

Adam Karczmarz and Jakub Łącki. Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 65:1-65:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{karczmarz_et_al:LIPIcs.ESA.2019.65, author = {Karczmarz, Adam and {\L}\k{a}cki, Jakub}, title = {{Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {65:1--65:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.65}, URN = {urn:nbn:de:0030-drops-111862}, doi = {10.4230/LIPIcs.ESA.2019.65}, annote = {Keywords: shortest paths, dynamic, incremental, decremental, directed graphs, hubs} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost 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~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(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.

Adam Karczmarz and Piotr Sankowski. Min-Cost Flow in Unit-Capacity Planar Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 66:1-66:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{karczmarz_et_al:LIPIcs.ESA.2019.66, author = {Karczmarz, Adam and Sankowski, Piotr}, title = {{Min-Cost Flow in Unit-Capacity Planar Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {66:1--66:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.66}, URN = {urn:nbn:de:0030-drops-111878}, doi = {10.4230/LIPIcs.ESA.2019.66}, annote = {Keywords: minimum-cost flow, minimum-cost circulation, planar graphs} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over Omega(n) operations, is O(log^2 n). Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs. It answers queries in O(1) time and processes edge deletions and contractions in O(log^2 n) amortized time. The previous best supported deletions and insertions in O(sqrt{n}) time.

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, and Eva Rotenberg. Decremental SPQR-trees for Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{holm_et_al:LIPIcs.ESA.2018.46, author = {Holm, Jacob and Italiano, Giuseppe F. and Karczmarz, Adam and Lacki, Jakub and Rotenberg, Eva}, title = {{Decremental SPQR-trees for Planar Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {46:1--46:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.46}, URN = {urn:nbn:de:0030-drops-95091}, doi = {10.4230/LIPIcs.ESA.2018.46}, annote = {Keywords: Graph embeddings, data structures, graph algorithms, planar graphs, SPQR-trees, triconnectivity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques.
Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra's algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds.
At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck.

Pawel Gawrychowski and Adam Karczmarz. Improved Bounds for Shortest Paths in Dense Distance Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2018.61, author = {Gawrychowski, Pawel and Karczmarz, Adam}, title = {{Improved Bounds for Shortest Paths in Dense Distance Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {61:1--61:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.61}, URN = {urn:nbn:de:0030-drops-90654}, doi = {10.4230/LIPIcs.ICALP.2018.61}, annote = {Keywords: shortest paths, dense distance graph, planar graph, Monge matrix} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in O(1) time. Moreover, it can report all the arising self-loops and parallel edges.
By applying the data structure, we can achieve optimal running times for decremental bridge detection, 2-edge connectivity, maximal 3-edge connected components, and the problem of finding a unique perfect matching for a static planar graph. Furthermore, we improve the running times of algorithms for several planar graph problems, including decremental 2-vertex and 3-edge connectivity, and we show that using our data structure in a black-box manner, one obtains conceptually simple optimal algorithms for computing MST and 5-coloring in planar graphs.

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, and Piotr Sankowski. Contracting a Planar Graph Efficiently. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{holm_et_al:LIPIcs.ESA.2017.50, author = {Holm, Jacob and Italiano, Giuseppe F. and Karczmarz, Adam and Lacki, Jakub and Rotenberg, Eva and Sankowski, Piotr}, title = {{Contracting a Planar Graph Efficiently}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {50:1--50:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.50}, URN = {urn:nbn:de:0030-drops-78755}, doi = {10.4230/LIPIcs.ESA.2017.50}, annote = {Keywords: planar graphs, algorithms, data structures, connectivity, coloring} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

A mergeable dictionary is a data structure storing a dynamic subset S of a totally ordered set U and supporting predecessor searches in S. Apart from insertions and deletions to S, we can both merge two arbitrarily interleaved dictionaries and split a given dictionary around some pivot x. We present an implementation of a mergeable dictionary matching the optimal amortized logarithmic bounds of Iacono and Ökzan [Iacono/Ökzan,ICALP'10]. However, our solution is significantly simpler. The proposed data structure can also be generalized to the case when the universe U is dynamic or infinite, thus addressing one issue of [Iacono/Ökzan,ICALP'10].

Adam Karczmarz. A Simple Mergeable Dictionary. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{karczmarz:LIPIcs.SWAT.2016.7, author = {Karczmarz, Adam}, title = {{A Simple Mergeable Dictionary}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.7}, URN = {urn:nbn:de:0030-drops-60285}, doi = {10.4230/LIPIcs.SWAT.2016.7}, annote = {Keywords: dictionary, mergeable, data structure, merge, split} }

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