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

Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the O(√n) barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved m^{1/ρ+o(1)} amortized update time with a O(log n)^{3ρ-2} factor in the approximation ratio, for any parameter ρ ≥ 1.
In this paper, we give the first constant-stretch fully dynamic distance oracle with small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of O(1/(ρ²))^{4/ρ} in amortized update time Õ(n^{ρ}), and query time Õ(n^{ρ/8}) for any constant parameter 0 < ρ < 1. Our algorithm is randomized and assumes an oblivious adversary.
A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution.

Sebastian Forster, Gramoz Goranci, Yasamin Nazari, and Antonis Skarlatos. Bootstrapping Dynamic Distance Oracles. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{forster_et_al:LIPIcs.ESA.2023.50, author = {Forster, Sebastian and Goranci, Gramoz and Nazari, Yasamin and Skarlatos, Antonis}, title = {{Bootstrapping Dynamic Distance Oracles}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {50:1--50:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.50}, URN = {urn:nbn:de:0030-drops-187031}, doi = {10.4230/LIPIcs.ESA.2023.50}, annote = {Keywords: Dynamic graph algorithms, Distance Oracles, Shortest Paths} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 22461 “Dynamic Graph Algorithms”, which took place from November 13 to November 18, 2022.
The field of dynamic graph algorithms studies algorithms for processing graphs that are changing over time. Formally, the goal is to process an interleaved sequence of update and query operations, where an update operation changes the input graph (e.g. inserts/deletes an edge), while the query operation is problem-specific and asks for some information about the current graph – for example, an s-t path, or a minimum spanning tree. The field has evolved rapidly over the past decade, and this Dagstuhl Seminar brought together leading researchers in dynamic algorithms and related areas of graph algorithms.

Aaron Bernstein, Shiri Chechik, Sebastian Forster, Tsvi Kopelowitz, Yasamin Nazari, and Nicole Wein. Dynamic Graph Algorithms (Dagstuhl Seminar 22461). In Dagstuhl Reports, Volume 12, Issue 11, pp. 45-65, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{bernstein_et_al:DagRep.12.11.45, author = {Bernstein, Aaron and Chechik, Shiri and Forster, Sebastian and Kopelowitz, Tsvi and Nazari, Yasamin and Wein, Nicole}, title = {{Dynamic Graph Algorithms (Dagstuhl Seminar 22461)}}, pages = {45--65}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Bernstein, Aaron and Chechik, Shiri and Forster, Sebastian and Kopelowitz, Tsvi and Nazari, Yasamin and Wein, Nicole}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.12.11.45}, URN = {urn:nbn:de:0030-drops-178354}, doi = {10.4230/DagRep.12.11.45}, annote = {Keywords: dynamic graphs, graph algorithms} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior.
In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that Ω(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k.

Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2023.20, author = {Bodwin, Greg and Dinitz, Michael and Nazari, Yasamin}, title = {{Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {20:1--20:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.20}, URN = {urn:nbn:de:0030-drops-175231}, doi = {10.4230/LIPIcs.ITCS.2023.20}, annote = {Keywords: Emulators, Fault Tolerance, Girth Conjecture} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Given a weighted undirected graph G = (V,E,w), a hopset H of hopbound β and stretch (1+ε) is a set of edges such that for any pair of nodes u, v ∈ V, there is a path in G ∪ H of at most β hops, whose length is within a (1+ε) factor from the distance between u and v in G. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of 2^{log^{Ω(1)} n} [Bernstein, FOCS'09; HKN, FOCS'14; Chechik, FOCS'18].
By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain (2k-1)(1+ε)-approximate all-pairs shortest paths (for any constant k ≥ 2), in Õ(n^{1/k}) amortized update time and O(k) query time. This improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS'18] that has a query time of O(log log(nW)), where W is the aspect ratio, and the amortized update time is n^{1/k}⋅(1/ε)^{Õ(√{log n})}). For sparse graphs our construction nearly matches the best known static running time / query time tradeoff.
We also obtain near-optimal bounds for maintaining approximate multi-source shortest paths and distance sketches, and get improved bounds for approximate single-source shortest paths. Our algorithms are randomized and our bounds hold with high probability against an oblivious adversary.

Jakub Łącki and Yasamin Nazari. Near-Optimal Decremental Hopsets with Applications. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lacki_et_al:LIPIcs.ICALP.2022.86, author = {{\L}\k{a}cki, Jakub and Nazari, Yasamin}, title = {{Near-Optimal Decremental Hopsets with Applications}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {86:1--86:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.86}, URN = {urn:nbn:de:0030-drops-164277}, doi = {10.4230/LIPIcs.ICALP.2022.86}, annote = {Keywords: Dynamic Algorithms, Data Structures, Shortest Paths, Hopsets} }

Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A k-spanner of a graph G is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of k, and a k-emulator is similar but not required to be a subgraph of G. A classic theorem by Althöfer et al. [Disc. Comp. Geom. '93] and Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners.
We complement our emulator upper bound with a lower bound construction that is essentially tight (within log n factors of the upper bound) when the stretch is 2k-1 and k is either a fixed odd integer or 2. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Vertex Fault-Tolerant Emulators. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2022.25, author = {Bodwin, Greg and Dinitz, Michael and Nazari, Yasamin}, title = {{Vertex Fault-Tolerant Emulators}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {25:1--25:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.25}, URN = {urn:nbn:de:0030-drops-156217}, doi = {10.4230/LIPIcs.ITCS.2022.25}, annote = {Keywords: Emulators, Fault Tolerance, Girth Conjecture} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph G=(V,E), a (β,ε)-hopset H with "hopbound" β, is a set of edges added to G such that for any pair of nodes u and v in G there is a path with at most β hops in G ∪ H with length within (1+ε) of the shortest path between u and v in G. Our hopsets are significantly sparser than the recent construction of [Censor-Hillel et al., 2019], that constructs a hopset of size Õ (n^{3/2}), but with a smaller polylogarithmic hopbound. On the other hand, the previously known construction of sparse hopsets with polylogarithmic hopbound in the Congested Clique model, proposed by [Elkin and Neiman, 2018; Elkin and Neiman, 2019; Elkin and Neiman, 2019], all require polynomial rounds.
One tool that we use is an efficient algorithm that constructs an l-limited neighborhood cover, that may be of independent interest. Finally, as a side result, we also give a hopset construction in a variant of the low-memory Massively Parallel Computation model, with improved running time over existing algorithms.

Yasamin Nazari. Sparse Hopsets in Congested Clique. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{nazari:LIPIcs.OPODIS.2019.34, author = {Nazari, Yasamin}, title = {{Sparse Hopsets in Congested Clique}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {34:1--34:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.34}, URN = {urn:nbn:de:0030-drops-118207}, doi = {10.4230/LIPIcs.OPODIS.2019.34}, annote = {Keywords: Hopsets, Congested Clique, Shortest Paths, Massively Parallel Computation} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. We provide efficient constructions in both of these models, but our core results are for MPC. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds.
Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use to construct distance sketches are an MPC construction of the hopsets of [Elkin and Neiman, 2016]. This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting.

Michael Dinitz and Yasamin Nazari. Massively Parallel Approximate Distance Sketches. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dinitz_et_al:LIPIcs.OPODIS.2019.35, author = {Dinitz, Michael and Nazari, Yasamin}, title = {{Massively Parallel Approximate Distance Sketches}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {35:1--35:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.35}, URN = {urn:nbn:de:0030-drops-118216}, doi = {10.4230/LIPIcs.OPODIS.2019.35}, annote = {Keywords: Distance Sketches, Massively Parallel Computation, Distance Oracles, Single-Source Shortest Paths} }

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Brief Announcement

**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

Data structures that allow efficient distance estimation have been extensively studied both in centralized models and classical distributed models. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds. Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use is an MPC construction of the hopsets of Elkin and Neiman (2016). This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting.

Michael Dinitz and Yasamin Nazari. Brief Announcement: Massively Parallel Approximate Distance Sketches. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 42:1-42:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dinitz_et_al:LIPIcs.DISC.2019.42, author = {Dinitz, Michael and Nazari, Yasamin}, title = {{Brief Announcement: Massively Parallel Approximate Distance Sketches}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {42:1--42:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.42}, URN = {urn:nbn:de:0030-drops-113491}, doi = {10.4230/LIPIcs.DISC.2019.42}, annote = {Keywords: Distance Sketches, Massively Parallel Computation, Congested Clique} }

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**Published in:** LIPIcs, Volume 95, 21st International Conference on Principles of Distributed Systems (OPODIS 2017)

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/ε) log n) rounds in the LOCAL model and with high probability finds a (1+ε)-approximation to the optimal LP solution for any 0 < ε ≤ 1, where D is the largest distance constraint.
While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

Michael Dinitz and Yasamin Nazari. Distributed Distance-Bounded Network Design Through Distributed Convex Programming. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dinitz_et_al:LIPIcs.OPODIS.2017.5, author = {Dinitz, Michael and Nazari, Yasamin}, title = {{Distributed Distance-Bounded Network Design Through Distributed Convex Programming}}, booktitle = {21st International Conference on Principles of Distributed Systems (OPODIS 2017)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-061-3}, ISSN = {1868-8969}, year = {2018}, volume = {95}, editor = {Aspnes, James and Bessani, Alysson and Felber, Pascal and Leit\~{a}o, Jo\~{a}o}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2017.5}, URN = {urn:nbn:de:0030-drops-86262}, doi = {10.4230/LIPIcs.OPODIS.2017.5}, annote = {Keywords: distributed algorithms, approximation algorithms, convex programming} }

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