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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.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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Complete Volume

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

LIPIcs, Volume 244, ESA 2022, Complete Volume

Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman. LIPIcs, Volume 244, ESA 2022, Complete Volume. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1-1406, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{chechik_et_al:LIPIcs.ESA.2022, title = {{LIPIcs, Volume 244, ESA 2022, Complete Volume}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {1--1406}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022}, URN = {urn:nbn:de:0030-drops-169374}, doi = {10.4230/LIPIcs.ESA.2022}, annote = {Keywords: LIPIcs, Volume 244, ESA 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Front Matter, Table of Contents, Preface, Conference Organization

Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman. Front Matter, Table of Contents, Preface, Conference Organization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 0:i-0:xxii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chechik_et_al:LIPIcs.ESA.2022.0, author = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {0:i--0:xxii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.0}, URN = {urn:nbn:de:0030-drops-169382}, doi = {10.4230/LIPIcs.ESA.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the replacement paths (RP) problem we are given a graph G and a shortest path P between two nodes s and t . The goal is to find for every edge e ∈ P, a shortest path from s to t that avoids e. The first result of this paper is a simple reduction from the RP problem to the problem of computing shortest cycles for all nodes on a shortest path.
Using this simple reduction we unify and extremely simplify two state of the art solutions for two different well-studied variants of the RP problem.
In the first variant (algebraic) we show that by using at most n queries to the Yuster-Zwick distance oracle [FOCS 2005], one can solve the the RP problem for a given directed graph with integer edge weights in the range [-M,M] in Õ(M n^ω) time . This improves the running time of the state of the art algorithm of Vassilevska Williams [SODA 2011] by a factor of log⁶n.
In the second variant (planar) we show that by using the algorithm of Klein for the multiple-source shortest paths problem (MSSP) [SODA 2005] one can solve the RP problem for directed planar graph with non negative edge weights in O (n log n) time. This matches the state of the art algorithm of Wulff-Nilsen [SODA 2010], but with arguably much simpler algorithm and analysis.

Shiri Chechik and Moran Nechushtan. Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 29:1-29:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2020.29, author = {Chechik, Shiri and Nechushtan, Moran}, title = {{Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {29:1--29:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.29}, URN = {urn:nbn:de:0030-drops-124365}, doi = {10.4230/LIPIcs.ICALP.2020.29}, annote = {Keywords: Fault tolerance, Distance oracle, Planar graph} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree K̂ rooted at a node s, and the goal is to output for every node t ∈ V and for every edge e in K̂ the length of the shortest path from s to t avoiding e.
We present an Õ(m√n + n²) time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed.
Our new upper bound for this problem matches the existing conditional combinatorial lower bounds. Hence, (assuming these conditional lower bounds) our result is essentially optimal and completes the picture of the SSRP problem in the combinatorial setting.
Our algorithm naturally extends to the case of small, rational edge weights. In the full version of the paper, we strengthen the existing conditional lower bounds in this case by showing that any O(mn^(1/2-ε)) time (combinatorial or algebraic) algorithm for some fixed ε > 0 yields a truly sub-cubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).

Shiri Chechik and Ofer Magen. Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 81:1-81:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2020.81, author = {Chechik, Shiri and Magen, Ofer}, title = {{Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {81:1--81:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.81}, URN = {urn:nbn:de:0030-drops-124886}, doi = {10.4230/LIPIcs.ICALP.2020.81}, annote = {Keywords: Fault tolerance, Replacement Paths, Combinatorial algorithms, Conditional lower bounds} }

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

In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant.
We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters.
Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem.

Shiri Chechik and Doron Mukhtar. Reachability and Shortest Paths in the Broadcast CONGEST Model. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chechik_et_al:LIPIcs.DISC.2019.11, author = {Chechik, Shiri and Mukhtar, Doron}, title = {{Reachability and Shortest Paths in the Broadcast CONGEST Model}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {11:1--11:13}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.11}, URN = {urn:nbn:de:0030-drops-113183}, doi = {10.4230/LIPIcs.DISC.2019.11}, annote = {Keywords: Distributed algorithms, Broadcast CONGEST, distance estimation, small diameter} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms.
For the replacement paths problem, let G = (V,E) be a directed unweighted graph with n vertices and m edges and let P be a shortest path from s to t in G. The replacement paths problem is to find for every edge e in P the shortest path from s to t avoiding e. Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of O~(m sqrt{n}). Here we provide the first deterministic algorithm for this problem, with the same O~(m sqrt{n}) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of O~(mn) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in O~(m sqrt{n}) time, and a deterministic algorithm for the k-simple shortest paths problem in O~(k m sqrt{n}) time (for any integer constant k > 0).
For the problem of distance sensitivity oracles, let G = (V,E) be a directed graph with real-edge weights. An f-Sensitivity Distance Oracle (f-DSO) gets as input the graph G=(V,E) and a parameter f, preprocesses it into a data-structure, such that given a query (s,t,F) with s,t in V and F subseteq E cup V, |F| <=f being a set of at most f edges or vertices (failures), the query algorithm efficiently computes the distance from s to t in the graph G \ F (i.e., the distance from s to t in the graph G after removing from it the failing edges and vertices F).
For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized f-DSOs. In particular, they presented a combinatorial f-DSO with O~(mn^{4-alpha}) preprocessing time and subquadratic O~(n^{2-2(1-alpha)/f}) query time, giving a tradeoff between preprocessing and query time for every value of 0 < alpha < 1. We derandomize this result and present a combinatorial deterministic f-DSO with the same asymptotic preprocessing and query time.

Noga Alon, Shiri Chechik, and Sarel Cohen. Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alon_et_al:LIPIcs.ICALP.2019.12, author = {Alon, Noga and Chechik, Shiri and Cohen, Sarel}, title = {{Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.12}, URN = {urn:nbn:de:0030-drops-105882}, doi = {10.4230/LIPIcs.ICALP.2019.12}, annote = {Keywords: replacement paths, distance sensitivity oracles, derandomization} }

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

We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic (2+epsilon)-approximate integral matching algorithm with small polylog worst-case update time. For the (2+epsilon)-approximation regime only a fractional fully-dynamic (2+epsilon)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.

Moab Arar, Shiri Chechik, Sarel Cohen, Cliff Stein, and David Wajc. Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arar_et_al:LIPIcs.ICALP.2018.7, author = {Arar, Moab and Chechik, Shiri and Cohen, Sarel and Stein, Cliff and Wajc, David}, title = {{Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {7:1--7:16}, 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.2018.7}, URN = {urn:nbn:de:0030-drops-90112}, doi = {10.4230/LIPIcs.ICALP.2018.7}, annote = {Keywords: Dynamic, Worst-case, Maximum Matching, Maximum Weight Matching} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m * log^*(n)) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(m * beta(m,n)), where beta(m,n) = min{k >= 1 | log^{(k)}n <= m/n } <= log^*(n) - log^*(m/n)+1. This is the first improvement for these problems in over 25 years. In particular, if m >= n * log^{(k)} * n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).

Shiri Chechik, Haim Kaplan, Mikkel Thorup, Or Zamir, and Uri Zwick. Bottleneck Paths and Trees and Deterministic Graphical Games. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 27:1-27:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chechik_et_al:LIPIcs.STACS.2016.27, author = {Chechik, Shiri and Kaplan, Haim and Thorup, Mikkel and Zamir, Or and Zwick, Uri}, title = {{Bottleneck Paths and Trees and Deterministic Graphical Games}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {27:1--27:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.27}, URN = {urn:nbn:de:0030-drops-57283}, doi = {10.4230/LIPIcs.STACS.2016.27}, annote = {Keywords: bottleneck paths, comparison model, deterministic graphical games} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G=(V,E) on m vertices and a small "dataset" D \subset V of size n << m, so that given a query point q \in V, one can quickly approximate dist(q,D) (the distance from q to its closest vertex in D) and find a vertex a \in D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices.
For planar graphs G with maximum degree Delta, we show how to efficiently construct a compact data structure -- of size ~O(n(Delta+1/epsilon)) -- that answers (1+epsilon)-NNS queries in time ~O(Delta+1/epsilon). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into l_2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Delta (in query time) are both essential.

Ittai Abraham, Shiri Chechik, Robert Krauthgamer, and Udi Wieder. Approximate Nearest Neighbor Search in Metrics of Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 20-42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{abraham_et_al:LIPIcs.APPROX-RANDOM.2015.20, author = {Abraham, Ittai and Chechik, Shiri and Krauthgamer, Robert and Wieder, Udi}, title = {{Approximate Nearest Neighbor Search in Metrics of Planar Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {20--42}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.20}, URN = {urn:nbn:de:0030-drops-52923}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.20}, annote = {Keywords: Data Structures, Nearest Neighbor Search, Planar Graphs, Planar Metrics, Planar Separator} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the distance relation via weighted sampling. Based on that, we present highly practical algorithms with strong statistical guarantees for fundamental problems. We show that the average distance (and hence the centrality) for all nodes in a graph can be estimated using O(epsilon^{-2}) single-source distance computations. For a set V of n points in a metric space, we show that after preprocessing which uses O(n) distance computations we can compute a weighted sample S subset of V of size O(epsilon^{-2}) such that the average distance from any query point v to V can be estimated from the distances from v to S. Finally, we show that for a set of points V in a metric space, we can estimate the average pairwise distance using O(n+epsilon^{-2}) distance computations. The estimate is based on a weighted sample of O(epsilon^{-2}) pairs of points, which is computed using O(n) distance computations. Our estimates are unbiased with normalized mean square error (NRMSE) of at most epsilon. Increasing the sample size by a O(log(n)) factor ensures that the probability that the relative error exceeds epsilon is polynomially small.

Shiri Chechik, Edith Cohen, and Haim Kaplan. Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 659-679, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chechik_et_al:LIPIcs.APPROX-RANDOM.2015.659, author = {Chechik, Shiri and Cohen, Edith and Kaplan, Haim}, title = {{Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {659--679}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.659}, URN = {urn:nbn:de:0030-drops-53291}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.659}, annote = {Keywords: Closeness Centrality; Average Distance; Metric Space; Weighted Sampling} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

A fully dynamic approximate distance oracle is a distance reporting data structure that supports dynamic insert edge and delete edge operations. In this paper we break a longstanding barrier in the design of fully dynamic all-pairs approximate distance oracles.
All previous results for this model incurred an amortized cost of at least Omega(n) per operation. We present the first construction that provides constant stretch and o(m) amortized update time. For graphs that are not too dense (where |E| = O(|V|^{2-delta}) for some delta>0 we break the O(n) barrier and provide the first construction with constant stretch and o(n) amortized cost.

Ittai Abraham, Shiri Chechik, and Kunal Talwar. Fully Dynamic All-Pairs Shortest Paths: Breaking the O(n) Barrier. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 1-16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{abraham_et_al:LIPIcs.APPROX-RANDOM.2014.1, author = {Abraham, Ittai and Chechik, Shiri and Talwar, Kunal}, title = {{Fully Dynamic All-Pairs Shortest Paths: Breaking the O(n) Barrier}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {1--16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.1}, URN = {urn:nbn:de:0030-drops-46857}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.1}, annote = {Keywords: Shortest Paths, Dynamic Algorithms} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In the {\em uncapacitated facility location} problem, given a graph,
a set of demands and opening costs, it is required to find a set of facilities $R$, so as to minimize the sum of the cost of opening the facilities in $R$ and the cost of assigning all node demands to open facilities.
This paper concerns the {\em robust fault-tolerant} version of the uncapacitated facility location problem (RFTFL). In this problem, one or more facilities might fail, and each demand should be supplied by the closest open facility that did not fail. It is required to find a set of facilities $R$, so as to minimize the sum of the cost of opening the facilities in $R$ and the cost of assigning all node demands to open facilities that did not fail, after the failure of up to $\alpha$ facilities.
We present a polynomial time algorithm that yields a 6.5-approximation
for this problem with at most one failure and a $1.5 + 7.5\alpha$-approximation for the problem with at most $\alpha > 1$ failures. We also show that the $RFTFL$ problem is NP-hard even on trees, and even in the case of a single failure.

Shiri Chechik and David Peleg. Robust Fault Tolerant Uncapacitated Facility Location. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 191-202, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chechik_et_al:LIPIcs.STACS.2010.2454, author = {Chechik, Shiri and Peleg, David}, title = {{Robust Fault Tolerant Uncapacitated Facility Location}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {191--202}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2454}, URN = {urn:nbn:de:0030-drops-24547}, doi = {10.4230/LIPIcs.STACS.2010.2454}, annote = {Keywords: Facility location, approximation algorithms, fault-tolerance} }

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