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**Published in:** LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)

We study the SUPPORTED model of distributed computing introduced by Schmid and Suomela [Schmid and Suomela, 2013], which generalizes the LOCAL and CONGEST models. In this framework, multiple instances of the same problem, differing from each other by the subnetwork to which they apply. recur over time, and need to be solved efficiently online. To do that, one may rely on an initial preprocessing phase for computing some useful information. This preprocessing phase makes it possible, in some cases, to obtain improved distributed algorithms, overcoming locality-based time lower bounds.
Our main contribution is to expand the class of problems to which the SUPPORTED model applies, by handling also multiple recurring instances of the same problem that differ from each other by some problem specific input, and not only the subnetwork to which they apply. We illustrate this by considering two extended problem classes. The first class, denoted PCS, concerns problems where client nodes of the network need to be served, and each recurring instance applies to some Partial Client Set. The second class, denoted PFO, concerns situations where each recurrent instance of the problem includes a partially fixed output, which needs to be completed to a full consistent solution.
Specifically, we propose some natural recurrent variants of the dominating set problem and the coloring problem that are of interest particularly in the distributed setting. For these problems, we show that information about the topology can be used to overcome locality-based lower bounds. We also categorize the round complexity of Locally Checkable Labellings in the SUPPORTED model for the simple case of paths. Finally we present some interesting open problems and some partial results towards resolving them.

Akanksha Agrawal, John Augustine, David Peleg, and Srikkanth Ramachandran. Local Recurrent Problems in the SUPPORTED Model. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.OPODIS.2023.22, author = {Agrawal, Akanksha and Augustine, John and Peleg, David and Ramachandran, Srikkanth}, title = {{Local Recurrent Problems in the SUPPORTED Model}}, booktitle = {27th International Conference on Principles of Distributed Systems (OPODIS 2023)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-308-9}, ISSN = {1868-8969}, year = {2024}, volume = {286}, editor = {Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.22}, URN = {urn:nbn:de:0030-drops-195124}, doi = {10.4230/LIPIcs.OPODIS.2023.22}, annote = {Keywords: Distributed Algorithms, LOCAL Model, SUPPORTED Model} }

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Complete Volume

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

LIPIcs, Volume 272, MFCS 2023, Complete Volume

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 1-1302, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{leroux_et_al:LIPIcs.MFCS.2023, title = {{LIPIcs, Volume 272, MFCS 2023, Complete Volume}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {1--1302}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023}, URN = {urn:nbn:de:0030-drops-185332}, doi = {10.4230/LIPIcs.MFCS.2023}, annote = {Keywords: LIPIcs, Volume 272, MFCS 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Front Matter, Table of Contents, Preface, Conference Organization

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{leroux_et_al:LIPIcs.MFCS.2023.0, author = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.0}, URN = {urn:nbn:de:0030-drops-185349}, doi = {10.4230/LIPIcs.MFCS.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

A singularly (near) optimal distributed algorithm is one that is (near) optimal in two criteria, namely, its time and message complexities. For synchronous CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general asynchronous CONGEST networks.
In this paper, we present a randomized distributed MST algorithm that, with high probability, computes an MST in asynchronous CONGEST networks and takes Õ(D^{1+ε} + √n) time and Õ(m) messages, where n is the number of nodes, m the number of edges, D is the diameter of the network, and ε > 0 is an arbitrarily small constant (both time and message bounds hold with high probability). Since Ω̃(D+√n) and Ω(m) are respective time and message lower bounds for distributed MST construction in the standard KT₀ model, our algorithm is message optimal (up to a polylog(n) factor) and almost time optimal (except for a D^ε factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all D = O(n^{1-ε})) and uses Õ(m) messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the KT₁ CONGEST model.
A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth Õ(D^{1+ε}) (for an arbitrarily small constant ε > 0) in Õ(D^{1+ε}) time and Õ(m) messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting. This tree construction may be of independent interest as it can also be used for efficiently performing basic tasks such as verified broadcast and convergecast in asynchronous networks.

Fabien Dufoulon, Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. An Almost Singularly Optimal Asynchronous Distributed MST Algorithm. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dufoulon_et_al:LIPIcs.DISC.2022.19, author = {Dufoulon, Fabien and Kutten, Shay and Moses Jr., William K. and Pandurangan, Gopal and Peleg, David}, title = {{An Almost Singularly Optimal Asynchronous Distributed MST Algorithm}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {19:1--19:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.19}, URN = {urn:nbn:de:0030-drops-172107}, doi = {10.4230/LIPIcs.DISC.2022.19}, annote = {Keywords: Asynchronous networks, Minimum Spanning Tree, Distributed Algorithm, Singularly Optimal} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The Distance Realization problem is defined as follows. Given an n × n matrix D of nonnegative integers, interpreted as inter-vertex distances, find an n-vertex weighted or unweighted graph G realizing D, i.e., whose inter-vertex distances satisfy dist_G(i,j) = D_{i,j} for every 1 ≤ i < j ≤ n, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of Distance Realization that was studied in the past is where each entry in the matrix D may contain a range of consecutive permissible values. We refer to this extension as Range Distance Realization (or Range-DR). Restricting each range to at most k values yields the problem k-Range Distance Realization (or k-Range-DR). The current paper introduces a new extension of Distance Realization, in which each entry D_{i,j} of the matrix may contain an arbitrary set of acceptable values for the distance between i and j, for every 1 ≤ i < j ≤ n. We refer to this extension as Set Distance Realization (Set-DR), and to the restricted problem where each entry may contain at most k values as k-Set Distance Realization (or k-Set-DR).
We first show that 2-Range-DR is NP-hard for unweighted graphs (implying the same for 2-Set-DR). Next we prove that 2-Set-DR is NP-hard for unweighted and weighted trees. We then explore Set-DR where the realization is restricted to the families of stars, paths, or cycles. For the weighted case, our positive results are that for each of these families there exists a polynomial time algorithm for 2-Set-DR. On the hardness side, we prove that 6-Set-DR is NP-hard for stars and 5-Set-DR is NP-hard for paths and cycles. For the unweighted case, our results are the same, except for the case of unweighted stars, for which k-Set-DR is polynomially solvable for any k.

Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz. Graph Realization of Distance Sets. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2022.13, author = {Bar-Noy, Amotz and Peleg, David and Perry, Mor and Rawitz, Dror}, title = {{Graph Realization of Distance Sets}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.13}, URN = {urn:nbn:de:0030-drops-168119}, doi = {10.4230/LIPIcs.MFCS.2022.13}, annote = {Keywords: Graph Realization, distance realization, network design} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analysing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2022.14, author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Peleg, David and Rawitz, Dror}, title = {{On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.14}, URN = {urn:nbn:de:0030-drops-168121}, doi = {10.4230/LIPIcs.MFCS.2022.14}, annote = {Keywords: Graph Realization, Bipartite Graphs, Degree Sequences, Graphic Sequences, Bigraphic Sequences, Approximate Realization, Multigraph Realization} }

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Invited Paper

**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

This paper addresses the classical problem of characterizing degree sequences that can be realized by a bipartite graph. For the simpler variant of the problem, where a partition of the sequence into the two sides of the bipartite graph is given as part of the input, a complete characterization was given by Gale and Ryser over 60 years ago. However, the general question, in which both the partition and the realizing graph need to be determined, is still open. This paper provides an overview of some of the known results on this problem in interesting special cases, including realizations by bipartite graphs and bipartite multigraphs.

Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On Realizing a Single Degree Sequence by a Bipartite Graph (Invited Paper). In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{barnoy_et_al:LIPIcs.SWAT.2022.1, author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Peleg, David and Rawitz, Dror}, title = {{On Realizing a Single Degree Sequence by a Bipartite Graph}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {1:1--1:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.1}, URN = {urn:nbn:de:0030-drops-161618}, doi = {10.4230/LIPIcs.SWAT.2022.1}, annote = {Keywords: Degree Sequences, Graph Realization, Bipartite Graphs, Graphic Sequences, Bigraphic Sequences, Multigraph Realization} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

The classical degree realization problem is defined as follows: Given a sequence d̄ = (d_1,…,d_n) of positive integers, construct an n-vertex graph in which each vertex u_i has degree d_i (or decide that no such graph exists). In this article, we present and study the related selected neighbor degree realization problem, which requires that each vertex u_i of G has a neighbor of degree d_i. We solve the problem when G is required to be acyclic (i.e., a forest), and present a sufficient and necessary condition for a given sequence to be realizable.

Amotz Bar-Noy, David Peleg, Dror Rawitz, and Elad Yehezkel. Selected Neighbor Degree Forest Realization. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{barnoy_et_al:LIPIcs.ISAAC.2021.27, author = {Bar-Noy, Amotz and Peleg, David and Rawitz, Dror and Yehezkel, Elad}, title = {{Selected Neighbor Degree Forest Realization}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.27}, URN = {urn:nbn:de:0030-drops-154609}, doi = {10.4230/LIPIcs.ISAAC.2021.27}, annote = {Keywords: network realization, graph algorithms, lower bound} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in asynchronous networks.
Kutten et al. (JACM 2015) presented a singularly near optimal randomized leader election algorithm for general synchronous networks that ran in O(D) time and used O(m log n) messages (where D, m, and n are the network’s diameter, number of edges and number of nodes, respectively) with high probability. Both bounds are near optimal (up to a logarithmic factor), since Ω(D) and Ω(m) are the respective lower bounds for time and messages for leader election even for synchronous networks and even for (Monte-Carlo) randomized algorithms. On the other hand, for general asynchronous networks, leader election algorithms are only known that are either time or message optimal, but not both. Kutten et al. (DISC 2020) presented a randomized asynchronous leader election algorithm that is singularly near optimal for complete networks, but left open the problem for general networks.
This paper shows that singularly near optimal (up to polylogarithmic factors) bounds can be achieved for general asynchronous networks. We present a randomized singularly near optimal leader election algorithm that runs in O(D + log² n) time and O(m log² n) messages with high probability. Our result is the first known distributed leader election algorithm for asynchronous networks that is near optimal with respect to both time and message complexity and improves over a long line of results including the classical results of Gallager et al. (ACM TOPLAS, 1983), Peleg (JPDC, 1989), and Awerbuch (STOC, 89).

Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. Singularly Near Optimal Leader Election in Asynchronous Networks. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kutten_et_al:LIPIcs.DISC.2021.27, author = {Kutten, Shay and Moses Jr., William K. and Pandurangan, Gopal and Peleg, David}, title = {{Singularly Near Optimal Leader Election in Asynchronous Networks}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.27}, URN = {urn:nbn:de:0030-drops-148294}, doi = {10.4230/LIPIcs.DISC.2021.27}, annote = {Keywords: Leader election, Singular optimality, Randomized algorithms, Asynchronous networks, Arbitrary graphs} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We study the Budgeted Dominating Set (BDS) problem on uncertain graphs, namely, graphs with a probability distribution p associated with the edges, such that an edge e exists in the graph with probability p(e). The input to the problem consists of a vertex-weighted uncertain graph 𝒢 = (V, E, p, ω) and an integer budget (or solution size) k, and the objective is to compute a vertex set S of size k that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of S. We refer to the problem as the Probabilistic Budgeted Dominating Set (PBDS) problem. In this article, we present the following results on the complexity of the PBDS problem.
1) We show that the PBDS problem is NP-complete even when restricted to uncertain trees of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is 𝖶[1]-hard for the budget parameter k, and under the Exponential time hypothesis it cannot be solved in n^o(k) time.
2) We show that if one is willing to settle for (1-ε) approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time.
3) We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is 𝖶[1]-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget k and the treewidth.
4) Finally, we extend some of our parameterized results to planar and apex-minor-free graphs.
Our first hardness proof (Thm. 1) makes use of the new problem of k-Subset Σ-Π Maximization (k-SPM), which we believe is of independent interest. We prove its NP-hardness by a reduction from the well-known k-SUM problem, presenting a close relationship between the two problems.

Keerti Choudhary, Avi Cohen, N. S. Narayanaswamy, David Peleg, and R. Vijayaragunathan. Budgeted Dominating Sets in Uncertain Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{choudhary_et_al:LIPIcs.MFCS.2021.32, author = {Choudhary, Keerti and Cohen, Avi and Narayanaswamy, N. S. and Peleg, David and Vijayaragunathan, R.}, title = {{Budgeted Dominating Sets in Uncertain Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {32:1--32:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.32}, URN = {urn:nbn:de:0030-drops-144723}, doi = {10.4230/LIPIcs.MFCS.2021.32}, annote = {Keywords: Uncertain graphs, Dominating set, NP-hard, PTAS, treewidth, planar graph} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses O(n) messages with high probability and runs in O(log² n) time (with high probability) to elect a unique leader. The O(n) message complexity should be contrasted with the Ω(n log n) lower bounds for the deterministic message complexity of leader election algorithms (regardless of time), proven by Korach, Moran, and Zaks (TCS, 1989) for asynchronous algorithms and by Afek and Gafni (SIAM J. Comput., 1991) for synchronous networks. Hence, our result also separates the message complexities of randomized and deterministic leader election. More importantly, our (randomized) time complexity of O(log² n) for obtaining the optimal O(n) message complexity is significantly smaller than the long-standing Θ̃(n) time complexity obtained by Afek and Gafni and by Singh (SIAM J. Comput., 1997) for message optimal (deterministic) election in asynchronous networks. Afek and Gafni also conjectured that Θ̃(n) time would be optimal for message-optimal asynchronous algorithms. Our result shows that randomized algorithms are significantly faster.
Turning to synchronous complete networks, Afek and Gafni showed an essentially singularly optimal deterministic algorithm with O(log n) time and O(n log n) messages. Ramanathan et al. (Distrib. Comput. 2007) used randomization to improve the message complexity, and showed a randomized algorithm with O(n) messages but still with O(log n) time (with failure probability O(1 / log^{Ω(1)}n)). Our second result shows that synchronous complete networks admit a tightly singularly optimal randomized algorithm, with O(1) time and O(n) messages (both bounds are optimal). Moreover, our algorithm’s time bound holds with certainty, and its message bound holds with high probability, i.e., 1-1/n^c for constant c.
Our results demonstrate that leader election can be solved in a simultaneously message and time-efficient manner in asynchronous complete networks using randomization. It is open whether this is possible in asynchronous general networks.

Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. Singularly Optimal Randomized Leader Election. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kutten_et_al:LIPIcs.DISC.2020.22, author = {Kutten, Shay and Moses Jr., William K. and Pandurangan, Gopal and Peleg, David}, title = {{Singularly Optimal Randomized Leader Election}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.22}, URN = {urn:nbn:de:0030-drops-131002}, doi = {10.4230/LIPIcs.DISC.2020.22}, annote = {Keywords: Leader election, Asynchronous systems, Randomized algorithms, Singularly optimal, Complete networks} }

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

We study a graph realization problem that pertains to degrees in vertex neighborhoods. The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles.
In this paper we introduce and explore the minimum degrees in vertex neighborhood profile as it is one of the most natural extensions of the classical degree profile to vertex neighboring degree profiles. Given a graph G = (V,E), the min-degree of a vertex v ∈ V, namely MinND(v), is given by min{deg(w) ∣ w ∈ N[v]}. Our input is a sequence σ = (d_𝓁^{n_𝓁}, ⋯ , d₁^{n₁}), where d_{i+1} > d_i and each n_i is a positive integer. We provide some necessary and sufficient conditions for σ to be realizable. Furthermore, under the restriction that the realization is acyclic, i.e., a tree or a forest, we provide a full characterization of realizable sequences, along with a corresponding constructive algorithm.
We believe our results are a crucial step towards understanding extremal neighborhood degree relations in graphs.

Amotz Bar-Noy, Keerti Choudhary, Avi Cohen, David Peleg, and Dror Rawitz. Minimum Neighboring Degree Realization in Graphs and Trees. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{barnoy_et_al:LIPIcs.ESA.2020.10, author = {Bar-Noy, Amotz and Choudhary, Keerti and Cohen, Avi and Peleg, David and Rawitz, Dror}, title = {{Minimum Neighboring Degree Realization in Graphs and Trees}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {10:1--10:15}, 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.2020.10}, URN = {urn:nbn:de:0030-drops-128765}, doi = {10.4230/LIPIcs.ESA.2020.10}, annote = {Keywords: Graph realization, neighborhood profile, graph algorithms, degree sequences} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles.
In this paper, we initiate the study of neighborhood degree profiles, wherein, our focus is on the natural problem of realizing maximum neighborhood degrees. More specifically, we ask the following question: "Given a sequence D of n non-negative integers 0≤ d₁≤ ⋯ ≤ d_n, does there exist a simple graph with vertices v₁,…, v_n such that for every 1≤ i ≤ n, the maximum degree in the neighborhood of v_i is exactly d_i?"
We provide in this work various results for maximum-neighborhood-degree for general n vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For closed as well as open neighborhood degree profiles, we provide a complete realizability criteria. We also provide tight bounds for the number of maximum neighbouring degree profiles of length n that are realizable. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time.

Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Graph Realizations: Maximum Degree in Vertex Neighborhoods. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{barnoy_et_al:LIPIcs.SWAT.2020.10, author = {Bar-Noy, Amotz and Choudhary, Keerti and Peleg, David and Rawitz, Dror}, title = {{Graph Realizations: Maximum Degree in Vertex Neighborhoods}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.10}, URN = {urn:nbn:de:0030-drops-122572}, doi = {10.4230/LIPIcs.SWAT.2020.10}, annote = {Keywords: Graph realization, neighborhood profile, extremum-degree} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat [Andrea Frosini and Maurice Nivat, 2007; Nivat, 2002]. Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value l_x, which we would like to determine. However, it might be that obtaining these values precisely (by a surgical probe) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods.
In this paper we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G, and the vector l in R^n assigns a value l_v to each node v. A probe centered at a vertex v will capture a window encompassing its entire neighborhood N[v], i.e., the outcome of a probe centered at v is P_v = sum_{w in N[v]} l_w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector l from the collection of probes, P^- = {P_v | v in V}.
We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector l yielding P). Let us assume that surgical probes (whose outcome at vertex v is the exact value of l_v) are technically available to us (yet are expensive or risky, and must be used sparingly). We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of standard probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP).
Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviors of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k-ary trees, paths, cycles, grids, tori and tubes.

Amotz Bar-Noy, Toni Böhnlein, Zvi Lotker, David Peleg, and Dror Rawitz. The Generalized Microscopic Image Reconstruction Problem. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barnoy_et_al:LIPIcs.ISAAC.2019.42, author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Lotker, Zvi and Peleg, David and Rawitz, Dror}, title = {{The Generalized Microscopic Image Reconstruction Problem}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.42}, URN = {urn:nbn:de:0030-drops-115382}, doi = {10.4230/LIPIcs.ISAAC.2019.42}, annote = {Keywords: Discrete mathematics, Combinatorics, Reconstruction algorithm, Image reconstruction, Graph spectra, Grid graphs} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We consider the problem of realizable interval-sequences. An interval sequence comprises of n integer intervals [a_i,b_i] such that 0 <= a_i <= b_i <= n-1, and is said to be graphic/realizable if there exists a graph with degree sequence, say, D=(d_1,...,d_n) satisfying the condition a_i <= d_i <= b_i, for each i in [1,n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdös-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n^2) time.
In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length p non-graphic sequence to a graphic one.

Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Efficiently Realizing Interval Sequences. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barnoy_et_al:LIPIcs.ISAAC.2019.47, author = {Bar-Noy, Amotz and Choudhary, Keerti and Peleg, David and Rawitz, Dror}, title = {{Efficiently Realizing Interval Sequences}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.47}, URN = {urn:nbn:de:0030-drops-115430}, doi = {10.4230/LIPIcs.ISAAC.2019.47}, annote = {Keywords: Graph realization, graphic sequence, interval sequence} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

This paper establishes tight bounds for the Minimum-weight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves simultaneously tilde ~O(|E|) messages and $tilde O(sqrt{n} + D) time, where |E| is the number of edges in the given graph G and D is G's diameter. On the negative side, we show that any MST verification algorithm must send Omega(|E|) messages and incur ~Omega(sqrt{n} + D) time in worst case.
Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of Omega(|E|) messages and
Omega(sqrt{n} + D) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously tilde O(|E|) messages and ´~O(sqrt{n} + D) time. Specifically, the best known time-optimal algorithm (using ~O(sqrt{n} + D) time) requires O(|E|+n^{3/2}) messages, and the best known message-optimal algorithm (using ~O(|E|) messages) requires O(n) time.
On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.

Liah Kor, Amos Korman, and David Peleg. Tight Bounds For Distributed MST Verification. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 69-80, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kor_et_al:LIPIcs.STACS.2011.69, author = {Kor, Liah and Korman, Amos and Peleg, David}, title = {{Tight Bounds For Distributed MST Verification}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {69--80}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.69}, URN = {urn:nbn:de:0030-drops-30000}, doi = {10.4230/LIPIcs.STACS.2011.69}, annote = {Keywords: distributed algorithms, distributed verification, labeling schemes, minimum-weight spanning tree} }

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

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of
$n$ points and $\delta$ is a distance function defined for these
points. Assume that $(V,\delta)$ has a constant doubling dimension
$d$ and assume that each point $p\in V$ has a disk of radius
$r(p)$ around it. The disk graph that corresponds to $V$ and
$r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices
are the points of $V$ and whose edge set includes a directed edge
from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In~\cite{PeRo08} we
presented an algorithm for constructing a $(1+\eps)$-spanner of
size $O(n/\eps^d \log M)$, where $M$ is the maximal radius $r(p)$.
The current paper presents two results. The first shows that the
spanner of~\cite{PeRo08} is essentially optimal, i.e., for metrics
of constant doubling dimension it is not possible to guarantee a
spanner whose size is independent of $M$. The second result shows
that by slightly relaxing the requirements and allowing a small
perturbation of the radius assignment, considerably better
spanners can be constructed. In particular, we show that if it is
allowed to use edges of the disk graph $I(V,E,r_{1+\eps})$, where
$r_{1+\eps}(p) = (1+\eps)\cdot r(p)$ for every $p\in V$, then it
is possible to get a $(1+\eps)$-spanner of size $O(n/\eps^d)$ for
$I(V,E,r)$. Our algorithm is simple and can be implemented
efficiently.

David Peleg and Liam Roditty. Relaxed Spanners for Directed Disk Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 609-620, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{peleg_et_al:LIPIcs.STACS.2010.2489, author = {Peleg, David and Roditty, Liam}, title = {{Relaxed Spanners for Directed Disk Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {609--620}, 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.2010.2489}, URN = {urn:nbn:de:0030-drops-24898}, doi = {10.4230/LIPIcs.STACS.2010.2489}, annote = {Keywords: Spanners, directed graphs} }

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