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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO'20). In this problem, the goal is to locally decide for each e ∈ E if it is in G' where G' is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ²). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n^{2/3}) queries for ϕ = Ω(1/n^{1/12}).
We then extend our result for (k, ϕ_in, ϕ_out)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕ_out n) for constant k and ϕ_in. This bound is almost optimal when ϕ_out = O(1/√n).

Reut Levi, Moti Medina, and Omer Tubul. Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{levi_et_al:LIPIcs.APPROX/RANDOM.2024.60, author = {Levi, Reut and Medina, Moti and Tubul, Omer}, title = {{Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {60:1--60:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.60}, URN = {urn:nbn:de:0030-drops-210537}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.60}, annote = {Keywords: Locally Computable Algorithms, Sublinear algorithms, Spanning Subgraphs, Clusterbale Graphs} }

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

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least n/2, where n denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model.
This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within O(log n) rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require Ω̃(n²) rounds, as shown by Bachrach et al. [PODC'19].
In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least n, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least n+1. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.

Noy Biton, Reut Levi, and Moti Medina. Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{biton_et_al:LIPIcs.MFCS.2023.19, author = {Biton, Noy and Levi, Reut and Medina, Moti}, title = {{Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {19:1--19:14}, 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.2023.19}, URN = {urn:nbn:de:0030-drops-185534}, doi = {10.4230/LIPIcs.MFCS.2023.19}, annote = {Keywords: the CONGEST model, Hamiltonian Path, Hamiltonian Cycle, Dirac graphs, Ore graphs, graph-algorithms} }

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

Ikenmeyer et al. (JACM'19) proved an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions. This raises the question: which classes of functions permit efficient hazard-free circuits?
In this work, we prove that circuit implementations of transducers with small state space are such a class. A transducer is a finite state machine that transcribes, symbol by symbol, an input string of length n into an output string of length n. We present a construction that transforms any function arising from a transducer into an efficient circuit of size 𝒪(n) computing the hazard-free extension of the function. More precisely, given a transducer with s states, receiving n input symbols encoded by l bits, and computing n output symbols encoded by m bits, the transducer has a hazard-free circuit of size n*m*2^{𝒪(s+𝓁)} and depth 𝒪(s*log(n) + 𝓁); in particular, if s, 𝓁,m ∈ 𝒪(1), size and depth are asymptotically optimal. In light of the strong hardness results by Ikenmeyer et al. (JACM'19), we consider this a surprising result.

Johannes Bund, Christoph Lenzen, and Moti Medina. Small Hazard-Free Transducers. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 32:1-32:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bund_et_al:LIPIcs.ITCS.2022.32, author = {Bund, Johannes and Lenzen, Christoph and Medina, Moti}, title = {{Small Hazard-Free Transducers}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {32:1--32:24}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.32}, URN = {urn:nbn:de:0030-drops-156281}, doi = {10.4230/LIPIcs.ITCS.2022.32}, annote = {Keywords: Hazard-Freeness, Parallel Prefix Computation, Finite State Transducers} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, G_U, is isomorphic to G_K which is given as an input to all the nodes in the network, or alternatively, only to a single node.
We first consider the decision variant of the problem in which the algorithm should distinguish the case where G_U and G_K are isomorphic from the case where G_U and G_K are not isomorphic. Specifically, if G_U and G_K are not isomorphic then w.h.p. at least one node should output reject and otherwise all nodes should output accept . We provide a randomized algorithm with O(n) rounds for the setting in which G_K is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is Ω̃(n²) rounds, where n denotes the number of nodes, which implies a separation between the randomized and the deterministic complexities of deciding GI . Our algorithm can be adapted to the semi-streaming model, where a single pass is performed and Õ(n) bits of space are used.
We then consider the property testing variant of the problem, where the algorithm is only required to distinguish the case that G_U and G_K are isomorphic from the case that G_U and G_K are far from being isomorphic (according to some predetermined distance measure). We show that every (possibly randomized) algorithm, requires Ω(D) rounds, where D denotes the diameter of the network. This lower bound holds even if all the nodes are given G_K as an input, and even if the message size is unbounded. We provide a randomized algorithm with an almost matching round complexity of O(D+(ε^{-1}log n)²) rounds that is suitable for dense graphs (namely, graphs with Ω(n²) edges).
We also show that with the same number of rounds it is possible that each node outputs its mapping according to a bijection which is an approximate isomorphism.
We conclude with simple simulation arguments that allow us to adapt centralized property testing algorithms and obtain essentially tight algorithms with round complexity Õ(D) for special families of sparse graphs.

Reut Levi and Moti Medina. Distributed Testing of Graph Isomorphism in the CONGEST Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{levi_et_al:LIPIcs.APPROX/RANDOM.2020.19, author = {Levi, Reut and Medina, Moti}, title = {{Distributed Testing of Graph Isomorphism in the CONGEST Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {19:1--19:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.19}, URN = {urn:nbn:de:0030-drops-126221}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.19}, annote = {Keywords: the CONGEST model, graph isomorphism, distributed property testing, distributed decision, graph algorithms} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

This paper presents a deterministic distributed algorithm for computing an f(1+epsilon) approximation of the well-studied minimum set cover problem, for any constant epsilon>0, in O(log (f Delta)/log log (f Delta)) rounds. Here, f denotes the maximum element frequency and Delta denotes the cardinality of the largest set. This f(1+epsilon) approximation almost matches the f-approximation guarantee of standard centralized primal-dual algorithms, which is known to be essentially the best possible approximation for polynomial-time computations. The round complexity almost matches the Omega(log (Delta)/log log (Delta)) lower bound of Kuhn, Moscibroda, Wattenhofer [JACM'16], which holds for even f=2 and for any poly(log Delta) approximation. Our algorithm also gives an alternative way to reproduce the time-optimal 2(1+epsilon)-approximation of vertex cover, with round complexity O(log Delta/log log Delta), as presented by Bar-Yehuda, Censor-Hillel, and Schwartzman [PODC'17] for weighted vertex cover. Our method is quite different and it can be viewed as a locality-optimal way of performing primal-dual for the more general case of set cover. We note that the vertex cover algorithm of Bar-Yehuda et al. does not extend to set cover (when f >= 3).

Guy Even, Mohsen Ghaffari, and Moti Medina. Distributed Set Cover Approximation: Primal-Dual with Optimal Locality. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{even_et_al:LIPIcs.DISC.2018.22, author = {Even, Guy and Ghaffari, Mohsen and Medina, Moti}, title = {{Distributed Set Cover Approximation: Primal-Dual with Optimal Locality}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.22}, URN = {urn:nbn:de:0030-drops-98114}, doi = {10.4230/LIPIcs.DISC.2018.22}, annote = {Keywords: Distributed Algorithms, Approximation Algorithms, Set Cover, Vertex Cover} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

In this paper we present distributed property-testing algorithms for graph properties in the CONGEST model, with emphasis on testing subgraph-freeness. Testing a graph property P means distinguishing graphs G = (V,E) having property P from graphs that are epsilon-far from having it, meaning that epsilon|E| edges must be added or removed from G to obtain a graph satisfying P.
We present a series of results, including:
- Testing H-freeness in O(1/epsilon) rounds, for any constant-sized graph H containing an edge (u,v) such that any cycle in H contain either u or v (or both). This includes all connected graphs over five vertices except K_5. For triangles, we can do even better when epsilon is not too small.
- A deterministic CONGEST protocol determining whether a graph contains a given tree as a subgraph in constant time.
- For cliques K_s with s >= 5, we show that K_s-freeness can be tested in O(m^(1/2-1/(s-2)) epsilon^(-1/2-1/(s-2))) rounds, where m is the number of edges in the network graph.
- We describe a general procedure for converting epsilon-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((log n)/epsilon)+f((log n)/epsilon) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an epsilon-tester for testing whether a graph is bipartite and testing whether a graph is cycle-free. Moreover, for cycle-freeness, we obtain a corrector of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property.
These protocols extend and improve previous results of [Censor-Hillel et al. 2016] and [Fraigniaud et al. 2016].

Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three Notes on Distributed Property Testing. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 15:1-15:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{even_et_al:LIPIcs.DISC.2017.15, author = {Even, Guy and Fischer, Orr and Fraigniaud, Pierre and Gonen, Tzlil and Levi, Reut and Medina, Moti and Montealegre, Pedro and Olivetti, Dennis and Oshman, Rotem and Rapaport, Ivan and Todinca, Ioan}, title = {{Three Notes on Distributed Property Testing}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {15:1--15:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.15}, URN = {urn:nbn:de:0030-drops-79847}, doi = {10.4230/LIPIcs.DISC.2017.15}, annote = {Keywords: Property testing, Property correcting, Distributed algorithms, CONGEST model} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space and randomness complexities of such samplers.
In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space and random bits, especially when only a small number of queries are actually issued. Instead, we propose to generate the graph on-the-fly, in response to queries, and therefore to require amounts of time, space, and random bits which are a function of the actual number of queries.
We focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) and the random recursive tree model. We give on-the-fly generation algorithms for both models. With probability 1-1/poly(n), each and every query is answered in polylog(n) time, and the increase in space and the number of random bits consumed by any single query are both polylog(n), where n denotes the number of vertices in the graph.
Our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph's nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their performance on such graphs.

Guy Even, Reut Levi, Moti Medina, and Adi Rosén. Sublinear Random Access Generators for Preferential Attachment Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{even_et_al:LIPIcs.ICALP.2017.6, author = {Even, Guy and Levi, Reut and Medina, Moti and Ros\'{e}n, Adi}, title = {{Sublinear Random Access Generators for Preferential Attachment Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.6}, URN = {urn:nbn:de:0030-drops-74242}, doi = {10.4230/LIPIcs.ICALP.2017.6}, annote = {Keywords: local computation algorithms, preferential attachment graphs, random recursive trees, sublinear algorithms} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

In this paper we improve the approximation ratio for the problem of scheduling packets on line networks with bounded buffers with the aim of maximizing the throughput. Each node in the network has a local buffer of bounded size B, and each edge (or link) can transmit a limited number c of packets in every time unit. The input to the problem consists of a set of packet requests, each defined by a source node, a destination node, and a release time. We denote by n the size of the network. A solution for this problem is a schedule that delivers (some of the) packets to their destinations without violating the capacity constraints of the network (buffers or edges). Our goal is to design an efficient algorithm that computes a schedule that maximizes the number of packets that arrive to their respective destinations.
We give a randomized approximation algorithm with constant approximation ratio for the case where the buffer-size to link-capacity ratio, B/c, does not depend on the input size. This improves over the previously best result of O(log^* n) [Räcke and Rosén SPAA 2009]. Our improvement is based on a new combinatorial lemma that we prove, stating, roughly speaking, that if packets are allowed to stay put in buffers only a limited number of time steps, 2d, where d is the longest source-destination distance, then the optimal solution is decreased by only a constant factor. This claim was not previously known in the integral (unsplitable, zero-one) case, and may find additional applications for routing and scheduling algorithms.
While we are not able to give the same improvement for the related problem when packets have hard deadlines, our algorithm does support "soft deadlines". That is, if packets have deadlines, we achieve a constant approximation ratio when the produced solution is allowed to miss deadlines by at most log n time units.

Guy Even, Moti Medina, and Adi Rosén. A Constant Approximation Algorithm for Scheduling Packets on Line Networks. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{even_et_al:LIPIcs.ESA.2016.40, author = {Even, Guy and Medina, Moti and Ros\'{e}n, Adi}, title = {{A Constant Approximation Algorithm for Scheduling Packets on Line Networks}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.40}, URN = {urn:nbn:de:0030-drops-63524}, doi = {10.4230/LIPIcs.ESA.2016.40}, annote = {Keywords: approximation algorithms, linear programming, randomized rounding, packet scheduling, admission control} }

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