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

We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. [George Giakkoupis et al., 2019] have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d = Ω(log n). The case of sub-logarithmic degree was left open, and is the main focus of this paper.
First, we observe that the equivalence shown in [George Giakkoupis et al., 2019] does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log² n/log log n), whereas randomized rumor spreading is completed in time Θ(log n), w.h.p. Next, we show a general upper bound of Õ(d ⋅ diam(G) + log³ n /d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(log n), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(log n⋅ log log n), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids (k ≥ 1 is constant), by adapting a technique by Kesten and Sidoravicius [Kesten and Sidoravicius, 2003; Kesten and Sidoravicius, 2005].

George Giakkoupis, Hayk Saribekyan, and Thomas Sauerwald. Spread of Information and Diseases via Random Walks in Sparse Graphs. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{giakkoupis_et_al:LIPIcs.DISC.2020.9, author = {Giakkoupis, George and Saribekyan, Hayk and Sauerwald, Thomas}, title = {{Spread of Information and Diseases via Random Walks in Sparse Graphs}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {9:1--9:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.9}, URN = {urn:nbn:de:0030-drops-130873}, doi = {10.4230/LIPIcs.DISC.2020.9}, annote = {Keywords: parallel random walks, information dissemination, infectious diseases} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Recently, Aspnes and Ruppert (DISC 2016) defined the following simple random experiment to determine the impact of concurrency on the performance of binary search trees: n randomly permuted keys arrive one at a time. When a new key arrives, it is first placed into a buffer of size c. Whenever the buffer is full, or when all keys have arrived, an adversary chooses one key from the buffer and inserts it into the binary search tree.
The ability of the adversary to choose the next key to insert among c buffered keys, models a distributed system, where up to c processes try to insert keys concurrently. Aspnes and Ruppert showed that the expected average depth of nodes in the resulting tree is O(log(n) + c) for a comparison-based adversary, which can only take the relative order of arrived keys into account. We generalize and strengthen this result. In particular, we allow an adversary that knows the actual values of all keys that have arrived, and show that the resulting expected average node depth is D_{avg}(n) + O(c), where D_{avg}(n) = 2ln(n) - Theta(1) is the expected average node depth of a random tree obtained in the standard unbuffered version of this experiment. Extending the bound by Aspnes and Ruppert to this stronger adversary model answers one of their open questions.

George Giakkoupis and Philipp Woelfel. An Improved Bound for Random Binary Search Trees with Concurrent Insertions. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{giakkoupis_et_al:LIPIcs.STACS.2018.37, author = {Giakkoupis, George and Woelfel, Philipp}, title = {{An Improved Bound for Random Binary Search Trees with Concurrent Insertions}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {37:1--37:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.37}, URN = {urn:nbn:de:0030-drops-85108}, doi = {10.4230/LIPIcs.STACS.2018.37}, annote = {Keywords: random binary search tree, buffer, average depth, concurrent data structures} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Plurality consensus considers a network of n nodes, each having one of k opinions. Nodes execute a (randomized) distributed protocol with the goal that all nodes adopt the plurality (the opinion initially supported by the most nodes). Communication is realized via the Gossip (or random phone call) model. A major open question has been whether there is a protocol for the complete graph that converges (w.h.p.) in polylogarithmic time and uses only polylogarithmic memory per node (local memory). We answer this question affirmatively.
We propose two protocols that need only mild assumptions on the bias in favor of the plurality. As an example of our results, consider the complete graph and an arbitrarily small constant multiplicative bias in favor of the plurality. Our first protocol achieves plurality consensus in O(log(k)*log(log(n))) rounds using log(k) + Theta(log(log(k))) bits of local memory. Our second protocol achieves plurality consensus in O(log(n)*log(log(n))) rounds using only log(k) + 4 bits of local memory. This disproves a conjecture by Becchetti et al. (SODA'15) implying that any protocol with local memory log(k)+O(1) has worst-case runtime Omega(k). We provide similar bounds for much weaker bias assumptions. At the heart of our protocols lies an undecided state, an idea introduced by Angluin et al. (Distributed Computing'08).

Petra Berenbrink, Tom Friedetzky, George Giakkoupis, and Peter Kling. Efficient Plurality Consensus, Or: the Benefits of Cleaning up from Time to Time. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 136:1-136:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berenbrink_et_al:LIPIcs.ICALP.2016.136, author = {Berenbrink, Petra and Friedetzky, Tom and Giakkoupis, George and Kling, Peter}, title = {{Efficient Plurality Consensus, Or: the Benefits of Cleaning up from Time to Time}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {136:1--136:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.136}, URN = {urn:nbn:de:0030-drops-62711}, doi = {10.4230/LIPIcs.ICALP.2016.136}, annote = {Keywords: plurality consensus, voting, majority, distributed, gossip} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In the voter model, each node of a graph has an opinion, and in every round each node chooses independently a random neighbour and adopts its opinion. We are interested in the consensus time, which is the first point in time where all nodes have the same opinion. We consider dynamic graphs in which the edges are rewired in every round (by an adversary) giving rise to the graph sequence G_1, G_2, ..., where we assume that G_i has conductance at least phi_i. We assume that the degrees of nodes don't change over time as one can show that the consensus time can become super-exponential otherwise. In the case of a sequence of d-regular graphs, we obtain asymptotically tight results. Even for some static graphs, such as the cycle, our results improve the state of the art. Here we show that the expected number of rounds until all nodes have the same opinion is bounded by O(m/(d_{min}*phi)), for any graph with m edges, conductance phi, and degrees at least d_{min}. In addition, we consider a biased dynamic voter model, where each opinion i is associated with a probability P_i, and when a node chooses a neighbour with that opinion, it adopts opinion i with probability P_i (otherwise the node keeps its current opinion). We show for any regular dynamic graph, that if there is an epsilon > 0 difference between the highest and second highest opinion probabilities, and at least Omega(log(n)) nodes have initially the opinion with the highest probability, then all nodes adopt w.h.p. that opinion. We obtain a bound on the convergence time, which becomes O(log(n)/phi) for static graphs.

Petra Berenbrink, George Giakkoupis, Anne-Marie Kermarrec, and Frederik Mallmann-Trenn. Bounds on the Voter Model in Dynamic Networks. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 146:1-146:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berenbrink_et_al:LIPIcs.ICALP.2016.146, author = {Berenbrink, Petra and Giakkoupis, George and Kermarrec, Anne-Marie and Mallmann-Trenn, Frederik}, title = {{Bounds on the Voter Model in Dynamic Networks}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {146:1--146:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.146}, URN = {urn:nbn:de:0030-drops-62901}, doi = {10.4230/LIPIcs.ICALP.2016.146}, annote = {Keywords: Voting, Distributed Computing, Conductance, Dynamic Graphs, Consensus} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple push-based protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudo-random generator, respectively. For several well-studied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability.
Previous explicit algorithms require Omega(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashing-based algorithm is within an O(log n)-factor of the theoretical minimum determined by [Giakkoupis and Woelfel, 2011].

George Giakkoupis, Thomas Sauerwald, He Sun, and Philipp Woelfel. Low Randomness Rumor Spreading via Hashing. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 314-325, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{giakkoupis_et_al:LIPIcs.STACS.2012.314, author = {Giakkoupis, George and Sauerwald, Thomas and Sun, He and Woelfel, Philipp}, title = {{Low Randomness Rumor Spreading via Hashing}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {314--325}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.314}, URN = {urn:nbn:de:0030-drops-34417}, doi = {10.4230/LIPIcs.STACS.2012.314}, annote = {Keywords: Parallel and Distributed Computing, Randomness, Rumor Spreading} }

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

We study the connection between the rate at which a rumor spreads throughout a graph and the conductance of the graph -- a standard measure of a graph's expansion properties.
We show that for any n-node graph with conductance phi, the classical PUSH-PULL algorithm distributes a rumor to all nodes of the graph in O(phi^(-1) log(n)) rounds with high probability (w.h.p.). This bound improves a recent result of Chierichetti, Lattanzi, and Panconesi [STOC 2010], and it is tight in the sense that there exist graphs where Omega(phi^(-1)log(n)) rounds of the PUSH-PULL algorithm are required to distribute a rumor w.h.p.
We also explore the PUSH and the PULL algorithms, and derive conditions that are both necessary and sufficient for the above upper bound to hold for those algorithms as well.
An interesting finding is that every graph contains a node such that the PULL algorithm takes O(phi^(-1) log(n)) rounds w.h.p. to distribute a rumor started at that node.
In contrast, there are graphs where the PUSH algorithm requires significantly more rounds for any start node.

George Giakkoupis. Tight bounds for rumor spreading in graphs of a given conductance. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 57-68, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{giakkoupis:LIPIcs.STACS.2011.57, author = {Giakkoupis, George}, title = {{Tight bounds for rumor spreading in graphs of a given conductance}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {57--68}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.57}, URN = {urn:nbn:de:0030-drops-29977}, doi = {10.4230/LIPIcs.STACS.2011.57}, annote = {Keywords: conductance, rumor spreading} }

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