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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.

Petra Berenbrink, Lukas Hintze, Hamed Hosseinpour, Dominik Kaaser, and Malin Rau. Dynamic Averaging Load Balancing on Arbitrary Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.ICALP.2023.18, author = {Berenbrink, Petra and Hintze, Lukas and Hosseinpour, Hamed and Kaaser, Dominik and Rau, Malin}, title = {{Dynamic Averaging Load Balancing on Arbitrary Graphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.18}, URN = {urn:nbn:de:0030-drops-180707}, doi = {10.4230/LIPIcs.ICALP.2023.18}, annote = {Keywords: Dynamic Load Balancing, Distributed Computing, Randomized Algorithms, Drift Analysis} }

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Complete Volume

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

LIPIcs, Volume 254, STACS 2023, Complete Volume

40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 1-1026, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{berenbrink_et_al:LIPIcs.STACS.2023, title = {{LIPIcs, Volume 254, STACS 2023, Complete Volume}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {1--1026}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023}, URN = {urn:nbn:de:0030-drops-176515}, doi = {10.4230/LIPIcs.STACS.2023}, annote = {Keywords: LIPIcs, Volume 254, STACS 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Front Matter, Table of Contents, Preface, Conference Organization

40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 0:i-0:xxii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.STACS.2023.0, author = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {0:i--0:xxii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.0}, URN = {urn:nbn:de:0030-drops-176525}, doi = {10.4230/LIPIcs.STACS.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

We study the consensus problem among n agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions.
We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called j-Majority: when activated, every agent samples j other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: (j+1)-Majority (for j > 1) converges stochastically faster than j-Majority for any initial opinion configuration. In our analysis we use Strassen’s Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [PODC 2017].

Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau. On the Hierarchy of Distributed Majority Protocols. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.OPODIS.2022.23, author = {Berenbrink, Petra and Coja-Oghlan, Amin and Gebhard, Oliver and Hahn-Klimroth, Max and Kaaser, Dominik and Rau, Malin}, title = {{On the Hierarchy of Distributed Majority Protocols}}, booktitle = {26th International Conference on Principles of Distributed Systems (OPODIS 2022)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-265-5}, ISSN = {1868-8969}, year = {2023}, volume = {253}, editor = {Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.23}, URN = {urn:nbn:de:0030-drops-176434}, doi = {10.4230/LIPIcs.OPODIS.2022.23}, annote = {Keywords: Consensus, Majority, Hierarchy, Stochastic Dominance, Population Protocols, Gossip Model, Strassen’s Theorem} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of n identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires O(log n) states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within O(n log n) interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly(n) parallel time w.h.p.
We use our clock to design a loosely-stabilizing protocol that solves the adaptive variant of the majority problem. We assume that the agents have either opinion A or B or they are undecided and agents can change their opinion at a rate of 1/n. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least Ω(log n) agents and a sufficiently large bias is present, then the protocol converges to a correct output within O(n log n) interactions and stays in a correct configuration for poly(n) interactions, w.h.p.

Petra Berenbrink, Felix Biermeier, Christopher Hahn, and Dominik Kaaser. Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berenbrink_et_al:LIPIcs.SAND.2022.7, author = {Berenbrink, Petra and Biermeier, Felix and Hahn, Christopher and Kaaser, Dominik}, title = {{Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.7}, URN = {urn:nbn:de:0030-drops-159493}, doi = {10.4230/LIPIcs.SAND.2022.7}, annote = {Keywords: Population Protocols, Phase Clocks, Loose Self-stabilization, Clock Synchronization, Majority, Adaptive} }

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Complete Volume

**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

LIPIcs, Volume 219, STACS 2022, Complete Volume

39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 1-1044, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{berenbrink_et_al:LIPIcs.STACS.2022, title = {{LIPIcs, Volume 219, STACS 2022, Complete Volume}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {1--1044}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022}, URN = {urn:nbn:de:0030-drops-158098}, doi = {10.4230/LIPIcs.STACS.2022}, annote = {Keywords: LIPIcs, Volume 219, STACS 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Front Matter, Table of Contents, Preface, Conference Organization

39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berenbrink_et_al:LIPIcs.STACS.2022.0, author = {Berenbrink, Petra and Monmege, Benjamin}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.0}, URN = {urn:nbn:de:0030-drops-158101}, doi = {10.4230/LIPIcs.STACS.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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

We consider the efficient simulation of population protocols. In the population model, we are given a system of n agents modeled as identical finite-state machines. In each step, two agents are selected uniformly at random to interact by updating their states according to a common transition function. We empirically and analytically analyze two classes of simulators for this model. First, we consider sequential simulators executing one interaction after the other. Key to the performance of these simulators is the data structure storing the agents' states. For our analysis, we consider plain arrays, binary search trees, and a novel Dynamic Alias Table data structure. Secondly, we consider batch processing to efficiently update the states of multiple independent agents in one step. For many protocols considered in literature, our simulator requires amortized sub-constant time per interaction and is fast in practice: given a fixed time budget, the implementation of our batched simulator is able to simulate population protocols several orders of magnitude larger compared to the sequential competitors, and can carry out 2^50 interactions among the same number of agents in less than 400s.

Petra Berenbrink, David Hammer, Dominik Kaaser, Ulrich Meyer, Manuel Penschuck, and Hung Tran. Simulating Population Protocols in Sub-Constant Time per Interaction. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{berenbrink_et_al:LIPIcs.ESA.2020.16, author = {Berenbrink, Petra and Hammer, David and Kaaser, Dominik and Meyer, Ulrich and Penschuck, Manuel and Tran, Hung}, title = {{Simulating Population Protocols in Sub-Constant Time per Interaction}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {16:1--16:22}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.16}, URN = {urn:nbn:de:0030-drops-128827}, doi = {10.4230/LIPIcs.ESA.2020.16}, annote = {Keywords: Population Protocols, Simulation, Random Sampling, Dynamic Alias Table} }

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

A population protocol is a sequence of pairwise interactions of n agents. During one interaction, two randomly selected agents update their states by applying a deterministic transition function. The goal is to stabilize the system at a desired output property. The main performance objectives in designing such protocols are small number of states per agent and fast stabilization time.
We present a fast population protocol for the exact-majority problem, which uses Theta(log n) states (per agent) and stabilizes in O(log^{5/3} n) parallel time (i.e., in O(n log^{5/3} n) interactions) in expectation and with high probability. Alistarh et al. [SODA 2018] showed that exact-majority protocols which stabilize in expected O(n^{1-Omega(1)}) parallel time and have the properties of monotonicity and output dominance require Omega(log n) states. Note that the properties mentioned above are satisfied by all known population protocols for exact majority, including ours. They also showed an O(log^2 n)-time exact-majority protocol with O(log n) states, which, prior to our work, was the fastest exact-majority protocol with polylogarithmic number of states. The standard design framework for majority protocols is based on O(log n) phases and requires that all agents are well synchronized within each phase, leading naturally to upper bounds of the order of log^2 n because of Theta(log n) synchronization time per phase. We show how this framework can be tightened with weak synchronization to break the O(log^2 n) upper bound of previous protocols.

Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Theta(log n) States. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berenbrink_et_al:LIPIcs.DISC.2018.10, author = {Berenbrink, Petra and Els\"{a}sser, Robert and Friedetzky, Tom and Kaaser, Dominik and Kling, Peter and Radzik, Tomasz}, title = {{A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Theta(log n) States}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {10:1--10:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.10}, URN = {urn:nbn:de:0030-drops-97999}, doi = {10.4230/LIPIcs.DISC.2018.10}, annote = {Keywords: Population Protocols, Randomized Algorithms, Majority} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

We provide a simple and efficient population protocol for leader election that uses O(log n) states and elects exactly one leader in O(n (log n)^2) interactions with high probability and in expectation. Our analysis is simple and based on fundamental stochastic arguments. Our protocol combines the tournament based leader elimination by Alistarh and Gelashvili, ICALP'15, with the synthetic coin introduced by Alistarh et al., SODA'17.

Petra Berenbrink, Dominik Kaaser, Peter Kling, and Lena Otterbach. Simple and Efficient Leader Election. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berenbrink_et_al:OASIcs.SOSA.2018.9, author = {Berenbrink, Petra and Kaaser, Dominik and Kling, Peter and Otterbach, Lena}, title = {{Simple and Efficient Leader Election}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {9:1--9:11}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.9}, URN = {urn:nbn:de:0030-drops-83029}, doi = {10.4230/OASIcs.SOSA.2018.9}, annote = {Keywords: population protocols, leader election, distributed, randomized} }

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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 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We consider plurality consensus in networks of n nodes. Initially, each node has one of k opinions. The nodes execute a (randomized) distributed protocol to agree on the plurality opinion (the opinion initially supported by the most nodes). In certain types of networks the nodes can be quite cheap and simple, and hence one seeks protocols that are not only time efficient but also simple and space efficient. Typically, protocols depend heavily on the employed communication mechanism, which ranges from sequential (only one pair of nodes communicates at any time) to fully parallel (all nodes communicate with all their neighbors at once) and everything in-between.
We propose a framework to design protocols for a multitude of communication mechanisms. We introduce protocols that solve the plurality consensus problem and are, with probability 1-o(1), both time and space efficient. Our protocols are based on an interesting relationship between plurality consensus and distributed load balancing. This relationship allows us to design protocols that generalize the state of the art for a large range of problem parameters.

Petra Berenbrink, Tom Friedetzky, Peter Kling, Frederik Mallmann-Trenn, and Chris Wastell. Plurality Consensus in Arbitrary Graphs: Lessons Learned from Load Balancing. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berenbrink_et_al:LIPIcs.ESA.2016.10, author = {Berenbrink, Petra and Friedetzky, Tom and Kling, Peter and Mallmann-Trenn, Frederik and Wastell, Chris}, title = {{Plurality Consensus in Arbitrary Graphs: Lessons Learned from Load Balancing}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {10:1--10:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.10}, URN = {urn:nbn:de:0030-drops-63610}, doi = {10.4230/LIPIcs.ESA.2016.10}, annote = {Keywords: Plurality Consensus, Distributed Computing, Load Balancing} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

A palindrome is defined as a string which reads forwards the same as backwards, like, for example, the string "racecar". In the Palindrome Problem, one tries to find all palindromes in a given string. In contrast, in the case of the Longest Palindromic Substring Problem, the goal is to find an arbitrary one of the longest palindromes in the string.
In this paper we present three algorithms in the streaming model for the the above problems, where at any point in time we are only allowed to use sublinear space. We first present a one-pass randomized algorithm that solves the Palindrome Problem. It has an additive error and uses square root of n space. We also give two variants of the algorithm which solve related and practical problems. The second algorithm determines the exact locations of all longest palindromes using two passes and square root of n space. The third algorithm is a one-pass randomized algorithm, which solves the Longest Palindromic Substring Problem. It has a multiplicative error using only O(log(n)) space.

Petra Berenbrink, Funda Ergün, Frederik Mallmann-Trenn, and Erfan Sadeqi Azer. Palindrome Recognition In The Streaming Model. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 149-161, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{berenbrink_et_al:LIPIcs.STACS.2014.149, author = {Berenbrink, Petra and Erg\"{u}n, Funda and Mallmann-Trenn, Frederik and Sadeqi Azer, Erfan}, title = {{Palindrome Recognition In The Streaming Model}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {149--161}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.149}, URN = {urn:nbn:de:0030-drops-44544}, doi = {10.4230/LIPIcs.STACS.2014.149}, annote = {Keywords: Palindromes, Streaming Model, Complementary Palindrome} }

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