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

This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT_1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., algorithms that have simultaneously optimal time and message complexity (up to polylogarithmic factors), for many fundamental problems in the standard KT_0 model (where nodes have only local knowledge of themselves and not their neighbors). The situation is less clear in the KT_1 model. In this paper, we present several new distributed algorithms in the KT_1 model that trade off between time and message complexity.
Our distributed algorithms are based on a uniform and general approach which involves constructing a sparsified spanning subgraph of the original graph - called a danner - that trades off the number of edges with the diameter of the sparsifier. In particular, a key ingredient of our approach is a distributed randomized algorithm that, given a graph G and any delta in [0,1], with high probability constructs a danner that has diameter O~(D + n^{1-delta}) and O~(min{m,n^{1+delta}}) edges in O~(n^{1-delta}) rounds while using O~(min{m,n^{1+delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. Using our danner construction, we present a family of distributed randomized algorithms for various fundamental problems that exhibit a trade-off between message and time complexity and that improve over previous results. Specifically, we show the following results (all hold with high probability) in the KT_1 model, which subsume and improve over prior bounds in the KT_1 model (King et al., PODC 2014 and Awerbuch et al., JACM 1990) and the KT_0 model (Kutten et al., JACM 2015, Pandurangan et al., STOC 2017 and Elkin, PODC 2017):
1) Leader Election, Broadcast, and ST. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,1].
2) MST and Connectivity. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. In particular, for delta = 0.5 we obtain a distributed MST algorithm that runs in optimal O~(D+sqrt{n}) rounds and uses O~(min{m,n^{3/2}}) messages. We note that this improves over the singularly optimal algorithm in the KT_0 model that uses O~(D+sqrt{n}) rounds and O~(m) messages.
3) Minimum Cut. O(log n)-approximate minimum cut can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5].
4) Graph Verification Problems such as Bipartiteness, Spanning Subgraph etc. These can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5].

Robert Gmyr and Gopal Pandurangan. Time-Message Trade-Offs in Distributed Algorithms. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gmyr_et_al:LIPIcs.DISC.2018.32, author = {Gmyr, Robert and Pandurangan, Gopal}, title = {{Time-Message Trade-Offs in Distributed Algorithms}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {32:1--32:18}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.32}, URN = {urn:nbn:de:0030-drops-98216}, doi = {10.4230/LIPIcs.DISC.2018.32}, annote = {Keywords: Randomized Algorithm, KT\underline1 Model, Sparsifier, MST, Singular Optimality} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Motivated by the problem of shape recognition by nanoscale computing agents, we investigate the problem of detecting the geometric shape of a structure composed of hexagonal tiles by a finite-state automaton robot. In particular, in this paper we consider the question of recognizing whether the tiles are assembled into a parallelogram whose longer side has length l = f(h), for a given function f(*), where h is the length of the shorter side. To determine the computational power of the finite-state automaton robot, we identify functions that can or cannot be decided when the robot is given a certain number of pebbles. We show that the robot can decide whether l = ah+b for constant integers a and b without any pebbles, but cannot detect whether l = f(h) for any function f(x) = omega(x). For a robot with a single pebble, we present an algorithm to decide whether l = p(h) for a given polynomial p(*) of constant degree. We contrast this result by showing that, for any constant k, any function f(x) = omega(x^(6k + 2)) cannot be decided by a robot with k states and a single pebble. We further present exponential functions that can be decided using two pebbles. Finally, we present a family of functions f_n(*) such that the robot needs more than n pebbles to decide whether l = f_n(h).

Robert Gmyr, Kristian Hinnenthal, Irina Kostitsyna, Fabian Kuhn, Dorian Rudolph, and Christian Scheideler. Shape Recognition by a Finite Automaton Robot. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gmyr_et_al:LIPIcs.MFCS.2018.52, author = {Gmyr, Robert and Hinnenthal, Kristian and Kostitsyna, Irina and Kuhn, Fabian and Rudolph, Dorian and Scheideler, Christian}, title = {{Shape Recognition by a Finite Automaton Robot}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {52:1--52:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.52}, URN = {urn:nbn:de:0030-drops-96347}, doi = {10.4230/LIPIcs.MFCS.2018.52}, annote = {Keywords: finite automata, shape recognition, computational geometry} }

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

We initiate the study of network monitoring algorithms in a class of hybrid networks in which the nodes are connected by an external network and an internal network (as a short form for externally and internally controlled network). While the external network lies outside of the control of the nodes (or in our case, the monitoring protocol running in them) and might be exposed to continuous changes, the internal network is fully under the control of the nodes. As an example, consider a group of users with mobile devices having access to the cell phone infrastructure. While the network formed by the WiFi connections of the devices is an external network (as its structure is not necessarily under the control of the monitoring protocol), the connections between the devices via the cell phone infrastructure represent an internal network (as it can be controlled by the monitoring protocol). Our goal is to continuously monitor properties of the external network with the help of the internal network. We present scalable distributed algorithms that efficiently monitor the number of edges, the average node degree, the clustering coefficient, the bipartiteness, and the weight of a minimum spanning tree. Their performance bounds demonstrate that monitoring the external network state with the help of an internal network can be done much more efficiently than just using the external network, as is usually done in the literature.

Robert Gmyr, Kristian Hinnenthal, Christian Scheideler, and Christian Sohler. Distributed Monitoring of Network Properties: The Power of Hybrid Networks. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 137:1-137:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gmyr_et_al:LIPIcs.ICALP.2017.137, author = {Gmyr, Robert and Hinnenthal, Kristian and Scheideler, Christian and Sohler, Christian}, title = {{Distributed Monitoring of Network Properties: The Power of Hybrid Networks}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {137:1--137:15}, 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.2017.137}, URN = {urn:nbn:de:0030-drops-73750}, doi = {10.4230/LIPIcs.ICALP.2017.137}, annote = {Keywords: Network Monitoring, Hybrid Networks, Overlay Networks} }

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