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

We consider evacuation of a group of n ≥ 2 autonomous mobile agents (or robots) from an unknown exit on an infinite line. The agents are initially placed at the origin of the line and can move with any speed up to the maximum speed 1 in any direction they wish and they all can communicate when they are co-located. However, the agents have different wireless communication abilities: while some are fully wireless and can send and receive messages at any distance, a subset of the agents are senders, they can only transmit messages wirelessly, and the rest are receivers, they can only receive messages wirelessly. The agents start at the same time and their communication abilities are known to each other from the start. Starting at the origin of the line, the goal of the agents is to collectively find a target/exit at an unknown location on the line while minimizing the evacuation time, defined as the time when the last agent reaches the target.
We investigate the impact of such a mixed communication model on evacuation time on an infinite line for a group of cooperating agents. In particular, we provide evacuation algorithms and analyze the resulting competitive ratio (CR) of the evacuation time for such a group of agents. If the group has two agents of two different types, we give an optimal evacuation algorithm with competitive ratio CR = 3+2√2. If there is a single sender or fully wireless agent, and multiple receivers we prove that CR ∈ [2+√5,5], and if there are multiple senders and a single receiver or fully wireless agent, we show that CR ∈ [3,5.681319]. Any group consisting of only senders or only receivers requires competitive ratio 9, and any other combination of agents has competitive ratio 3.

Jurek Czyzowicz, Ryan Killick, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, Jaroslav Opatrny, Denis Pankratov, and Sunil Shende. Group Evacuation on a Line by Agents with Different Communication Abilities. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{czyzowicz_et_al:LIPIcs.ISAAC.2021.57, author = {Czyzowicz, Jurek and Killick, Ryan and Kranakis, Evangelos and Krizanc, Danny and Narayanan, Lata and Opatrny, Jaroslav and Pankratov, Denis and Shende, Sunil}, title = {{Group Evacuation on a Line by Agents with Different Communication Abilities}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {57:1--57:24}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.57}, URN = {urn:nbn:de:0030-drops-154903}, doi = {10.4230/LIPIcs.ISAAC.2021.57}, annote = {Keywords: Agent, Communication, Evacuation, Mobile, Receiver, Search, Sender} }

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

Consider a set of n mobile computational entities, called robots, located and operating on a continuous cycle C (e.g., the perimeter of a closed region of R^2) of arbitrary length l. The robots are identical, can only see their current location, have no location awareness, and cannot communicate at a distance. In this weak setting, we study the classical problems of gathering (GATHER), requiring all robots to meet at a same location; and election (ELECT), requiring all robots to agree on a single one as the "leader". We investigate how to solve the problems depending on the amount of knowledge (exact, upper bound, none) the robots have about their number n and about the length of the cycle l. Cost of the algorithms is analyzed with respect to time and number of random bits. We establish a variety of new results specific to the continuous cycle - a geometric domain never explored before for GATHER and ELECT in a mobile robot setting; compare Monte Carlo and Las Vegas algorithms; and obtain several optimal bounds.

Paola Flocchini, Ryan Killick, Evangelos Kranakis, Nicola Santoro, and Masafumi Yamashita. Gathering and Election by Mobile Robots in a Continuous Cycle. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{flocchini_et_al:LIPIcs.ISAAC.2019.8, author = {Flocchini, Paola and Killick, Ryan and Kranakis, Evangelos and Santoro, Nicola and Yamashita, Masafumi}, title = {{Gathering and Election by Mobile Robots in a Continuous Cycle}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {8:1--8:19}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.8}, URN = {urn:nbn:de:0030-drops-115044}, doi = {10.4230/LIPIcs.ISAAC.2019.8}, annote = {Keywords: Cycle, Election, Gathering, Las Vegas, Monte Carlo, Randomized Algorithm} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can collaborate in the search, but can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of d, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least 9d - o(d); a simple doubling strategy achieves this time bound. It was shown recently in [Chrobak et al., 2015] that k >= 2 robots traveling at unit speed also require at least 9d group search time.
We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance x at constant speed s is given by s^2 x, as motivated by energy consumption models in physics and engineering. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple c of the distance d and the speed of the robots is bounded by b. Motivation for this study is that for the case when robots must complete the search in 9d time with maximum speed one (b=1; c=9), a single robot requires at least 9d energy, while for two robots, all previously proposed algorithms consume at least 28d/3 energy.
When the robots have bounded memory and can use only a constant number of fixed speeds, we generalize an algorithm described in [Baeza-Yates and Schott, 1995; Chrobak et al., 2015] to obtain a family of algorithms parametrized by pairs of b,c values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. In particular, for each such pair, we determine optimal (and in some cases nearly optimal) algorithms inducing the lowest possible energy consumption.
We also propose a novel search algorithm that simultaneously achieves search time 9d and consumes energy 8.42588d. Our result shows that two robots can search on the line in optimal time 9d while consuming less total energy than a single robot within the same search time. Our algorithm uses robots that have unbounded memory, and a finite number of dynamically computed speeds. It can be generalized for any c, b with cb=9, and consumes energy 8.42588b^2d.

Jurek Czyzowicz, Konstantinos Georgiou, Ryan Killick, Evangelos Kranakis, Danny Krizanc, Manuel Lafond, Lata Narayanan, Jaroslav Opatrny, and Sunil Shende. Energy Consumption of Group Search on a Line. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 137:1-137:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{czyzowicz_et_al:LIPIcs.ICALP.2019.137, author = {Czyzowicz, Jurek and Georgiou, Konstantinos and Killick, Ryan and Kranakis, Evangelos and Krizanc, Danny and Lafond, Manuel and Narayanan, Lata and Opatrny, Jaroslav and Shende, Sunil}, title = {{Energy Consumption of Group Search on a Line}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {137:1--137:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.137}, URN = {urn:nbn:de:0030-drops-107138}, doi = {10.4230/LIPIcs.ICALP.2019.137}, annote = {Keywords: Evacuation, Exit, Line, Face-to-face Communication, Robots, Search} }

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**Published in:** LIPIcs, Volume 125, 22nd International Conference on Principles of Distributed Systems (OPODIS 2018)

Two anonymous robots placed at different positions on an infinite line need to rendezvous. Each robot possesses a clock which it uses to time its movement. However, the robot's individual parameters in the form of their walking speed and time unit may or may not be the same for both robots. We study the feasibility of rendezvous in different scenarios, in which some subsets of these parameters are not the same. As the robots are anonymous, they execute the same algorithm and when both parameters are identical the rendezvous is infeasible. We propose a universal algorithm, such that the robots are assured of meeting in finite time, in any case when at least one of the parameters is not equal for both robots.

Jurek Czyzowicz, Ryan Killick, and Evangelos Kranakis. Linear Rendezvous with Asymmetric Clocks. In 22nd International Conference on Principles of Distributed Systems (OPODIS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 125, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{czyzowicz_et_al:LIPIcs.OPODIS.2018.25, author = {Czyzowicz, Jurek and Killick, Ryan and Kranakis, Evangelos}, title = {{Linear Rendezvous with Asymmetric Clocks}}, booktitle = {22nd International Conference on Principles of Distributed Systems (OPODIS 2018)}, pages = {25:1--25:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-098-9}, ISSN = {1868-8969}, year = {2019}, volume = {125}, editor = {Cao, Jiannong and Ellen, Faith and Rodrigues, Luis and Ferreira, Bernardo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2018.25}, URN = {urn:nbn:de:0030-drops-100855}, doi = {10.4230/LIPIcs.OPODIS.2018.25}, annote = {Keywords: anonymous, asymmetric clock, infinite line, rendezvous, mobile robot, speed, competitive ratio} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in 1 + (2 pi)/3 + sqrt{3} ~~ 4.8264 time units and this is optimal [Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most 4.81854. Furthermore, we show that any strategy for saving the queen requires time at least 3 + pi/6 + sqrt{3}/2 ~~ 4.3896 in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to 3.8327 for two servants, and 3.3738 for three servants. Finally we show lower bounds for these cases of 3.6307 (two servants) and 3.2017 (three servants). The case of n >= 4 is the subject of an independent study by Queen Daniela's Royal Scientific Team.

Jurek Czyzowicz, Konstantinos Georgiou, Ryan Killick, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, Jaroslav Opatrny, and Sunil Shende. God Save the Queen. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{czyzowicz_et_al:LIPIcs.FUN.2018.16, author = {Czyzowicz, Jurek and Georgiou, Konstantinos and Killick, Ryan and Kranakis, Evangelos and Krizanc, Danny and Narayanan, Lata and Opatrny, Jaroslav and Shende, Sunil}, title = {{God Save the Queen}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {16:1--16:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.16}, URN = {urn:nbn:de:0030-drops-88074}, doi = {10.4230/LIPIcs.FUN.2018.16}, annote = {Keywords: Algorithm, Evacuation, Exit, Disk, Wireless Communication, Queen, Priority, Robots, Search, Servants, Trajectory} }

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