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

Consider search on an infinite line involving an autonomous robot starting at the origin of the line and an oblivious moving target at initial distance d ≥ 1 from it. The robot can change direction and move anywhere on the line with constant maximum speed 1 while the target is also moving on the line with constant speed v > 0 but is unable to change its speed or direction. The goal is for the robot to catch up to the target in as little time as possible.
The classic case where v = 0 and the target’s initial distance d is unknown to the robot is the well-studied "cow-path problem". Alpert and Gal [Steve Alpern and Shmuel Gal, 2003] gave an optimal algorithm for the case where a target with unknown initial distance d is moving away from the robot with a known speed v < 1. In this paper we design and analyze search algorithms for the remaining possible knowledge situations, namely, when d and v are known, when v is known but d is unknown, when d is known but v is unknown, and when both v and d are unknown. Furthermore, for each of these knowledge models we consider separately the case where the target is moving away from the origin and the case where it is moving toward the origin. We design algorithms and analyze competitive ratios for all eight cases above. The resulting competitive ratios are shown to be optimal when the target is moving towards the origin as well as when v is known and the target is moving away from the origin.

Jared Coleman, Evangelos Kranakis, Danny Krizanc, and Oscar Morales-Ponce. Line Search for an Oblivious Moving Target. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{coleman_et_al:LIPIcs.OPODIS.2022.12, author = {Coleman, Jared and Kranakis, Evangelos and Krizanc, Danny and Morales-Ponce, Oscar}, title = {{Line Search for an Oblivious Moving Target}}, booktitle = {26th International Conference on Principles of Distributed Systems (OPODIS 2022)}, pages = {12:1--12:19}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.12}, URN = {urn:nbn:de:0030-drops-176325}, doi = {10.4230/LIPIcs.OPODIS.2022.12}, annote = {Keywords: Infinite Line, Knowledge, Oblivious, Robot, Search, Search-Time, Speed, Target} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We consider evacuation of two robots from an Exit placed at an unknown location on the perimeter of a unit (radius) disk. The robots can move with max speed 1 and start at the center of the disk at the same time. We consider a new communication model, known as the SR model, in which the robots have communication faults as follows: one of the robots is a Sender and can only send wirelessly at any distance, while the other is a Receiver in that it can only receive wirelessly from any distance. The communication status of each robot is known to the other robot. In addition, both robots can exchange messages when they are co-located, which is known as Face-to-Face (F2F) model.
There have been several studies in the literature concerning the evacuation time when both robots may employ either F2F or Wireless (WiFi) communication. The SR communication model diverges from these two in that the two robots themselves have differing communication capabilities. We study the evacuation time, namely the time it takes until the last robot reaches the Exit, and show that the evacuation time in the SR model is strictly between the F2F and the WiFi models. The main part of our technical contribution is also an evacuation algorithm in which two cooperating robots accomplish the task in worst-case time at most π+2. Interesting features of the proposed algorithm are the asymmetry inherent in the resulting trajectories, as well as that the robots do not move at full speed for the entire duration of their trajectories.

Konstantinos Georgiou, Nikos Giachoudis, and Evangelos Kranakis. Evacuation from a Disk for Robots with Asymmetric Communication. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{georgiou_et_al:LIPIcs.ISAAC.2022.19, author = {Georgiou, Konstantinos and Giachoudis, Nikos and Kranakis, Evangelos}, title = {{Evacuation from a Disk for Robots with Asymmetric Communication}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.19}, URN = {urn:nbn:de:0030-drops-173047}, doi = {10.4230/LIPIcs.ISAAC.2022.19}, annote = {Keywords: Communication, Cycle, Evacuation, Receiver, Sender, Mobile Agents} }

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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.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.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.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.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.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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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We consider the problem of fault-tolerant parallel search on an infinite line by n robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed 1 and can communicate in wireless mode among themselves. However, among the n robots, there are f robots that exhibit byzantine faults. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient verification that the target has been found. We aim to design algorithms that minimize the value of S_d (n, f), the time to find a target at a distance d from the origin by n robots among which f are faulty. We give several different algorithms whose running time depends on the ratio f/n, the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

Jurek Czyzowicz, Konstantinos Georgiou, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, Jaroslav Opatrny, and Sunil Shende. Search on a Line by Byzantine Robots. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 27:1-27:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{czyzowicz_et_al:LIPIcs.ISAAC.2016.27, author = {Czyzowicz, Jurek and Georgiou, Konstantinos and Kranakis, Evangelos and Krizanc, Danny and Narayanan, Lata and Opatrny, Jaroslav and Shende, Sunil}, title = {{Search on a Line by Byzantine Robots}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {27:1--27:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.27}, URN = {urn:nbn:de:0030-drops-67972}, doi = {10.4230/LIPIcs.ISAAC.2016.27}, annote = {Keywords: Cow path problem, Parallel search, Mobile robots, Wireless communication, Byzantine faults} }

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