. .. ,

.. .. <-2?,

.. .. Arbitrary-angles,

. .. Conclusion, 112 1. The function ReduceRectangle(R) returns a straight rectangle Rec, such that Rec ? R, and either R \

, The agent traveled a distance of at most 21(P erimeter(R) ? P erimeter(Rec)) during the execution of ReduceRectangle(R)

, All other necessary objects will be defined within the proof. For the notation, refer to Fig. 6.2. Consider the execution of function ReduceRectangle(R) starting at the center p of R

?. P1, The variable N ewRectangle is set to a straight rectangle R such that R ? R , and either R \ R does not contain ? k (z) or the agent has seen the treasure

?. P2, The inequality P erimeter(R ) ? P erimeter(R ) ? 2 holds

?. P3, The agent traveled a distance of at most 21(P erimeter(R) ? P erimeter(R )) during the execution of function ReduceRectangle(R)

, Note that the points defined in lines 4 to 6 (in particular the points A,B, d and e) exist and ABde is a straight rectangle such that ABde ? R in view of the fact that (R , (L 1 , x 1 )) is a basic configuration. Moreover, since z ? (L 1 , x 1 ), we have ? k (z) ? (L 1 , x 1 ). However, edCD ? (L 1 , x 1 ) ? [de] and R \ ABde = edCD, So we have (R \ ABde) ? (L 1 , x 1 ) = ? and Property P1 is satisfied. Property P2 also holds because

, For every integer i ? 3 and for every triangle T of Slicing(i), there is at least one tile t of T iling(4?(i) ? 2), such that t ? T and one side of t is included in a side of S. For the base case i = 3

, included in t , for all integers r < r . Hence H 3 is true. Assume that H j is true for some integer j ? 3 and let us prove that H j+1 is also true. included in L. By the inductive hypothesis, there exists a tile t of T iling(4?(j) ? 2) such that t ? T and one side of t is included in L. For any integers r < r , every tile of T iling(r) contains side length of a tile of T iling(4?(j + 1) ? 2), ? 2 ? 10. Moreover, each side of every tile t of T iling(r) contains at least one side of a tile of T iling(r )

S. Abiteboul, H. Kaplan, and T. Milo, Compact Labeling Schemes for Ancestor Queries, Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, vol.365529, pp.547-556, 2001.

N. Agmon and D. Peleg, Fault-Tolerant Gathering Algorithms for Autonomous Mobile Robots, SIAM J Comput, vol.36, issue.1, pp.56-82, 2006.

O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe et al., Generalized Self-Approaching Curves, Discrete Appl. Math, vol.109, pp.3-24, 2001.

M. Alcantara, A. Castañeda, D. Flores-peñaloza, and S. Rajsbaum, The Topology of Look-Compute-Move Robot Wait-Free Algorithms with Hard Termination, Distrib. Comput, vol.32, pp.235-255, 2019.

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovász, and C. Rackoff, Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems, 20th Annual Symposium on Foundations of Computer Science, pp.218-223, 1979.

S. Alpern, The Rendezvous Search Problem, SIAM J Control Optim, vol.33, issue.3, pp.363-0129, 1995.

S. Alpern, Rendezvous Search: A Personal Perspective, Oper. Res, vol.50, pp.772-795, 2002.

K. Altisen, P. Corbineau, and S. Devismes, A Framework for Certified Self-Stabilization, Log. Methods Comput. Sci, vol.13, p.14, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01272158

K. Altisen, S. Devismes, S. Dubois, and F. Petit, Introduction to Distributed Self-Stabilizing Algorithms, Synthesis Lectures on Distributed Computing Theory, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02178659

H. Attiya, A. Bar-noy, and D. Dolev, Sharing Memory Robustly in Message-Passing Systems, J ACM 42, vol.1, pp.124-142, 1995.

R. A. Baeza-yates, J. C. Culberson, and G. J. Rawlins, Searching in the Plane, Inf Comput, vol.106, pp.234-252, 1993.

T. Balabonski, A. Delga, L. Rieg, S. Tixeuil, and X. Urbain, Synchronous Gathering without Multiplicity Detection: A Certified Algorithm, Theory Comput Syst, vol.63, pp.200-218, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01491813

E. Bampas, J. Czyzowicz, L. Gasieniec, D. Ilcinkas, and A. Labourel, Almost Optimal Asynchronous Rendezvous in Infinite Multidimensional Grids, Distributed Computing, 24th International Symposium, DISC 2010, vol.6343, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00534263

, , pp.297-311, 2010.

M. Barborak and M. Malek, The Consensus Problem in Fault-Tolerant Computing, Comput Surv, vol.25, pp.171-220, 1993.

L. Barrière, P. Flocchini, E. M. Barrameda, and N. Santoro, Uniform Scattering of Autonomous Mobile Robots in a Grid, Int J Found Comput Sci, vol.22, pp.679-697, 2011.

V. Baston and S. Gal, Rendezvous Search When Marks Are Left at the Starting Points, Nav. Res. Logist. NRL, vol.48, 2001.

A. Beck and D. J. Newman, Yet More on the Linear Search Problem, Israel J. Math, vol.8, issue.4, pp.1565-8511, 1970.

A. Bonato and R. Nowakowski, The Game of Cops and Robbers on Graphs, vol.61, 2011.

S. Bouchard, M. Bournat, Y. Dieudonné, S. Dubois, and F. Petit, Asynchronous Approach in the Plane: A Deterministic Polynomial Algorithm, 31st International Symposium on Distributed Computing, vol.91, pp.1-8, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01672916

S. Bouchard, M. Bournat, Y. Dieudonné, S. Dubois, and F. Petit, Approche asynchrone dans le plan : un algorithme déterministe polynomial, ALGOTEL 2018 -20èmes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications. ALGOTEL, 2018.

S. Bouchard, M. Bournat, Y. Dieudonné, S. Dubois, and F. Petit, Asynchronous Approach in the Plane: A Deterministic Polynomial Algorithm". In: Distrib. Comput, vol.32, pp.317-337, 2019.

S. Bouchard, Y. Dieudonné, and B. Ducourthial, Byzantine Gathering in Networks, Structural Information and Communication Complexity -22nd International Colloquium, vol.9439, pp.179-193, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01330420

S. Bouchard, Y. Dieudonné, and B. Ducourthial, Byzantine Gathering in Networks, Distrib. Comput, vol.29, pp.435-457, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01330420

S. Bouchard, Y. Dieudonné, and B. Ducourthial, Rassemblement byzantin dans les réseaux, 2018.

S. Bouchard, Y. Dieudonné, and A. Lamani, Byzantine Gathering in Polynomial Time, 45th International Colloquium on Automata, Languages, and Programming, vol.107, pp.1-147, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01965743

S. Bouchard, Y. Dieudonné, A. Pelc, and F. Petit, Deterministic Treasure Hunt in the Plane with Angular Hints, 29th International Symposium on Algorithms and Computation, vol.123, pp.1-48, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01970990

S. Bouchard, Y. Dieudonné, A. Pelc, and F. Petit, On Deterministic Rendezvous at a Node of Agents with Arbitrary Velocities, Inf Process Lett, vol.133, pp.39-43, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01701786

S. Bouchard, Y. Dieudonné, A. Pelc, and F. Petit, Trouver un trésor plus rapidement avec des conseils angulaires, ALGOTEL, 2019.

M. Bournat, S. Dubois, and F. Petit, Gracefully Degrading Gathering in Dynamic Rings, Stabilization, Safety, and Security of Distributed Systems -20th International Symposium, vol.11201, pp.349-364, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01790554

W. Quirijn, T. Bouts, A. Castermans, M. J. Van-goethem, W. Van-kreveld et al., Competitive Searching for a Line on a Line Arrangement, 29th International Symposium on Algorithms and Computation, vol.123, pp.1-49, 2018.

Q. Bramas and S. Tixeuil, Brief Announcement: Probabilistic Asynchronous Arbitrary Pattern Formation, Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, pp.443-445, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01349087

A. Casteigts, P. Flocchini, W. Quattrociocchi, and N. Santoro, Time-Varying Graphs and Dynamic Networks, IJPEDS 27, vol.5, pp.387-408, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00854287

J. Chalopin, Y. Dieudonné, A. Labourel, and A. Pelc, Rendezvous in Networks in Spite of Delay Faults, Distrib. Comput, vol.29, pp.187-205, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01330426

J. Chalopin, E. Godard, Y. Métivier, and R. Ossamy, Mobile Agent Algorithms Versus Message Passing Algorithms, Principles of Distributed Systems, 10th International Conference, vol.4305, pp.187-201, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00308127

T. H. Chung, G. A. Hollinger, and V. Isler, Search and Pursuit-Evasion in Mobile Robotics -A Survey, Auton Robots, vol.31, pp.299-316, 2011.

S. Cicerone, G. D. Stefano, and A. Navarra, Asynchronous Arbitrary Pattern Formation: The Effects of a Rigorous Approach, Distrib. Comput, vol.32, pp.91-132, 2019.

M. Cieliebak, P. Flocchini, G. Prencipe, and N. Santoro, Distributed Computing by Mobile Robots: Gathering, SIAM J Comput, vol.41, pp.829-879, 2012.

R. Cohen, P. Fraigniaud, D. Ilcinkas, A. Korman, and D. Peleg, Label-Guided Graph Exploration by a Finite Automaton, ACM Trans Algorithms, vol.4, pp.1-42, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00339772

R. Cohen and D. Peleg, Convergence of Autonomous Mobile Robots with Inaccurate Sensors and Movements, SIAM J Comput, vol.38, pp.276-302, 2008.

A. Collins, J. Czyzowicz, L. Gasieniec, and A. Labourel, Tell Me Where I Am So I Can Meet You Sooner, Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, vol.6199, pp.502-514, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00866392

J. Czyzowicz, K. Georgiou, E. Kranakis, D. Krizanc, L. Narayanan et al., Search on a Line by Byzantine Robots, 27th International Symposium on Algorithms and Computation, vol.64, pp.1-27, 2016.

J. Czyzowicz, A. Kosowski, and A. Pelc, How to Meet When You Forget: Log-Space Rendezvous in Arbitrary Graphs, Distrib. Comput, vol.25, pp.165-178, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00725875

J. Czyzowicz, A. Pelc, and A. Labourel, How to Meet Asynchronously (Almost) Everywhere". In: ACM Trans Algorithms, vol.8, p.14, 2012.

D. Gianlorenzo, G. D. Angelo, A. Stefano, and . Navarra, Gathering on Rings under the Look-Compute-Move Model, Distrib. Comput, vol.27, pp.255-285, 2014.

S. Das, D. Dereniowski, A. Kosowski, and P. Uznanski, Rendezvous of Distance-Aware Mobile Agents in Unknown Graphs, Structural Information and Communication Complexity -21st International Colloquium, vol.8576, pp.295-310, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00986103

X. Défago, M. Gradinariu, S. Messika, and P. R. Parvédy, Fault-Tolerant and Self-Stabilizing Mobile Robots Gathering, Distributed Computing, 20th International Symposium, vol.4167, pp.46-60, 2006.

E. D. Demaine, P. Sándor, S. Fekete, and . Gal, Online Searching with Turn Cost, Theor Comput Sci, vol.361, pp.342-355, 2006.

A. Dessmark, P. Fraigniaud, R. Dariusz, A. Kowalski, and . Pelc, Deterministic Rendezvous in Graphs, Algorithmica, vol.46, pp.69-96, 2006.

Y. Dieudonné and A. Pelc, Deterministic Polynomial Approach in the Plane, Distrib. Comput, vol.28, pp.111-129, 2015.

Y. Dieudonné and A. Pelc, Anonymous Meeting in Networks, Algorithmica 74, vol.2, pp.908-946, 2016.

Y. Dieudonné, A. Pelc, and D. Peleg, Gathering Despite Mischief, ACM Trans Algorithms 11, vol.1, p.28, 2014.

Y. Dieudonné, A. Pelc, and V. Villain, How to Meet Asynchronously at Polynomial Cost". In: SIAM J Comput, vol.44, pp.844-867, 2015.

Y. Dieudonné and F. Petit, Self-Stabilizing Gathering with Strong Multiplicity Detection, Theor Comput Sci, vol.428, pp.47-57, 2012.

H. T. , T. Doan, F. Bonnet, and K. Ogata, Model Checking of Robot Gathering, 21st International Conference on Principles of Distributed Systems, vol.95, pp.1-12, 2017.

S. Dolev, Self-Stabilization, 2000.
URL : https://hal.archives-ouvertes.fr/inria-00627780

Y. Emek, T. Langner, D. Stolz, J. Uitto, and R. Wattenhofer, How Many Ants Does It Take to Find the Food?, In: Theor Comput Sci, vol.608, pp.255-267, 2015.

R. Fagin, J. Y. Halpern, Y. Moses, and M. Vardi, Reasoning About Knowledge, vol.533, pp.978-978, 2004.

P. Flocchini, G. Prencipe, and N. Santoro, Distributed Computing by Oblivious Mobile Robots, Synthesis Lectures on Distributed Computing Theory, 2012.

, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, vol.11340, 2019.

P. Flocchini, G. Prencipe, N. Santoro, and P. Widmayer, Gathering of Asynchronous Robots with Limited Visibility, Theor Comput Sci, vol.337, pp.147-168, 2005.

P. Fraigniaud, C. Gavoille, D. Ilcinkas, and A. Pelc, Distributed Computing with Advice: Information Sensitivity of Graph Coloring, Distrib. Comput, vol.21, pp.395-403, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00339878

P. Fraigniaud, D. Ilcinkas, and A. Pelc, Tree Exploration with Advice, Inf Comput, vol.206, pp.1276-1287, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00341624

P. Fraigniaud and A. Pelc, Deterministic Rendezvous in Trees with Little Memory, Distributed Computing, 22nd International Symposium, vol.5218, pp.242-256, 2008.

P. Fraigniaud and A. Pelc, Delays Induce an Exponential Memory Gap for Rendezvous in Trees, Trans Algorithms, vol.9, p.24, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00922693

G. Matthew-fricke, J. P. Hecker, A. D. Griego, L. T. Tran, and M. E. Moses, A Distributed Deterministic Spiral Search Algorithm for Swarms, 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2016, pp.4430-4436, 2016.

B. Grünbaum, Partitions of Mass-Distributions and of Convex Bodies by Hyperplanes, In: Pacific J. Math, vol.10, pp.1257-1261, 1960.

S. Guilbault and A. Pelc, Gathering Asynchronous Oblivious Agents with Local Vision in Regular Bipartite Graphs, Theor Comput Sci, vol.509, pp.86-96, 2013.

J. Y. Halpern, Computer Science and Game Theory: A Brief Survey, 2007.

T. Izumi, S. Souissi, Y. Katayama, N. Inuzuka, X. Défago et al., The Gathering Problem for Two Oblivious Robots with Unreliable Compasses, SIAM J Comput, vol.41, issue.1, pp.26-46, 2012.

A. Jez and J. Lopuszanski, On the Two-Dimensional Cow Search Problem, Inf Process Lett, vol.109, pp.543-547, 2009.

M. Kao, J. H. Reif, and S. R. Tate, Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem, Inf Comput, vol.131, pp.63-79, 1996.

M. Katz, A. Nir, A. Katz, D. Korman, and . Peleg, Labeling Schemes for Flow and Connectivity, SIAM J Comput, vol.34, pp.23-40, 2004.

D. Kempe, J. M. Kleinberg, and A. Kumar, Connectivity and Inference Problems for Temporal Networks, J Comput Syst Sci, vol.64, pp.820-842, 2002.

A. Kosowski and A. Navarra, Graph Decomposition for Memoryless Periodic Exploration, Algorithmica 63, vol.1, pp.26-38, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00725922

D. Kotz and R. S. Gray, Mobile Agents and the Future of the Internet, Oper. Syst. Rev, vol.33, pp.7-13, 1999.

M. Koucký, Universal Traversal Sequences with Backtracking, J Comput Syst Sci, vol.65, pp.717-726, 2002.

R. Dariusz, A. Kowalski, and . Malinowski, How to Meet in Anonymous Network, Theoretical Computer Science. Structural Information and Communication Complexity (SIROCCO 2006), vol.399, pp.304-3975, 2008.

E. Kranakis, D. Krizanc, E. Markou, A. Pagourtzis, and F. Ramírez, Different Speeds Suffice for Rendezvous of Two Agents on Arbitrary Graphs, SOFSEM 2017: Theory and Practice of Computer Science -43rd International Conference on Current Trends in Theory and Practice of Computer Science, vol.10139, pp.79-90, 2017.

E. Kranakis, N. Santoro, C. Sawchuk, and D. Krizanc, Mobile Agent Rendezvous in a Ring, 23rd International Conference on Distributed Computing Systems (ICDCS 2003), pp.592-599, 2003.

E. Langetepe, On the Optimality of Spiral Search, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp.1-12, 2010.

E. Langetepe, Searching for an Axis-Parallel Shoreline, Theor Comput Sci, vol.447, pp.85-99, 2012.

T. Langner, B. Keller, J. Uitto, and R. Wattenhofer, Overcoming Obstacles with Ants, 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Ed. by Emmanuelle Anceaume, Christian Cachin, and Maria Gradinariu Potop-Butucaru, vol.46, pp.1-9, 2016.

M. Latapy, T. Viard, and C. Magnien, Stream Graphs and Link Streams for the Modeling of Interactions over Time, In: Soc. Netw Anal. Min, vol.8, issue.1, p.29, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01665084

G. A. , D. Luna, P. Flocchini, L. Pagli, G. Prencipe et al., Gathering in Dynamic Rings, Structural Information and Communication Complexity -24th International Colloquium, vol.10641, 2017.

, , pp.339-355, 2017.

N. A. Lynch, Distributed Algorithms, pp.978-979, 1996.

L. Gianluca-de-marco, E. Gargano, D. Kranakis, A. Krizanc, U. Pelc et al., Asynchronous Deterministic Rendezvous in Graphs". In: Theor Comput Sci, vol.355, issue.3, pp.315-326, 2006.

T. Masuzawa and H. Kakugawa, Self-Stabilization in Spite of Frequent Changes of Networks: Case Study of Mutual Exclusion on Dynamic Rings, Self-Stabilizing Systems, 7th International Symposium, vol.3764, pp.183-197, 2005.

A. Miller and A. Pelc, Fast Rendezvous with Advice, In: Theor Comput Sci, vol.608, pp.190-198, 2015.

A. Miller and A. Pelc, Time versus Cost Tradeoffs for Deterministic Rendezvous in Networks, Distrib. Comput, vol.29, pp.51-64, 2016.

N. Nisse and D. Soguet, Graph Searching with Advice, Theor Comput Sci 410, vol.14, pp.1307-1318, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00423450

L. Pagli, G. Prencipe, and G. Viglietta, Getting Close without Touching: Near-Gathering for Autonomous Mobile Robots, Distrib. Comput, vol.28, pp.333-349, 2015.

P. Panaite and A. Pelc, Exploring Unknown Undirected Graphs, J Algorithms, vol.33, pp.281-295, 1999.

C. Marshall, R. E. Pease, L. Shostak, and . Lamport, Reaching Agreement in the Presence of Faults, J ACM 27, vol.2, pp.228-234, 1980.

A. Pelc, Deterministic Rendezvous in Networks: A Comprehensive Survey, vol.59, pp.331-347, 2012.

E. Rasmusen, Games and Information: An Introduction to Game Theory, vol.559, pp.978-979, 2006.

O. Reingold, Undirected Connectivity in Log-Space, J ACM, vol.55, p.24, 2008.

C. Thomas and . Schelling, The Strategy of Conflict

A. Ta, -. Shma, and U. Zwick, Deterministic Rendezvous, Treasure Hunts, and Strongly Universal Exploration Sequences, ACM Trans Algorithms, vol.10, p.15, 2014.

K. Spieser and E. Frazzoli, The Cow-Path Game: A Competitive Vehicle Routing Problem, Proceedings of the 51th IEEE Conference on Decision and Control, CDC 2012, pp.6513-6520, 2012.

I. Suzuki and M. Yamashita, Distributed Anonymous Mobile Robots: Formation of Geometric Patterns, SIAM J Comput, vol.28, pp.1347-1363, 1999.

M. Thorup and U. Zwick, Approximate Distance Oracles, J ACM 52, vol.1, pp.1-24, 2005.

Y. Yamauchi, T. Izumi, and S. Kamei, Mobile Agent Rendezvous on a Probabilistic Edge Evolving Ring, Third International Conference on Networking and Computing, ICNC 2012, pp.103-112, 2012.

Y. Yamauchi, T. Uehara, S. Kijima, and M. Yamashita, Plane Formation by Synchronous Mobile Robots in the Three-Dimensional Euclidean Space, J ACM, vol.64, p.43, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01206131