H. Attiya and J. Welch, Distributed computing: fundamentals, simulations and advance topics, 2004.
DOI : 10.1002/0471478210

M. Bauderon, Y. Métivier, M. Mosbah, and A. Sellami, Graph Relabelling Systems A Tool for Encoding, Proving, Studying and Visualizing Distributed Algorithms, Electronic Notes in Theoretical Computer Science, vol.51, issue.1, pp.93-107, 2001.
DOI : 10.1016/S1571-0661(04)00194-X
URL : https://hal.archives-ouvertes.fr/hal-00307013

J. Bourgeois and S. C. Goldstein, Distributed Intelligent MEMS: Progresses and Perspectives, IEEE Systems Journal, vol.9, issue.3, pp.1057-1068, 2015.
DOI : 10.1109/JSYST.2013.2281124
URL : https://hal.archives-ouvertes.fr/hal-00932051

Z. J. Butler, K. Kotay, D. Rus, and K. Tomita, Generic Decentralized Control for Lattice-Based Self-Reconfigurable Robots, The International Journal of Robotics Research, vol.23, issue.9, pp.919-937, 2004.
DOI : 10.1023/A:1026544419097

J. J. Daymude, R. Gmyr, A. W. Richa, C. Scheideler, and T. Strothmann, Improved leader election for self-organizing programmable matter, ALGOSENSORS 2017: Algorithms for Sensor Systems, pp.127-140, 2017.

Z. Derakhshandeh, S. Dolev, R. Gmyr, A. W. Richa, C. Scheideler et al., Brief announcement, Proceedings of the 26th ACM symposium on Parallelism in algorithms and architectures, SPAA '14, pp.220-222, 2014.
DOI : 10.1145/2612669.2612712

Z. Derakhshandeh, R. Gmyr, A. W. Richa, C. Scheideler, and T. Strothmann, An Algorithmic Framework for Shape Formation Problems in Self-Organizing Particle Systems, Proceedings of the Second Annual International Conference on Nanoscale Computing and Communication, NANOCOM' 15, pp.1-21, 2015.
DOI : 10.1145/2800795.2800829

Z. Derakhshandeh, R. Gmyr, T. Strothmann, R. A. Bazzi, A. W. Richa et al., Leader Election and Shape Formation with Self-organizing Programmable Matter, DNA, vol.2015, pp.117-132, 2015.
DOI : 10.1007/978-3-319-21999-8_8

Z. Derakhshandeh, R. Gmyr, A. W. Richa, C. Scheideler, and T. Strothmann, , pp.289-299, 2016.

Z. Derakhshandeh, R. Gmyr, A. Porter, A. W. Richa, C. Scheideler et al., On the runtime of universal coating for Programmable Matter, pp.148-164, 2016.

Z. Derakhshandeh, R. Gmyr, A. W. Richa, C. Scheideler, and T. Strothmann, Universal coating for programmable matter, Theoretical Computer Science, vol.671, pp.56-68, 2017.
DOI : 10.1016/j.tcs.2016.02.039

G. Fertin, E. Godard, and A. Raspaud, Acyclic and k-distance coloring of the grid, Information Processing Letters, vol.87, issue.1, pp.51-58, 2003.
DOI : 10.1016/S0020-0190(03)00232-1
URL : https://hal.archives-ouvertes.fr/hal-00307787

G. A. Di-luna, P. Flocchini, N. Santoro, and G. Viglietta, Yamauchi, Shape Formation by Programmable Particles, 2017.

H. Lakhlef, H. Mabed, and J. Bourgeois, Optimization of the logical topology for mobile MEMS networks, Journal of Network and Computer Applications, vol.42, pp.163-177, 2014.
DOI : 10.1016/j.jnca.2014.02.014

A. Itai and M. Rodeh, Symmetry breaking in distributed networks, Information and Computation, vol.88, issue.1, pp.150-158, 1981.
DOI : 10.1016/0890-5401(90)90004-2

P. Jacko and S. , Distance Coloring of the Hexagonal Lattice, Discussiones Mathematicae Graph Theory, vol.25, issue.1-2, pp.151-166, 2005.
DOI : 10.7151/dmgt.1269

F. Kramer and H. Kramer, A survey on the distance-colouring of graphs, Discrete Math, pp.422-426, 2008.

A. Naz, B. Piranda, J. Bourgeois, and S. C. Goldstein, A distributed self-reconfiguration algorithm for cylindrical lattice-based modular robots, 2016 IEEE 15th International Symposium on Network Computing and Applications (NCA), pp.254-263, 2016.
DOI : 10.1109/NCA.2016.7778628

A. Mazurkiewicz, Distributed enumeration, Information Processing Letters, vol.61, issue.5, pp.233-239, 1997.
DOI : 10.1016/S0020-0190(97)00022-7

B. Piranda and J. Bourgeois, Geometrical Study of a Quasi-spherical Module for Building Programmable Matter, Distributed Autonomous Robotic Systems, pp.347-400, 2018.

P. Rosenstiehl, J. Fiksel, and A. Holliger, Intelligent graphs, Graph theory and computing, pp.219-265, 1972.

A. Sevcikova, Distant chromatic number of the planar graphs, 2001.

T. Tucci, B. Piranda, and J. Bourgeois, Efficient scene encoding for programmable matter selfreconfiguration algorithms, SAC, pp.256-261, 2017.

M. Yim, W. Shen, B. Salemi, D. Rus, M. Moll et al., Modular Self-Reconfigurable Robot Systems [Grand Challenges of Robotics], IEEE Robotics & Automation Magazine, vol.14, issue.1, pp.43-52, 2007.
DOI : 10.1109/MRA.2007.339623

, Proof of Theorem 1 and bound on the complexity of the S-contraction algorithm

, The three lemmas presented in this appendix are used in order to prove Theorem 1. In the following lemma, we describe how to determine, in the context of programmable matter