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Geometric computations by broadcasting automata

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Abstract

In this paper we introduce and apply a novel approach for self-organisation, partitioning and pattern formation on the non-oriented grid environment. The method is based on the generation of nodal patterns in the environment via sequences of discrete waves. The power of the primitives is illustrated by giving solutions to two geometric problems using the broadcast automata model arranged in an integer grid (a square lattice) formation. In this model automata cannot directly observe their neighbours’ state and can only communicate with neighbouring automata through the non-oriented broadcast of messages from a finite alphabet. In particular we show linear time algorithms for the problem of finding the centre of a digital disk starting from any point on the border of the disc and the problem of electing a set of automata that form the inscribed square of such a digital disk. Possible generalizations and applications of techniques based on nodal patterns and the construction of different discrete wave interference pictures are discussed in the conclusion.

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Notes

  1. It is also possible to show, in a more or less straightforward way, that broadcasting automata on \({\mathbb{Z}^n}\) (for any \({n > 0}\)), with a single initial source of transmission, two radii of broadcasting (1 and 1.5) and a large alphabet of messages, can simulate a Turing Machine.

  2. As there are a finite number of passing of waves which all time-bounded by at most \({O(D)},\) the algorithm cannot exceed linear growth by \({D}\).

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Martin, R., Nickson, T. & Potapov, I. Geometric computations by broadcasting automata. Nat Comput 11, 623–635 (2012). https://doi.org/10.1007/s11047-012-9330-0

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