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The Search Complexity of Collaborative Swarms in Expanding \(\mathbf{Z}^{2}\) Grid Regions

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Swarms and Network Intelligence in Search

Part of the book series: Studies in Computational Intelligence ((SCI,volume 729))

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Abstract

In this paper we study the strengths and limitations of collaborative swarms of simple agents. In particular, we discuss the efficient use of simple drones, or “ant robots” for covering a connected region on the \(\mathbf{Z}^{2}\) grid, whose area is unknown in advance, and which expands at a given rate, where n is the initial size of the connected region. We show that regardless of the algorithm used, and the robots’ hardware and software specifications, the minimal number of robots required in order for such coverage to be possible is \(\varOmega ({\sqrt{n}})\). In addition, we show that when the region expands at a sufficiently slow rate, a team of \(\varTheta (\sqrt{n})\) robots could cover it in at most \(O(n^{2} \ln n)\) time. This completion time can even be achieved by myopic robots, with no ability to directly communicate with each other, and where each robot is equipped with a memory of size O(1) bits w.r.t the size of the region (therefore, the robots cannot maintain maps of the terrain, nor plan complete paths). Regarding the coverage of non-expanding regions in the grid, we improve the current best known result of \(O(n^{2})\) by demonstrating an algorithm that guarantees such a coverage with completion time of \(O(\frac{1}{k} n^{1.5} + n)\) in the worst case, and faster for shapes of perimeter length which is shorter than O(n).

This chapter is based on work previously published in parts in [16, 17, 19].

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Notes

  1. 1.

    For counting purposes the agents must be equipped with counters that can store the number of agents in their immediate vicinity. This can of course be implemented using \(O(\log k)\) memory. However, throughout the proof of Lemma 5 in [85] it is shown that the maximal number of agents that may simultaneously reside in the same tile at any given moment is upper bounded by O(1). Therefore, counting the agents in the immediate vicinity can be done using counters of O(1) bits.

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Authors and Affiliations

  1. MIT Media Lab, Massachusetts Institute of Technology, Cambridge, MA, USA

    Yaniv Altshuler & Alex Pentland

  2. Computer Science Department, Israeli Institute of Technology (Technion), Haifa, Israel

    Alfred M. Bruckstein

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  1. Yaniv Altshuler
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Altshuler, Y., Pentland, A., Bruckstein, A.M. (2018). The Search Complexity of Collaborative Swarms in Expanding \(\mathbf{Z}^{2}\) Grid Regions. In: Swarms and Network Intelligence in Search. Studies in Computational Intelligence, vol 729. Springer, Cham. https://doi.org/10.1007/978-3-319-63604-7_5

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