Accelerating Amoebots via Reconfigurable Circuits

Abstract

We consider an extension to the geometric amoebot model that allows amoebots to form so-called circuits. Given a connected amoebot structure, a circuit is a subgraph formed by the amoebots that permits the instant transmission of signals. We show that such an extension allows for significantly faster solutions to a variety of problems related to programmable matter. More specifically, we provide algorithms for leader election, consensus, compass alignment, chirality agreement and shape recognition. Leader election can be solved in Θ(log n) rounds, w.h.p., consensus in O(1) rounds and both, compass alignment and chirality agreement, can be solved in O(log n) rounds, w.h.p. For shape recognition, the amoebots have to decide whether the amoebot structure forms a particular shape. We show how the amoebots can detect a parallelogram with linear and polynomial side ratio within Θ(log n) rounds, w.h.p. Finally, we show that the amoebots can detect a shape composed of triangles within O(1) rounds, w.h.p.

The Structural Power of Reconfigurable Circuits in the Amoebot Model

“…In this paper, we assume that all amoebots share a common compass orientation and chirality. This is reasonable since Feldmann et al [7] showed that all amoebots are able to come to an agreement within the considered extension (see Section 1.4).…”

“…In the reconfigurable circuit extension [7], each edge between two neighboring amoebots u and v is replaced by k edges called external links with endpoints called pins, for some constant k ≥ 1 that is the same for all amoebots. For each of these links, one pin is owned by u while the other pin is owned by v. In this paper, we assume that neighboring amoebots have a common labeling of their incident external links.…”

“…The reconfigurable circuit extension was introduced by Feldmann et al [7]. They have proposed solutions for leader election (see Section 2), consensus, compass alignment, chirality agreement, and various shape recognition problems.…”

“…Inspired by the nervous and muscular system, Feldmann et al [7] introduced a reconfigurable circuit extension to the amoebot model with the goal of significantly accelerating fundamental problems like leader election and shape transformation. As a first step, they showed that leader election, consensus, compass alignment, chirality agreement, and various…”

“…Figure 1 Feldmann et al [7] have proposed joint movements to the amoebot model where an expanding amoebot is capable of pushing other amoebots away from it, and a contracting amoebot is capable of pulling other amoebots towards it. (a) and (b) show a joint expansion (from left to right) resp.…”