We introduce the problem of shape replication in the Wang tile self-assembly model. Given an input shape, we consider the problem of designing a self-assembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA implementations of Wang tiles, we consider a model in which tiles consisting of DNA or RNA can be dynamically added in a sequence of stages. We further permit the addition of RNase enzymes capable of disintegrating RNA tiles. Under this model, we show that arbitrary genus-0 shapes can be replicated infinitely many times using only O(1) distinct tile types and O(1) stages. Further, we show how to replicate precisely n copies of a shape using O(log n) stages and O(1) tile types.

Abstract-Modular robots consist of many small units that attach together and can perform local motions. By combining these motions, we can achieve a reconfiguration of the global shape. The term modular comes from the idea of grouping together a fixed number of units into a module, which behaves as a larger individual component.Recently, a fair amount of research has focused on Crystalline robots, whose units (and modules) fit on a cubic lattice. When the proper module size is formed, these robots can reconfigure in linear time within a rather physically restrictive model, or in O(log n) time in a more unrestricted theoretical model.In this paper, we show that the results for Crystalline robots also apply to two other modular robots: M-TRAN and Molecube. The common requirement, for each robot type, is that a fixed number of units combine to create modules of specified shapes. In this way, we are able to simulate the actions of Crystalline modules. Previous reconfiguration bounds thus transfer automatically, as long as the robots are composed of the module shapes that we specify.

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ε = 1/2 Ω(n) .

In this paper we propose a novel algorithm that, given a source robot S and a target robot T , reconfigures S into T . Both S and T are robots composed of n atoms arranged in 2 × 2 × 2 meta-modules. The reconfiguration involves a total of O (n) atomic operations (expand, contract, attach, detach) and is performed in O (n) parallel steps. This improves on previous reconfiguration algorithms [D. Rus, M. Vona, Crystalline robots: Self-reconfiguration with compressible unit modules, Autonomous Robots 10 (1) (2001) 107-124; S. Vassilvitskii, M. Yim, J. Suh, A complete, local and parallel reconfiguration algorithm for cube style modular robots, in: Proc. of the IEEE Intl. ], which require O (n 2 ) parallel steps. Our algorithm is in-place; that is, the reconfiguration takes place within the union of the bounding boxes of the source and target robots. We show that the algorithm can also be implemented in a synchronous, distributed fashion.

Abstract. For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k "walls", represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.