We envision programmable matter consisting of systems of computationally limited devices (which we call particles) that are able to self-organize in order to achieve a desired collective goal without the need for central control or external intervention. Central problems for these particle systems are shape formation and coating problems. In this paper, we present a universal shape formation algorithm which takes an arbitrary shape composed of a constant number of equilateral triangles of unit size and lets the particles build that shape at a scale depending on the number of particles in the system. Our algorithm runs in O(√n) asynchronous execution rounds, where $n$ is the number of particles in the system, provided we start from a well-initialized configuration of the particles. This is optimal in a sense that for any shape deviating from the initial configuration, any movement strategy would require Ω(√n) rounds in the worst case (over all asynchronous activations of the particles). Our algorithm relies only on local information (e.g., particles do not have ids, nor do they know n, or have any sort of global coordinate system), and requires only a constant-size memory per particle.

Imagine that we had a piece of matter that can change its physical properties like shape, density, conductivity, or color in a programmable fashion based on either user input or autonomous sensing. This is the vision behind what is commonly known as programmable matter. Many proposals have already been made for realizing programmable matter, ranging from DNA tiles, shape-changing molecules, and cells created via synthetic biology to reconfigurable modular robotics. We are particularly interested in programmable matter consisting of simple elements called particles that can compute, bond, and move, and the feasibility of solving fundamental problems relevant for programmable matter with these particles. As a model for that programmable matter, we will use a general form of the amoebot model first proposed in SPAA 2014, and as examples of fundamental problems we will focus on leader election and shape formation. For shape formation, we investigate the line formation problem, i.e. we are searching for a local-control protocol so that for any connected structure of particles, the particles will eventually form a line.

The term programmable matter refers to matter which has the ability to change its physical properties (shape, density, moduli, conductivity, optical properties, etc.) in a programmable fashion, based upon user input or autonomous sensing. This has many applications like smart materials, autonomous monitoring and repair, and minimal invasive surgery, so there is a high relevance of this topic to industry and society in general. While programmable matter has just been science fiction more than two decades ago, a large amount of research activities can now be seen in this field in the recent years. Often programmable matter is envisioned, as a very large number of small locally interacting computational \emph{particles}. We propose the Amoebot model, a new model which builds upon this vision of programmable matter. Inspired by the behavior of amoeba, the Amoebot model offers a versatile framework to model self-organizing particles and facilitates rigorous algorithmic research in the area of programmable matter.

The orthogonal frequency division multiplexing (OFDM) technology provides an opportunity for efficient resource utilization in optical networks. It allows allocation of multiple sub-carriers to meet traffic demands of varying size. Utilizing OFDM technology, a spectrum efficient and scalable optical transport network called SLICE was proposed recently. The SLICE architecture enables sub-wavelength, super-wavelength resource allocation and multiple rate data traffic that results in efficient use of spectrum. However, the benefit is accompanied by additional complexities in resource allocation. In SLICE architecture, in order to minimize utilized spectrum, one has to solve the routing and spectrum allocation (RSA) problem, a generalization of the routing and wavelength allocation (RWA) problem. In this paper, we focus our attention to the on-line version of RSA problem and provide an algorithm for the ring network with a competitive ratio of min{O(log(dmax)), O(log(k))} where k is the total number of requests and dmaxis the maximum demand in terms of the number of sub-carriers. Moreover, we provide a heuristic for the network with arbitrary topology and measure the effectiveness of the heuristic with extensive simulation.

The orthogonal frequency division multiplexing (OFDM) technology provides an opportunity for efficient resource utilization in optical networks. It allows allocation of multiple sub-carriers to meet traffic demands of varying size. Utilizing OFDM technology, a spectrum efficient and scalable optical transport network called SLICE was proposed recently. The SLICE architecture enables sub-wavelength, super-wavelength resource allocation and multiple rate data traffic that results in efficient use of spectrum. However, the benefit is accompanied by additional complexities in resource allocation. In SLICE architecture, in order to minimize the utilized spectrum, one has to solve the routing and spectrum allocation problem (RSA). In this paper, we focus our attention to RSA and (i) prove that RSA is NP-complete even when the optical network topology is as simple as a chain or a ring, (ii) provide approximation algorithms for RSA when the network topology is a binary tree or a ring, (iii) provide a heuristic for the network with arbitrary topology and measure the effectiveness of the heuristic with extensive simulation. Simulation results demonstrate that our heuristic significantly outperforms several other heuristics proposed recently for RSA.