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We introduce a communication model for hybrid networks, where nodes have access to two different communication modes: a local mode where (like in traditional networks) communication is only possible between specific pairs of nodes, and a global mode where (like in overlay networks) communication between any pair of nodes is possible. This can be motivated, for instance, by wireless networks in which we combine direct device-to-device communication (e.g., using WiFi) with communication via a shared infrastructure (like base stations, the cellular network, or satellites).Typically, communication over short-range connections is cheaper and can be done at a much higher rate. Hence, we are focusing here on the LOCAL model (in which the nodes can exchange an unbounded amount of information in each round) for the local connections while for the global communication we assume the so-called node-capacitated clique model, where in each round every node can exchange only O(log n)-bit messages with just O(log n) other nodes. However, our model for hybrid networks is very general so that it can also capture many other scenarios, like the congested clique model.In order to explore the power of combining local and global communication, we study the complexity of computing shortest paths in the graph given by the local connections. We show that our model allows the development of algorithms that are significantly faster than what can be done by using either local or global communication only.We specifically show the following results. For the all-pairs shortest paths problem (APSP), we show that an exact solution can be computed in timeÕ n 2/3 1 and that approximate solutions can be computed in timeΘ √ n . For the single-source shortest paths problem (SSSP), we show that an exact solution can be computed in timeÕ √ SPD , where SPD denotes the shortest path diameter. We further show that a 1+o(1) -approximate solution can be computed in timeÕ n 1/3 . Additionally, we show that for every constant ε > 0, it is possible to compute an O(1)-approximate solution in timeÕ(n ε ).1 Note that theÕ(·)-notation hides factors that are polylogarithmic in n.Many existing communication networks exploit a combination of multiple communication modes to maximize cost-efficiency and throughput. As a prominent example, hybrid datacenter networks combine highspeed optical or wireless circuit switching technologies with traditional electronic packet switches to offer higher throughput at lower cost [14,21]. In the Internet, dynamic multipoint VPNs can be set up to connect different branches of an organization by combining leased lines (offering them quality-of-service guarantees for their mission-critical traffic) with standard, best-effort VPN connections (for their lower-priority traffic) [35]. Alternatively, an organization may also set up a so-called hybrid WAN by combining their own communication infrastructure with connections via the Internet [38]. Finally, the emerging 5G standard promises to allow smartphones to not only communicate via the ce...

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We envision programmable matter as a system of nano-scale agents (called particles) with very limited computational capabilities that move and compute collectively to achieve a desired goal. We use the geometric amoebot model as our computational framework, which assumes particles move on the triangular lattice. Motivated by the problem of shape sealing whose goal is to seal an object using as little resources as possible, we investigate how a particle system can self-organize to form an object's convex hull. We give a fully distributed, local algorithm for convex hull formation and prove that it runs in O(B + H log H) asynchronous rounds, where B is the length of the object's boundary and H is the length of the object's convex hull. Our algorithm can be extended to also form the object's ortho-convex hull, which requires the same number of particles but additionally minimizes the enclosed space within the same asymptotic runtime.