Convex Hull Formation for Programmable Matter

Abstract

We envision programmable matter as a system of nanoscale agents (called particles) with very limited computational capabilities that move and compute collectively to achieve a desired goal. Motivated by the problem of sealing an object using minimal resources, we show how a particle system can self-organize to form an object’s convex hull. We give a distributed, local algorithm for convex hull formation and prove that it runs in ${\cal O}(B)$ asynchronous rounds, where $B$ is the length of the object’s boundary. Within the same asymptotic runtime, this algorithm can be extended to also form the object’s (weak) O-hull, which uses the same number of particles but minimizes the area enclosed by the hull. Our algorithms are the first to compute convex hulls with distributed entities that have strictly local sensing, constant-size memory, and no shared sense of orientation or coordinates. Ours is also the first distributed approach to computing restricted-orientation convex hulls. This approach involves coordinating particles as distributed memory; thus, as a supporting but independent result, we present and analyze an algorithm for organizing particles with constant-size memory as distributed binary counters that efficiently support increments, decrements, and zero-tests — even as the particles move.

Publication
Proceedings of the 21st International Conference on Distributed Computing and Networking
Joshua J. Daymude
Joshua J. Daymude
Assistant Professor, Computer Science

I am a Christian and assistant professor in computer science studying collective emergent behavior and programmable matter through the lens of distributed computing, stochastic processes, and bio-inspired algorithms. I also love gaming and playing music.

Robert Gmyr
Robert Gmyr
Research Software Development Engineer
Kristian Hinnenthal
Kristian Hinnenthal
PhD Student, Computer Science
Irina Kostitsyna
Irina Kostitsyna
Professor of Mathematics and Computer Science
Christian Scheideler
Christian Scheideler
Professor of Computer Science
Andréa W. Richa
Andréa W. Richa
Professor of Computer Science

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