Convex Hull Formation for Programmable Matter

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 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 $\mathcal{O}(B)$ 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) $\mathcal{O}$-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

I/O-optimal algorithms on grid graphs

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.Comment: 12 pages' extended abstract plus 12 pages' appendix with details, proofs and calculations. Has not been published in and is currently not under review of any conference or journa