TY - GEN

T1 - Convex Hull Formation for Programmable Matter

AU - Daymude, Joshua J.

AU - Gmyr, Robert

AU - Hinnenthal, Kristian

AU - Kostitsyna, Irina

AU - Scheideler, Christian

AU - Richa, Andréa W.

N1 - Funding Information:
Daymude and Richa are supported in part by the National Science Foundation under awards CCF-1422603, CCF-1637393, and CCF-1733680. Hinnenthal and Scheideler are supported by the German Research Foundation (DFG) under Project SCHE 1592/6-1.

PY - 2020

Y1 - 2020

N2 - 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 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.

AB - 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 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.

KW - computational geometry

KW - convex hull

KW - distributed algorithms

KW - programmable matter

KW - restricted-orientation geometry

KW - self-organization

UR - http://www.scopus.com/inward/record.url?scp=85098011751&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85098011751&partnerID=8YFLogxK

U2 - 10.1145/3369740.3372916

DO - 10.1145/3369740.3372916

M3 - Conference contribution

AN - SCOPUS:85098011751

SN - 9781450377515

T3 - ACM International Conference Proceeding Series

BT - ACM International Conference Proceeding Series

PB - Association for Computing Machinery

T2 - 21st International Conference on Distributed Computing and Networking, ICDCN 2020

Y2 - 4 January 2020 through 7 January 2020

ER -