Minimum algorithm sizes for self-stabilizing gathering and related problems of autonomous mobile robots

We investigate a swarm of autonomous mobile robots in the Euclidean plane. A robot has a function called {\em target function} to determine the destination point from the robots' positions. All robots in the swarm conventionally take the same target function, but there is apparent limitation in problem-solving ability. We allow the robots to take different target functions. The number of different target functions necessary and sufficient to solve a problem $\Pi$ is called the {\em minimum algorithm size} (MAS) for $\Pi$. We establish the MASs for solving the gathering and related problems from {\bf any} initial configuration, i.e., in a {\bf self-stabilizing} manner. We show, for example, for $1 \leq c \leq n$, there is a problem $\Pi_c$ such that the MAS for the $\Pi_c$ is $c$, where $n$ is the size of swarm. The MAS for the gathering problem is 2, and the MAS for the fault tolerant gathering problem is 3, when $1 \leq f (< n)$ robots may crash, but the MAS for the problem of gathering all robot (including faulty ones) at a point is not solvable (even if all robots have distinct target functions), as long as a robot may crash

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three dimensional space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon