Compatibility of convergence algorithms for autonomous mobile robots

We investigate autonomous mobile robots in the Euclidean plane. A robot has a function called target function to decide the destination from the robots' positions, and operates in Look-Compute-Move cycles, i.e., identifies the robots' positions, computes the destination by the target function, and then moves there. Robots may have different target functions. Let $\Phi$ and $\Pi$ be a set of target functions and a problem, respectively. If the robots whose target functions are chosen from $\Phi$ always solve $\Pi$, we say that $\Phi$ is compatible with respect to $\Pi$. If $\Phi$ is compatible with respect to $\Pi$, every target function $\phi \in \Phi$ is an algorithm for $\Pi$ (in the conventional sense). Note that even if both $\phi$ and $\phi'$ are algorithms for $\Pi$, $\{ \phi, \phi' \}$ may not be compatible with respect to $\Pi$. From the view point of compatibility, we investigate the convergence, the fault tolerant ($n,f$)-convergence (FC($f$)), the fault tolerant ($n,f$)-convergence to $f$ points (FC($f$)-PO), the fault tolerant ($n,f$)-convergence to a convex $f$-gon (FC($f$)-CP), and the gathering problems, assuming crash failures. As a result, we see that these problems are classified into three groups: The convergence, the FC(1), the FC(1)-PO, and the FC($f$)-CP compose the first group: Every set of target functions which always shrink the convex hull of a configuration is compatible. The second group is composed of the gathering and the FC($f$)-PO for $f \geq 2$: No set of target functions which always shrink the convex hull of a configuration is compatible. The third group, the FC($f$) for $f \geq 2$, is placed in between. Thus, the FC(1) and the FC(2), the FC(1)-PO and the FC(2)-PO, and the FC(2) and the FC(2)-PO are respectively in different groups, despite that the FC(1) and the FC(1)-PO are in the first group

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