Yuichi Asahiro | Searcharxiv

arXiv2023

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

This paper investigates a swarm of autonomous mobile robots in the Euclidean plane, under the semi-synchronous ($\cal SSYNC$) scheduler. Each robot has a target function to determine a destination point from the robots' positions. All robots in the swarm take the same target function conventionally. We allow the robots to take different target functions, and investigate the effects of the number of distinct target functions on the problem-solving ability, regarding target function as a resource to solve a problem like time. Specifically, we are interested in how many distinct target functions are necessary and sufficient to solve a problem $Π$. The number of distinct target functions necessary and sufficient to solve $Π$ is called the minimum algorithm size (MAS) for $Π$. The MAS is defined to be $\infty$, if $Π$ is unsolvable even for the robots with unique target functions. We show that the problems form an infinite hierarchy with respect to their MASs; for each integer $c > 0$ and $\infty$, the set of problems whose MAS is $c$ is not empty, which implies that target function is a resource irreplaceable, e.g., with time. We propose MAS as a natural measure to measure the complexity of a problem. We establish the MASs for solving the gathering and related problems from any initial configuration, i.e., in a self-stabilizing manner. For example, the MAS for the gathering problem is 2. It is 3, for the problem of gathering {\bf all non-faulty} robots at a single point, regardless of the number $(< n)$ of crash failures. It is however $\infty$, for the problem of gathering all robots at a single point, in the presence of at most one crash failure.

cs.DC↗

arXiv2023

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. Robots may have different target functions. If the robots whose target functions are chosen from a set $Φ$ of target functions always solve a problem $Π$, we say that $Φ$ is compatible with respect to $Π$. If $Φ$ is compatible with respect to $Π$, every target function $ϕ\in Φ$ is an algorithm for $Π$. Even if both $ϕ$ and $ϕ'$ are algorithms for $Π$, $\{ ϕ, ϕ' \}$ may not be compatible with respect to $Π$. 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. Obtained results classify these problems into three groups: The convergence, FC(1), FC(1)-PO, and 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 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, FC($f$) for $f \geq 2$, is placed in between. Thus, FC(1) and FC(2), FC(1)-PO and FC(2)-PO, and FC(2) and FC(2)-PO are respectively in different groups, despite that FC(1) and FC(1)-PO are in the first group.

cs.DC↗