Compatibility of convergence algorithms for autonomous mobile robots

Abstract

We investigate a swarm of anonymous oblivious mobile robots under the semi-synchronous ($\mathrm{SSYNC}$) scheduler. Each 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 Φ and Π be a set of target functions and a problem, respectively. If the robots whose target functions are chosen from Φ always solve Π, we say that Φ is compatible with respect to Π. If Φ is compatible with respect to Π, every target function $\varphi \in \Phi$ is an algorithm for Π (in the conventional sense). Note that even if both ϕ and ${\varphi }^{\prime }$ are algorithms for Π, $\{\varphi ,{\varphi }^{\prime }\}$ 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. 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 $f\ge 2$: No set of target functions which always shrink the convex hull of a configuration is compatible. The third group, the FC(f) for $f\ge 2$, 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.