Stand-Up Indulgent Gathering on Rings

TL;DR

The paper addresses crash-tolerant gathering (SUIG) and rendezvous (SUIR) for disoriented, oblivious robots on ring networks with unlimited visibility and no multiplicity detection. It provides a precise FSYNC-based algorithm for SUIG when $k>3$ or ($k>2$ and $n$ is odd) starting from node-edge symmetric, non-periodic configurations, and builds a framework of symmetry-aware predicates (NE(k), L3, L4, L5, P, QNE, etc.) and moves (M_main, M_secondary, M_L2, M_same, M_opp) to guarantee progress and crash resilience. The correctness is proved via a sequence of geometric lemmas about axes of symmetry and a transition diagram showing reduction from NE(k) to L3 and then to gathered configuration, with dedicated handling for crash scenarios. The results delineate solvability boundaries (e.g., SUIR in FSYNC under specific symmetry, impossibility in SSYNC, and limitations when the ring size is a multiple of 3 for SUIG with small $k$) and offer a constructive algorithm with potential extensions to other topologies and detection capabilities. Overall, the work advances understanding of crash-tolerant coordination in constrained discrete environments and provides techniques that can be useful for symmetric, distributed robot swarms in ring-like networks.

Abstract

We consider a collection of $k \geq 2$ robots that evolve in a ring-shaped network without common orientation, and address a variant of the crash-tolerant gathering problem called the \emph{Stand-Up Indulgent Gathering} (SUIG): given a collection of robots, if no robot crashes, robots have to meet at the same arbitrary location, not known beforehand, in finite time; if one robot or more robots crash on the same location, the remaining correct robots gather at the location of the crashed robots. We aim at characterizing the solvability of the SUIG problem without multiplicity detection capability.

Figures (15)

• Figure 1: An instance of an initial configuration with $k=3$ when $n\mod 3 = 0$
• Figure 2: An instance of a node-edge symmetric configuration
• Figure 3: An instance of a $\{r,r'\}$-quasi-node-edge symmetric configuration
• Figure 4: Overview of the transitions between configurations. Dashed lines represent transitions that occur only when a crashed robot is ordered to move. Loops are shown to occur only a finite number of times, see the corresponding lemmas for more details.
• Figure 5: Only the secondary robots are ordered to move towards the target node.

Theorems & Definitions (51)

• theorem 1: REA2008
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4
• proof
• definition 1