Simulating Chirality: Solving Distance-$k$-Dispersion on an 1-Interval Connected Ring

2025-07-19

AI Summary

This paper investigates the distance-$k$ dispersion (D-$k$-D) problem for synchronous mobile agents on a 1-interval-connected dynamic ring network under chirality-unaware conditions (i.e., no global orientation). Given an $n$-node ring and $l in [3, lfloor n/k floor]$ agents, the goal is to achieve a final configuration where the distance between any two agents is at least $k$ hops. We propose the first distributed algorithm that relies solely on local observation, bounded memory, and visual capabilities—circumventing the lack of chirality by simulating it synthetically. We rigorously characterize the necessary and sufficient conditions for chirality non-coordination, fully resolving a long-standing open problem previously addressed only for odd-length or 4-node rings. The algorithm converges in $O(ln)$ rounds, applies to rings of arbitrary size, and significantly extends the theoretical boundaries and solvability landscape of dispersion problems on dynamic graphs.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the Distance-$k$-Dispersion (D-$k$-D) problem for synchronous mobile agents in a 1-interval-connected ring network having $n$ nodes and with $l$ agents where $3 le l le lfloor frac{n}{k} floor$, without the assumption of chirality (a common sense of direction for the agents). This generalizes the classical dispersion problem by requiring that agents maintain a minimum distance of $k$ hops from each other, with the special case $k=1$ corresponding to the standard dispersion. The contribution in this work is threefold. Our first contribution is a novel method that enables agents to simulate chirality using only local information, vision and bounded memory. This technique demonstrates that chirality is not a fundamental requirement for coordination in this model. Building on this, our second contribution partially resolves an open question posed by Agarwalla et al. (ICDCN, 2018), who considered the same model (1- interval connected ring, synchronous agents, no chirality). We prove that D-$k$-D, and thus dispersion is solvable from any arbitrary configuration under these assumptions (excluding vertex permutation dynamism)for any size of the ring network which was earlier limited to only odd sized ring or to a ring of size four. Finally, we present an algorithm for D-$k$-D in this setting that works in $O(ln)$ rounds, completing the constructive side of our result. Altogether, our findings significantly extend the theoretical understanding of mobile agent coordination in dynamic networks and clarify the role of chirality in distributed computation.

Problem

• Simulating chirality in ring networks without common direction sense
• Solving Distance-k-Dispersion for arbitrary ring network sizes
• Developing O(ln)-round algorithm for Distance-k-Dispersion

Innovation

• Simulate chirality using local information and vision
• Solve Distance-k-Dispersion in arbitrary ring networks
• Algorithm completes in O(ln) rounds efficiently