Time Optimal Distance-$k$-Dispersion on Dynamic Ring

TL;DR

The paper introduces Distance-$k$-Dispersion (D-$k$-D), a generalization of dispersion for $l$ robots on an $n$-node network, focusing on a 1-interval-connected dynamic ring with a rooted initial configuration. It proves that a fully synchronous scheduler is necessary and presents a deterministic FSync algorithm, D-$k$-D DynamicRing, that achieves a valid configuration in $Θ(n)$ rounds, thereby establishing time-optimality. The solution relies on two subroutines, Spread and ReconstructChain, and demonstrates correctness across chain/non-chain configurations under chirality; it also shows how to handle non-chiral settings via a preprocessing step. The work is the first to study D-$k$-D on dynamic rings, with potential extensions to arbitrary networks and scenarios with limited visibility, making a notable contribution to the theory of distributed robot dispersion and synchronization.

Abstract

Dispersion by mobile agents is a well studied problem in the literature on computing by mobile robots. In this problem, $l$ robots placed arbitrarily on nodes of a network having $n$ nodes are asked to relocate themselves autonomously so that each node contains at most $\lfloor \frac{l}{n}\rfloor$ robots. When $l\le n$, then each node of the network contains at most one robot. Recently, in NETYS'23, Kaur et al. introduced a variant of dispersion called \emph{Distance-2-Dispersion}. In this problem, $l$ robots have to solve dispersion with an extra condition that no two adjacent nodes contain robots. In this work, we generalize the problem of Dispersion and Distance-2-Dispersion by introducing another variant called \emph{Distance-$k$-Dispersion (D-$k$-D)}. In this problem, the robots have to disperse on a network in such a way that shortest distance between any two pair of robots is at least $k$ and there exist at least one pair of robots for which the shortest distance is exactly $k$. Note that, when $k=1$ we have normal dispersion and when $k=2$ we have D-$2$-D. Here, we studied this variant for a dynamic ring (1-interval connected ring) for rooted initial configuration. We have proved the necessity of fully synchronous scheduler to solve this problem and provided an algorithm that solves D-$k$-D in $Θ(n)$ rounds under a fully synchronous scheduler. So, the presented algorithm is time optimal too. To the best of our knowledge, this is the first work that considers this specific variant.

Figures (7)

• Figure 1: $v_x$ and $v_y$ are consecutive occupied nodes in the arc $(v_x,v_z)_{\mathcal{D}}$ of length $5$ in direction $\mathcal{D}$. Black dots represent the nodes on the ring and blue boxes represent the robots.
• Figure 2: Chain configuration where $(v_{i-1},v_j)_{\mathcal{D}}$ is a chain in clockwise direction $\mathcal{D}$ and $(v_{i+1},v_0)_{\mathcal{D}'}$ is a chain in counter clockwise direction for $k=3$.
• Figure 3: Different types of chain configuration for $k=3$
• Figure 4: Different types of non-chain configuration for $k=3$
• Figure 5: Target Configuration of D-$k$-D where $k=3$

Theorems & Definitions (30)

• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• corollary thmcountercorollary
• definition thmcounterdefinition: Configuration
• definition thmcounterdefinition: Arc
• definition thmcounterdefinition: Distance between two nodes
• definition thmcounterdefinition: Consecutive occupied nodes