Rendezvous on a Known Dynamic Point on a Finite Unoriented Grid

In this paper, we have considered two fully synchronous $\mathcal{OBLOT}$ robots having no agreement on coordinates entering a finite unoriented grid through a door vertex at a corner, one by one. There is a resource that can move around the grid synchronously with the robots until it gets co-located along with at least one robot. Assuming the robots can see and identify the resource, we consider the problem where the robots must meet at the location of this dynamic resource within finite rounds. We name this problem "Rendezvous on a Known Dynamic Point". Here, we have provided an algorithm for the two robots to gather at the location of the dynamic resource. We have also provided a lower bound on time for this problem and showed that with certain assumption on the waiting time of the resource on a single vertex, the algorithm provided is time optimal. We have also shown that it is impossible to solve this problem if the scheduler considered is semi-synchronous

Space and move-optimal Arbitrary Pattern Formation on infinite rectangular grid by Oblivious Robot Swarm

Arbitrary Pattern Formation (APF) is a fundamental coordination problem in swarm robotics. It requires a set of autonomous robots (mobile computing units) to form any arbitrary pattern (given as input) starting from any initial pattern. The APF problem is well-studied in both continuous and discrete settings. This work concerns the discrete version of the problem. A set of robots is placed on the nodes of an infinite rectangular grid graph embedded in a euclidean plane. The movements of the robots are restricted to one of the four neighboring grid nodes from its current position. The robots are autonomous, anonymous, identical, and homogeneous, and operate Look-Compute-Move cycles. Here we have considered the classical $\mathcal{OBLOT}$ robot model, i.e., the robots have no persistent memory and no explicit means of communication. The robots have full unobstructed visibility. This work proposes an algorithm that solves the APF problem in a fully asynchronous scheduler under this setting assuming the initial configuration is asymmetric. The considered performance measures of the algorithm are space and number of moves required for the robots. The algorithm is asymptotically move-optimal. A definition of the space-complexity is presented here. We observe an obvious lower bound $\mathcal{D}$ of the space complexity and show that the proposed algorithm has the space complexity $\mathcal{D}+4$. On comparing with previous related works, we show that this is the first proposed algorithm considering $\mathcal{OBLOT}$ robot model that is asymptotically move-optimal and has the least space complexity which is almost optimal

Asynchronous Gathering of Robots with Finite Memory on a Circle under Limited Visibility

Consider a set of $n$ mobile entities, called robots, located and operating on a continuous circle, i.e., all robots are initially in distinct locations on a circle. The \textit{gathering} problem asks to design a distributed algorithm that allows the robots to assemble at a point on the circle. Robots are anonymous, identical, and homogeneous. Robots operate in a deterministic Look-Compute-Move cycle within the circular path. Robots agree on the clockwise direction. The robot's movement is rigid and they have limited visibility $\pi$, i.e., each robot can only see the points of the circle which is at an angular distance strictly less than $\pi$ from the robot. Di Luna \textit{et al}. [DISC'2020] provided a deterministic gathering algorithm of oblivious and silent robots on a circle in semi-synchronous (\textsc{SSync}) scheduler. Buchin \textit{et al}. [IPDPS(W)'2021] showed that, under full visibility, $\mathcal{OBLOT}$ robot model with \textsc{SSync} scheduler is incomparable to $\mathcal{FSTA}$ robot (robots are silent but have finite persistent memory) model with asynchronous (\textsc{ASync}) scheduler. Under limited visibility, this comparison is still unanswered. So, this work extends the work of Di Luna \textit{et al}. [DISC'2020] under \textsc{ASync} scheduler for $\mathcal{FSTA}$ robot model

Arbitrary Pattern Formation on a Continuous Circle by Oblivious Robot Swarm

In the field of distributed system, Arbitrary Pattern Formation (APF) problem is an extensively studied problem. The purpose of APF is to design an algorithm to move a swarm of robots to a particular position on an environment (discrete or continuous) such that the swarm can form a specific but arbitrary pattern given previously to every robot as an input. In this paper the solvability of the APF problem on a continuous circle has been discussed for a swarm of oblivious and silent robots without chirality under a semi synchronous scheduler. Firstly a class of configurations called \textit{Formable Configuration}($FC$) has been provided which is necessary to solve the APF problem on a continuous circle. Then considering the initial configuration to be an $FC$, an deterministic and distributed algorithm has been provided that solves the APF problem for $n$ robots on a continuous circle of fixed radius within $O(n)$ epochs without collision

Time Optimal Gathering of Robots on an Infinite Triangular Grid with Limited Visibility

This work deals with the problem of gathering of $n$ oblivious mobile entities, called robots, with limited visibility, at a point (not known beforehand) placed on an infinite triangular grid. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is arbitrary pattern formation where full visibility is considered (\cite{C21}). The other one considers seven robots with 2- hop visibility that form a hexagon with one robot in the center of the hexagon in a collision-less environment under a fully synchronous scheduler (\cite{ShibataOS00K21}). In this work, we first show that gathering on a triangular grid with 1-hop vision of robots is not possible even under a fully synchronous scheduler if the robots do not agree on any axis. So one axis agreement has been considered in this work (i.e., the robots agree on a direction and its orientation). We have also showed that the lower bound for time is $\Omega(n)$ epochs when $n$ number of robots are gathering on an infinite triangular grid. An algorithm is then presented where a swarm of $n$ number of robots with 1-hop visibility can gather within $O(n)$ epochs under a semi-synchronous scheduler. So the algorithm presented here is time optimal

Move and Time Optimal Arbitrary Pattern Formation by Asynchronous Robots on Infinite Grid

The \textsc{Arbitrary Pattern Formation} (\textsc{Apf}) is a widely studied in distributed computing for swarm robots. This problem asks to design a distributed algorithm that allows a team of identical, autonomous mobile robots to form any arbitrary pattern given as input. This paper considers that the robots are operating on a two-dimensional infinite grid. Robots are initially positioned on distinct grid points forming an asymmetric configuration (no two robots have the same snapshot). They operate under a fully asynchronous scheduler and do not have any access to a global coordinate system, but they will align the axes of their local coordinate systems along the grid lines. The previous work dealing with \textsc{Apf} problem solved it in $O(\mathcal{D}^2k)$ robot movements under similar conditions, where $\mathcal{D}$ is the side of the smallest square that can contain both initial and target configuration and, $k$ is the number of robots. Let $\mathcal{D}'=\max\{\mathcal{D},k\}$. This paper presents two algorithms of \textsc{Apf} on an infinite grid. The first algorithm solves the \textsc{Apf} problem using $O(\mathcal{D}')$ asymptotically move optimal. The second algorithm solves the \textsc{Apf} problem in $O(\mathcal{D}')$ epochs, which we show is asymptotically optimal