Uniform deployment of mobile agents in asynchronous rings

In this paper, we consider the uniform deployment problem of mobile agents in asynchronous unidirectional rings, which requires the agents to uniformly spread in the ring. The uniform deployment problem is in striking contrast to the rendezvous problem which requires the agents to meet at the same node. While rendezvous aims to break the symmetry, uniform deployment aims to attain the symmetry. It is well known that the symmetry breaking is difficult in distributed systems and the rendezvous problem cannot be solved from some initial configurations. Hence, we are interested in clarifying what difference the uniform deployment problem has on the solvability and the number of agent moves compared to the rendezvous problem. We consider two problem settings, with knowledge of k (or n) and without knowledge of k or n where k is the number of agents and n is the number of nodes. First, we consider agents with knowledge of k (or n since k and n can be easily obtained if one of them is given). In this case, we propose two algorithms. The first algorithm solves the uniform deployment problem with termination detection. This algorithm requires O(k log n) memory space per agent, O(n) time, and O(kn) total moves. The second algorithm also solves the uniform deployment problem with termination detection. This algorithm reduces the memory space per agent to O(log n), but uses O(n log k) time, and requires O(kn) total moves. Both algorithms are asymptotically optimal in terms of total moves since there are some initial configurations such that agents re- quire Ω(kn) total moves to solve the problem. Next, we consider agents with no knowledge of k or n. In this case, we show that, when termination detection is required, there exists no algorithm to solve the uniform deployment problem. For this reason, we consider the relaxed uniform deployment problem that does not require termination detection, and we propose an algorithm to solve the relaxed uniform deployment problem. This algorithm requires O((k/l) log(n/l)) memory space per agent, O(n/l) time, and O(kn/l) total moves when the initial configuration has symmetry degree l. This means that the algorithm can solve the problem more efficiently when the initial configuration has higher symmetric degree (i.e., is closer to uniform deployment). Note that all the proposed algorithms achieve uniform deployment from any initial configuration, which is a striking difference from the rendezvous problem because the rendezvous problem is not solvable from some initial configurations

Space-Efficient Uniform Deployment of Mobile Agents in Asynchronous Unidirectional Rings

In this paper, we consider the uniform deployment problem of mobile agents in asynchronous unidirectional ring networks. This problem requires agents to spread uniformly in the network. In this paper, we focus on the memory space per agent required to solve the problem. We consider two problem settings. The first setting assumes that agents have no multiplicity detection, that is, agents cannot detect whether another agent is staying at the same node or not. In this case, we show that each agent requires Ω(log n) memory space to solve the problem, where n is the number of nodes. In addition, we propose an algorithm to solve the problem with O(k+log n) memory space per agent, where k is the number of agents. The second setting assumes that each agent is equipped with the weak multiplicity detection, that is, agents can detect another agent staying at the same node, but cannot learn the exact number. Then, we show that the memory space per agent can be reduced to O(log k+log log n) . To the best of our knowledge, this is the first research considering the effect of the multiplicity detection on memory space required to solve problems.25th International Colloquium (SIROCCO 2018), June 18-21, 2018, Ma\u27ale HaHamisha, Israe

Space-efficient uniform deployment of mobile agents in asynchronous unidirectional rings

In this paper, we consider the uniform deployment problem of mobile agents in asynchronous unidirectional ring networks. This problem requires agents to spread uniformly in the network. In this paper, we focus on the memory space per agent required to solve the problem. We consider two problem settings. The first setting assumes that agents have no multiplicity detection, that is, agents cannot detect whether another agent is staying at the same node or not. In this case, we show that each agent requires memory space to solve the problem, where n is the number of nodes. In addition, we propose an algorithm to solve the problem with memory space per agent, where k is the number of agents. The second setting assumes that each agent is equipped with the weak multiplicity detection, that is, agents can detect whether another agent is staying at the same node or not, but cannot get any other information about the number of the agents. Then, we show that the memory space per agent can be reduced to. To the best of our knowledge, this is the first research considering the effect of the multiplicity detection on memory space required to solve problems

Uniform multi-agent deployment on a ring

AbstractWe consider two variants of the task of spreading a swarm of agents uniformly on a ring graph. Ant-like oblivious agents having limited capabilities are considered. The agents are assumed to have little memory, they all execute the same algorithm and no direct communication is allowed between them. Furthermore, the agents do not possess any global information. In particular, the size of the ring (n) and the number of agents in the swarm (k) are unknown to them. The agents are assumed to operate on an unweighted ring graph. Every agent can measure the distance to his two neighbors on the ring, up to a limited range of V edges.The first task considered, is dynamical (i.e. in motion) uniform deployment on the ring. We show that if either the ring is unoriented, or the visibility range is less than ⌊n/k⌋, this is an impossible mission for the agents. Then, for an oriented ring and V≥⌈n/k⌉, we propose an algorithm which achieves the deployment task in optimal time. The second task discussed, called quiescent spread, requires the agents to spread uniformly over the ring and stop moving. We prove that under our model, in which every agent can measure the distance only to his two neighbors, this task is impossible. Subsequently, we propose an algorithm which achieves quiescent but only almost uniform spread.The algorithms we present are scalable and robust. In case the environment (the size of the ring) or the number of agents changes during the run, the swarm adapts and re-deploys without requiring any outside interference

Near-Optimal Dispersion on Arbitrary Anonymous Graphs

Given an undirected, anonymous, port-labeled graph of n memory-less nodes, m edges, and degree ?, we consider the problem of dispersing k ? n robots (or tokens) positioned initially arbitrarily on one or more nodes of the graph to exactly k different nodes of the graph, one on each node. The objective is to simultaneously minimize time to achieve dispersion and memory requirement at each robot. If all k robots are positioned initially on a single node, depth first search (DFS) traversal solves this problem in O(min{m,k?}) time with ?(log(k+?)) bits at each robot. However, if robots are positioned initially on multiple nodes, the best previously known algorithm solves this problem in O(min{m,k?}? log ?) time storing ?(log(k+?)) bits at each robot, where ? ? k/2 is the number of multiplicity nodes in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in O(min{m,k?}) time with ?(log(k+?)) bits at each robot, improving the time bound of the best previously known algorithm by O(log ?) and matching asymptotically the single-source DFS traversal bounds. This is the first algorithm for dispersion that is optimal in both time and memory in arbitrary anonymous graphs of constant degree, ? = O(1). Furthermore, the result holds in both synchronous and asynchronous settings

Semi-Uniform Deployment of Mobile Robots in Perfect ℓ-ary Trees

In this paper, we consider the problem of semi-uniform deployment for mobile robots in perfect ℓ-ary trees, where every intermediate node has ℓ children, and all leaf nodes have the same depth. This problem requires robots to spread in the tree so that, for some positive integer d and some fixed integer s(0≤s≤d−1), each node of depth s+dj (j≥0) is occupied by a robot. In other words, after semi-uniform deployment is achieved, nodes of depth s,s+d,s+2d,… are occupied by a robot. Robots have an infinite visibility range but are opaque, that is, robot ri cannot observe some robot rj if there exists another robot rk in the path between ri and rj. In addition, each robot can emit a light color visible to itself and other robots, taken from a set of κ colors, at each time step. Then, we clarify the relationship between the number of available light colors and the solvability of the semi-uniform deployment problem. First, we consider robots with the minimum number of available light colors, that is, robots with κ=1 (in this case, robots are oblivious). In this setting, we show that there is no collision-free algorithm to solve the semi-uniform deployment problem with explicit termination. Next, we relax the number of available light colors, that is, we consider robots with κ=2. In this setting, we propose a collision-free algorithm that can solve the problem with explicit termination. Thus, our algorithm is optimal with respect to the number of light colors. In addition, to the best of our knowledge, this paper is the first to report research considering (a variant of) uniform deployment in graphs other than rings or grids.2021 Ninth International Symposium on Computing and Networking, CANDAR 2021, 23-26 November, 2021, Virtual Conferenc

Position discovery for a system of bouncing robots

International audienceA collection of n anonymous mobile robots is deployed on a unit-perimeter ring or a unit-length line segment. Every robot starts moving at constant speed, and bounces each time it meets any other robot or segment endpoint, changing its walk direction. We study the problem of position discovery, in which the task of each robot is to detect the presence and the initial positions of all other robots. The robots cannot communicate or perceive information about the environment in any way other than by bouncing nor they have control over their walks which are determined by their initial positions and their starting directions. Each robot has a clock allowing it to observe the times of its bounces. We give complete characterizations of all initial configurations for both the ring and the segment in which no position detection algorithm exists and we design optimal position detection algorithms for all feasible configurations