Dispersion, Capacitated Nodes, and the Power of a Trusted Shepherd

In this paper, we look at and expand the problems of dispersion and Byzantine dispersion of mobile robots on a graph, introduced by Augustine and Moses~Jr.~[ICDCN~2018] and by Molla, Mondal, and Moses~Jr.~[ALGOSENSORS~2020], respectively, to graphs where nodes have variable capacities. We use the idea of a single shepherd, a more powerful robot that will never act in a Byzantine manner, to achieve fast Byzantine dispersion, even when other robots may be strong Byzantine in nature. We also show the benefit of a shepherd for dispersion on capacitated graphs when no Byzantine robots are present

Fault-Tolerant Dispersion of Mobile Robots

We consider the mobile robot dispersion problem in the presence of faulty robots (crash-fault). Mobile robot dispersion consists of $k\leq n$ robots in an $n$-node anonymous graph. The goal is to ensure that regardless of the initial placement of the robots over the nodes, the final configuration consists of having at most one robot at each node. In a crash-fault setting, up to $f \leq k$ robots may fail by crashing arbitrarily and subsequently lose all the information stored at the robots, rendering them unable to communicate. In this paper, we solve the dispersion problem in a crash-fault setting by considering two different initial configurations: i) the rooted configuration, and ii) the arbitrary configuration. In the rooted case, all robots are placed together at a single node at the start. The arbitrary configuration is a general configuration (a.k.a. arbitrary configuration in the literature) where the robots are placed in some $l<k$ clusters arbitrarily across the graph. For the first case, we develop an algorithm solving dispersion in the presence of faulty robots in $O(k^2)$ rounds, which improves over the previous $O(f\cdot\text{min}(m,k\Delta))$-round result by \cite{PS021}. For the arbitrary configuration, we present an algorithm solving dispersion in $O((f+l)\cdot\text{min}(m, k \Delta, k^2))$ rounds, when the number of edges $m$ and the maximum degree $\Delta$ of the graph is known to the robots

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

Run for Cover: Dominating Set via Mobile Agents

Research involving computing with mobile agents is a fast-growing field, given the advancement of technology in automated systems, e.g., robots, drones, self-driving cars, etc. Therefore, it is pressing to focus on solving classical network problems using mobile agents. In this paper, we study one such problem -- finding small dominating sets of a graph $G$ using mobile agents. Dominating set is interesting in the field of mobile agents as it opens up a way for solving various robotic problems, e.g., guarding, covering, facility location, transport routing, etc. In this paper, we first present two algorithms for computing a {\em minimal dominating set}: (i) an $O(m)$ time algorithm if the robots start from a single node (i.e., gathered initially), (ii) an $O(\ell\Delta\log(\lambda)+n\ell+m)$ time algorithm, if the robots start from multiple nodes (i.e., positioned arbitrarily), where $m$ is the number of edges and $\Delta$ is the maximum degree of $G$, $\ell$ is the number of clusters of the robot initially and $\lambda$ is the maximum ID-length of the robots. Then we present a $\ln (\Delta)$ approximation algorithm for the {\em minimum} dominating set which takes $O(n\Delta\log (\lambda))$ rounds

Fast Deterministic Gathering with Detection on Arbitrary Graphs: The Power of Many Robots

Over the years, much research involving mobile computational entities has been performed. From modeling actual microscopic (and smaller) robots, to modeling software processes on a network, many important problems have been studied in this context. Gathering is one such fundamental problem in this area. The problem of gathering k robots, initially arbitrarily placed on the nodes of an n-node graph, asks that these robots coordinate and communicate in a local manner, as opposed to global, to move around the graph, find each other, and settle down on a single node as fast as possible. A more difficult problem to solve is gathering with detection, where once the robots gather, they must subsequently realize that gathering has occurred and then terminate. In this paper, we propose a deterministic approach to solve gathering with detection for any arbitrary connected graph that is faster than existing deterministic solutions for even just gathering (without the requirement of detection) for arbitrary graphs. In contrast to earlier work on gathering, it leverages the fact that there are more robots present in the system to achieve gathering with detection faster than those previous papers that focused on just gathering. The state of the art solution for deterministic gathering [Ta-Shma and Zwick, TALG, 2014] takes O˜(n 5 log ℓ) rounds, where ℓ is the smallest label among robots and O˜ hides a polylog factor. We design a deterministic algorithm for gathering with detection with the following trade-offs depending on how many robots are present: (i) when k ≥ ⌊n/2⌋ + 1, the algorithm takes O(n 3 ) rounds, (ii) when k ≥ ⌊n/3⌋+1, the algorithm takes O(n 4 log n) rounds, and (iii) otherwise, the algorithm takes O˜(n 5 ) rounds. The algorithm is not required to know k, but only