Bastian Degener

A tight runtime bound for synchronous gathering of autonomous robots with limited visibility

The problem of gathering n autonomous robots in the Euclidean plane at one (not predefined) point is well-studied under various restrictions on the capabilities of the robots and in several time models. However, only very few runtime bounds are known. We consider the scenario of local algorithms in which the robots can only observe their environment within a fixed viewing range and have to base their decision where to move in the next step solely on the relative positions of the robots within their viewing range. Such local algorithms have to guarantee that the (initially connected) unit disk graph defined by the viewing range of the robots stays connected at all times. In this paper, we focus on the synchronous setting in which all robots are activated concurrently. Ando et al. [2] presented an algorithm where a robot essentially moves to the center of the smallest enclosing circle of the robots in its viewing range and showed that this strategy performs gathering of the robots in finite time. However, no bounds on the number of rounds needed by the algorithm are known. We present a lower bound of ©(n2) for the number of rounds as well as a matching upper bound of O(n2) and thereby obtain a tight runtime analysis of the algorithm of Θ(n).

A new approach for analyzing convergence algorithms for mobile robots

Given a set of n mobile robots in the d-dimensional Euclidean space, the goal is to let them converge to a single not predefined point. The challenge is that the robots are limited in their capabilities. Robots can, upon activation, compute the positions of all other robots using an individual affine coordinate system. The robots are indistinguishable, oblivious and may have different affine coordinate systems. A very general discrete time model assumes that robots are activated in arbitrary order. Further, the computation of a new target point may happen much earlier than the movement, so that the movement is based on outdated information about other robots positions. Time is measured as the number of rounds, where a round ends as soon as each robot has moved at least once. In [6], the Center of Gravity is considered as target function, convergence was proven, and the number of rounds needed for halving the diameter of the convex hull of the robots positions was shown to be O(n2) and Ω(n). We present an easy-to-check property of target functions that guarantee convergence and yields upper time bounds. This property intuitively says that when a robot computes a new target point, this point is significantly within the current axes aligned minimal box containing all robots. This property holds, e.g., for the above-mentioned target function, and improves the above O(n2) to an asymptotically optimal O(n) upper bound. Our technique also yields a constant time bound for a target function that requires all robots having identical coordinate axes.

Collisionless gathering of robots with an extent

Gathering n mobile robots in one single point in the Euclidean plane is a widely studied problem from the area of robot formation problems. Classically, the robots are assumed to have no physical extent, and they are able to share a position with other robots. We drop these assumptions and investigate a similar problem for robots with (a spherical) extent: the goal is to gather the robots as close together as possible. More exactly, we want the robots to form a sphere with minimum radius around a predefined point. We propose an algorithm for this problem which synchronously moves the robots towards the center of the sphere unless they block each other. In this case, if possible, the robots spin around the center of the sphere. We analyze this algorithm experimentally in the plane. If R is the distance of the farthest robot to the center of the sphere, the simulations indicate a runtime which is linear in n and R. Additionally, we prove a theoretic upper bound for the runtime of O(nR) for a discrete version of the problem. Simulations also suggest a runtime of O(n + R) for the discrete version.

A local O(n 2 ) gathering algorithm

The gathering problem, where

A continuous, local strategy for constructing a short chain of mobile robots

n

Survey: A survey on relay placement with runtime and approximation guarantees

autonomous robots with restricted capabilities are required to meet in a single point of the plane, is widely studied. We consider the case that robots are limited to see only robots within a bounded vicinity and present an algorithm achieving gathering in O(n2) rounds in expectation. A round consists of a movement of all robots, in random order. All previous algorithms with a proven time bound assume global view on the configuration of all robots.

Kinetic Facility Location

We are given an arbitrarily shaped chain of n robots with fixed end points in the plane. We assume that each robot can only see its two neighbors in the chain, which have to be within its viewing range. The goal is to move the robots to the straight line between the end points. Each robot has to base the decision where to move on the relative positions of its neighbors only. Such local strategies considered until now are based on discrete rounds, where a round consists of a movement of each robot. In this paper, we initiate the study of continuous local strategies: The robots may perpetually observe the relative positions of their neighbors, and may perpetually adjust their speed and direction in response to these observations. We assume a speed limit for the robots, that we normalize to one, which corresponds to the viewing range. Our contribution is a continuous, local strategy that needs time

Linear and Competitive Strategies for Continuous Robot Formation Problems

{\mathcal O}(min\{n, (OPT+d) \log(n)\})

A local, distributed constant-factor approximation algorithm for the dynamic facility location problem

. Here d is the distance between the two stationary end points, and OPT is the time needed by an optimal global strategy. Our strategy has the property that the robot which reaches its destination last always moves with maximum speed. Thus, the same bound as above also holds for the distance travelled.

The Kinetic Facility Location Problem

We discuss aspects and variants of the fundamental problem of relay placement: given a set of immobile terminals in the Euclidean plane, place a number of relays with limited viewing range such that the result is a low-cost communication infrastructure between the terminals. We first consider the problem from a global point of view. The problem here is similar to forming discrete Steiner tree structures. Then we investigate local variants of the problem, assuming mobile relays that must decide where to move based only on information from their local environment. We give a local algorithm for the general problem, but we show that no local algorithm can achieve good approximation factors for the number of relays. The following two restricted variants each address different aspects of locality. First we provide each relay with knowledge of two fixed neighbors, such that the relays form a chain between two terminals. The goal here is to let the relays move to the line segment between the terminals as fast as possible. Then we focus on the aspect of neighbors that are not fixed, but which may change over time. In return, we relax the objective function from geometric structures to just forming a single point. The goal in all our local formation problems is to use relays that are as limited as possible with respect to memory, sensing capabilities and so on. We focus on algorithms for which we can prove performance guarantees such as upper bounds on the required runtime, maximum traveled distances of the relays and approximation factors for the solution.