Section: New Results

Mobile Agents

More efficient periodic traversal in anonymous undirected graphs

Participants : David Ilcinkas, Ralf Klasing.

In [15] , we consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all $n$ nodes of an input simple, connected, undirected graph in a periodic manner. Graphs are assumed to be anonymous, that is, nodes are unlabeled. While visiting a node, the agent may distinguish between the edges incident to it; for each node $v$, the endpoints of the edges incident to $v$ are uniquely identified by different integer labels called port numbers. We are interested in algorithms for assigning the port numbers together with traversal algorithms for agents using these port numbers to obtain short traversal periods.

Periodic graph exploration is unsolvable if the port numbers are set arbitrarily; see Budach (1978). However, surprisingly small periods can be achieved by carefully assigning the port numbers. Dobrev et al. (2005) described an algorithm for assigning port numbers and an oblivious agent (i.e., an agent with no memory) using it, such that the agent explores any graph with $n$ nodes within the period $10n$. When the agent has access to a constant number of memory bits, the optimal length of the period was proved in Gasieniec et al. (2008) to be no more than $3.75n-2$ (using a different assignment of the port numbers and a different traversal algorithm). In our work, we improve both these bounds. More precisely, we show how to achieve a period length of at most $(4+\frac{1}{3})n-4$ for oblivious agents and a period length of at most $3.5n-2$ for agents with constant memory. To obtain our results, we introduce a new, fast graph decomposition technique called a three-layer partition that may also be useful for solving other graph problems in the future. Finally, we present the first non-trivial lower bound, $2.8n-2$, on the period length for the oblivious case.

Gathering of Robots on Anonymous Grids without Multiplicity Detection

Participant : Ralf Klasing.

In [28] , we study the gathering problem on grid networks. A team of robots placed at different nodes of a grid have to meet at some node and remain there. Robots operate in Look-Compute-Move cycles; in one cycle, a robot perceives the current configuration in terms of occupied nodes (Look), decides whether to move towards one of its neighbors (Compute), and in the positive case makes the computed move instantaneously (Move). Cycles are performed asynchronously for each robot. The problem has been deeply studied for the case of ring networks. However, the known techniques used on rings cannot be directly extended to grids. Moreover, on rings, another assumption concerning the so-called multiplicity detection capability was required in order to accomplish the gathering task. That is, a robot is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one.

In our work, we provide a full characterization about gatherable configurations for grids. In particular, we show that in this case, the multiplicity detection is not required. Very interestingly, sometimes the problem appears trivial, as it is for the case of grids with both odd sides, while sometimes the involved techniques require new insights with respect to the well-studied ring case. Moreover, our results reveal the importance of a structure like the grid that allows to overcome the multiplicity detection with respect to the ring case.