Section: New Results

Mobile Agents

Collision-Free Network Exploration

Participants : Ralf Klasing, Adrian Kosowski, Dominik Pajak.

A set of mobile agents is placed at different nodes of a $n$-node network. The agents synchronously move along the network edges in a collision-free way, i.e., in no round may two agents occupy the same node. In each round, an agent may choose to stay at its currently occupied node or to move to one of its neighbors. An agent has no knowledge of the number and initial positions of other agents. We are looking for the shortest possible time required to complete the collision-free network exploration, i.e., to reach a configuration in which each agent is guaranteed to have visited all network nodes and has returned to its starting location. In [34] , we first consider the scenario when each mobile agent knows the map of the network, as well as its own initial position. We establish a connection between the number of rounds required for collision-free exploration and the degree of the minimum-degree spanning tree of the graph. We provide tight (up to a constant factor) lower and upper bounds on the collision-free exploration time in general graphs, and the exact value of this parameter for trees. For our second scenario, in which the network is unknown to the agents, we propose collision-free exploration strategies running in $O\left(n^2\right)$ rounds for tree networks and in $O(n^{5}logn)$ rounds for general networks.

Deterministic Rendezvous of Asynchronous Bounded-Memory Agents in Polygonal Terrains

Participant : Adrian Kosowski.

In [22] , we deal with a more geometric variant of the rendezvous problem. Two mobile agents, modeled as points starting at different locations of an unknown terrain, have to meet. The terrain is a polygon with polygonal holes. We consider two versions of this rendezvous problem: exact RV, when the points representing the agents have to coincide at some time, and $ϵ$-RV, when these points have to get at distance less than $ϵ$ in the terrain. In any terrain, each agent chooses its trajectory, but the movements of the agent on this trajectory are controlled by an adversary that may, e.g., speed up or slow down the agent. Agents have bounded memory: their computational power is that of finite state machines. Our aim is to compare the feasibility of exact and of $ϵ$-RV when agents are anonymous vs. when they are labeled. We show classes of polygonal terrains which distinguish all the studied scenarios from the point of view of feasibility of rendezvous. The features which influence the feasibility of rendezvous include symmetries present in the terrains, boundedness of their diameter, and the number of vertices of polygons in the terrains.

Optimal Patrolling of Fragmented Boundaries

Participant : Adrian Kosowski.

Mobile agents in geometric scenarios are also studied in [33] , where a set of mobile robots is deployed on a simple curve of finite length, composed of a finite set of vital segments separated by neutral segments. The robots have to patrol the vital segments by perpetually moving on the curve, without exceeding their maximum speed. The quality of patrolling is measured by the idleness, i.e., the longest time period during which any vital point on the curve is not visited by any robot. Given a configuration of vital segments, our goal is to provide algorithms describing the movement of the robots along the curve so as to minimize the idleness. Our main contribution is a proof that the optimal solution to the patrolling problem is attained either by the cyclic strategy, in which all the robots move in one direction around the curve, or by the partition strategy, in which the curve is partitioned into sections which are patrolled separately by individual robots. These two fundamental types of strategies were studied in the past in the robotics community in different theoretical and experimental settings. However, to our knowledge, this is the first theoretical analysis proving optimality in such a general scenario.

Fast Collaborative Graph Exploration

Participants : Adrian Kosowski, Dominik Pajak, Przemyslaw Uznanski.

In [35] , we study the following scenario of online graph exploration. A team of $k$ agents is initially located at a distinguished vertex $r$ of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with $n$ vertices at a distance of at most $D$ from $r$ in time $O\left(D\right)$, using a team of agents of polynomial size $k=D{n}^{1+ϵ}<n^{2+ϵ}$, for any $ϵ>0$. Our strategy works in the local communication model, without knowledge of global parameters such as $n$ or $D$. We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large $k$. For any constant $c>1$, we show that in the global communication model, a team of $k=D{n}^c$ agents can always complete exploration in $D(1+\frac{1}{c-1}+o\left(1\right))$ time steps, whereas at least $D(1+\frac{1}{c}-o\left(1\right))$ steps are sometimes required. In the local communication model, $D(1+\frac{2}{c-1}+o\left(1\right))$ steps always suffice to complete exploration, and at least $D(1+\frac{2}{c}-o\left(1\right))$ steps are sometimes required. This shows a clear separation between the global and local communication models.

A $\tilde{O}\left(n⌃2\right)$ Time-Space Trade-off for Undirected s-t Connectivity

Participant : Adrian Kosowski.

The work [43] makes use of the Metropolis-type walks due to Nonaka et al. (2010) to provide a faster solution to the $S$-$T$-connectivity problem in undirected graphs (USTCON). As the main result of this research, we propose a family of randomized algorithms for USTCON which achieves a time-space product of $S·T=\tilde{O}\left(n^2\right)$ in graphs with $n$ nodes and $m$ edges (where the $\tilde{O}$-notation disregards poly-logarithmic terms). This improves the previously best trade-off of $\tilde{O}\left(nm\right)$, due to Feige (1995). Our algorithm consists in deploying several short Metropolis-type walks, starting from landmark nodes distributed using the scheme of Broder et al. (1994) on a modified input graph. In particular, we obtain an algorithm running in time $\tilde{O}(n+m)$ which is, in general, more space-efficient than both BFS and DFS. Finally, we show how to fine-tune the Metropolis-type walk so as to match the performance parameters (e.g., average hitting time) of the unbiased random walk for any graph, while preserving a worst-case bound of $\tilde{O}\left(n^2\right)$ on cover time.

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

Participants : Ralf Klasing, Adrian Kosowski, Dominik Pajak.

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. In [42] , we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the $n$-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks.

We show that on the ring the rotor-router with $k$ agents admits a cover time of between $\Theta (n^2/k^2)$ in the best case and $\Theta (n^2/logk)$ in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of cover time for $k$ random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between $\Theta (n^2/(k^2/{log}^{2}k))$ and $\Theta (n^2/logk)$.

Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every $\Theta (n/k)$ steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of $k$ random walks. All our results hold up to a polynomially large number of agents ($1\le k<n^{1/11}$).

Efficient Exploration of Anonymous Undirected Graphs

Participant : Ralf Klasing.

In [41] , we consider the problem of exploring an anonymous undirected graph using an oblivious robot. The studied exploration strategies are designed so that the next edge in the robot's walk is chosen using only local information. We present some current developments in the area. In particular, we focus on recent work on equitable strategies and on the multi-agent rotor-router.

Gathering radio messages in the path

Participant : Ralf Klasing.

In [19] , we address the problem of gathering information in one node (sink) of a radio network where interference constraints are present: when a node transmits, it produces interference in an area bigger than the area in which its message can actually be received. The network is modeled by a graph; a node is able to transmit one unit of information to the set of vertices at distance at most $dt$ in the graph, but when doing so it generates interferences that do not allow nodes at distance up to $di$ ($di\ge dt$) to listen to other transmissions. We are interested in finding a gathering protocol, that is an ordered sequence of rounds (each round consists of non-interfering simultaneous transmissions) such that $w\left(u\right)$ messages are transmitted from any node $u$ to a fixed node called the sink. Our aim is to find a gathering protocol with the minimum number of rounds (called gathering time). In [19] , we focus on the specific case where the network is a path with the sink at an end vertex of the path and where the traffic is unitary ($w\left(u\right)=1$ for all $u$); indeed this simple case appears to be already very difficult. We first give a new lower bound and a protocol with a gathering time that differ only by a constant independent of the length of the path. Then we present a method to construct incremental protocols. An incremental protocol for the path on $n+1$ vertices is obtained from a protocol for $n$ vertices by adding new rounds and new calls to some rounds but without changing the calls of the original rounds. We show that some of these incremental protocols are optimal for many values of $dt$ and $di$ (in particular when $dt$ is prime). We conjecture that this incremental construction always gives optimal protocols. Finally, we derive an approximation algorithm when the sink is placed in an arbitrary vertex in the path.

Computing Without Communicating: Ring Exploration by Asynchronous Oblivious Robots

Participant : David Ilcinkas.

In [24] , we consider the problem of exploring an anonymous unoriented ring by a team of $k$ identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimensional plane, but (with one exception) has not been investigated before for networks. Our results imply that, although these weak capabilities of robots render the problem considerably more difficult, ring exploration by a small team of robots is still possible. We first show that, when $k$ and $n$ are not co-prime, the problem is not solvable in general, e.g., if $k$ divides $n$ there are initial placements of the robots for which gathering is impossible. We then prove that the problem is always solvable provided that $n$ and $k$ are co-prime, for $k\ge 17$, by giving an exploration algorithm that always terminates, starting from arbitrary initial configurations. Finally, we consider the minimum number $\rho \left(n\right)$ of robots that can explore a ring of size $n$. As a consequence of our positive result we show that $\rho \left(n\right)$ is $O(logn)$. We additionally prove that $\Omega (logn)$ robots are necessary for infinitely many $n$.

Worst-case optimal exploration of terrains with obstacles

Participant : David Ilcinkas.

A mobile robot represented by a point moving in the plane has to explore an unknown flat terrain with impassable obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point $p$ of the terrain explores (sees) all points $q$ of the terrain for which the segment $pq$ belongs to the terrain, and the limited vision, when we require additionally that the distance between $p$ and $q$ is at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm, called its complexity, is measured by the length of the trajectory of the robot.

For unlimited vision we show in [21] an exploration algorithm with complexity $O(P+D\sqrt{k})$, where $P$ is the total perimeter of the terrain (including perimeters of obstacles), $D$ is the diameter of the convex hull of the terrain, and $k$ is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity $O(P+A+\sqrt{Ak})$, where $A$ is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains (if one of the parameters $A$ or $k$ is known) or for c-fat terrains, where $c$ is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain $𝒯$ with obstacles is $c$-fat if $R/r\le c$, where $R$ is the radius of the smallest disc containing $𝒯$ and $r$ is the radius of the largest disc contained in $𝒯$.) We also prove a matching lower bound $\Omega (P+A+\sqrt{Ak})$ on the complexity of exploration for limited vision, even if the terrain is known to the robot.

Exploration of the $T$-Interval-Connected Dynamic Graphs: the Case of the Ring

Participants : David Ilcinkas, Ahmed Wade.

In [40] , we study the $T$-interval-connected dynamic graphs from the point of view of the time necessary and sufficient for their exploration by a mobile entity (agent). A dynamic graph (more precisely, an evolving graph) is $T$-interval-connected ($T\ge 1$) if, for every window of $T$ consecutive time steps, there exists a connected spanning subgraph that is stable (always present) during this period. This property of connection stability over time was introduced by Kuhn, Lynch and Oshman (STOC 2010). We focus on the case when the underlying graph is a ring of size $n$, and we show that the worst-case time complexity for the exploration problem is $2n-T-\Theta \left(1\right)$ time units if the agent knows the dynamics of the graph, and $n+\frac{n}{max\{1,T-1\}}(\delta -1)\pm \Theta \left(\delta \right)$ time units otherwise, where $\delta$ is the maximum time between two successive appearances of an edge.

Time vs. space trade-offs for rendezvous in trees

Participant : Adrian Kosowski.

In [23] , we consider the rendezvous problem, in which two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. We obtain a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with $k$ memory bits, we show that optimal rendezvous time is $\Theta (n+n^2/k)$ in $n$-node trees. More precisely, if $k\ge clogn$, for some constant $c$, we design agents accomplishing rendezvous in arbitrary trees of size $n$ (unknown to the agents) in time $O(n+n^2/k)$, starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time $o(n+n^2/k)$, even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a $n$-node line.