Black Hole Search with Finite Automata Scattered in a Synchronous Torus

We consider the problem of locating a black hole in synchronous anonymous networks using finite state agents. A black hole is a harmful node in the network that destroys any agent visiting that node without leaving any trace. The objective is to locate the black hole without destroying too many agents. This is difficult to achieve when the agents are initially scattered in the network and are unaware of the location of each other. Previous studies for black hole search used more powerful models where the agents had non-constant memory, were labelled with distinct identifiers and could either write messages on the nodes of the network or mark the edges of the network. In contrast, we solve the problem using a small team of finite-state agents each carrying a constant number of identical tokens that could be placed on the nodes of the network. Thus, all resources used in our algorithms are independent of the network size. We restrict our attention to oriented torus networks and first show that no finite team of finite state agents can solve the problem in such networks, when the tokens are not movable. In case the agents are equipped with movable tokens, we determine lower bounds on the number of agents and tokens required for solving the problem in torus networks of arbitrary size. Further, we present a deterministic solution to the black hole search problem for oriented torus networks, using the minimum number of agents and tokens

Broadcasting with Mobile Agents in Dynamic Networks

We study the standard communication problem of broadcast for mobile agents moving in a network. The agents move autonomously in the network and can communicate with other agents only when they meet at a node. In this model, broadcast is a communication primitive for information transfer from one agent, the source, to all other agents. Previous studies of this problem were restricted to static networks while, in this paper, we consider the problem in dynamic networks modelled as an evolving graph. The dynamicity of the graph is unknown to the agents; in each round an adversary selects which edges of the graph are available, and an agent can choose to traverse one of the available edges adjacent to its current location. The only restriction on the adversary is that the subgraph of available edges in each round must span all nodes; in other words the evolving graph is constantly connected. The agents have global visibility allowing them to see the location of other agents in the graph and move accordingly. Depending on the topology of the underlying graph, we determine how many agents are necessary and sufficient to solve the broadcast problem in dynamic networks. While two agents plus the source are sufficient for ring networks, much larger teams of agents are necessary for denser graphs such as grid graphs and hypercubes, and finally for complete graphs of n nodes at least n-2 agents plus the source are necessary and sufficient. We show lower bounds on the number of agents and provide some algorithms for solving broadcast using the minimum number of agents, for various topologies

Exclusive graph searching vs. pathwidth

International audienceIn Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network. The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. Blin et al. defined the Exclusive Graph Searching where searchers cannot " jump " and no node can be occupied by more than one searcher. In this paper, we study the complexity of this new variant. We show that the problem is NP-hard in planar graphs with maximum degree 3 and it can be solved in linear-time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. Hence, the computational complexities of monotone Exclusive Graph Searching and Pathwidth cannot be compared. This is the first variant of Graph Searching for which such a difference is proved

Tight Bounds for Black Hole Search with Scattered Agents in Synchronous Rings

We study the problem of locating a particularly dangerous node, the so-called black hole in a synchronous anonymous ring network with mobile agents. A black hole is a harmful stationary process residing in a node of the network and destroying destroys all mobile agents visiting that node without leaving any trace. We consider the more challenging scenario when the agents are identical and initially scattered within the network. Moreover, we solve the problem with agents that have constant-sized memory and carry a constant number of identical tokens, which can be placed at nodes of the network. In contrast, the only known solutions for the case of scattered agents searching for a black hole, use stronger models where the agents have non-constant memory, can write messages in whiteboards located at nodes or are allowed to mark both the edges and nodes of the network with tokens. This paper solves the problem for ring networks containing a single black hole. We are interested in the minimum resources (number of agents and tokens) necessary for locating all links incident to the black hole. We present deterministic algorithms for ring topologies and provide matching lower and upper bounds for the number of agents and the number of tokens required for deterministic solutions to the black hole search problem, in oriented or unoriented rings, using movable or unmovable tokens

How to Place Efficiently Guards and Paintings in an Art Gallery

In the art gallery problem the goal is to place guards (as few as possible) in a polygon so that a maximal area of the polygon is covered. We address here a closely related problem: how to place paintings and guards in an art gallery so that the total value of guarded paintings is a maximum. More formally, a simple polygon is given along with a set of paintings. Each painting, has a length and a value. We study how to place at the same time: i) a given number of guards on the boundary of the polygon and ii) paintings on the boundary of the polygon so that the total value of guarded paintings is maximum. We investigate this problem for a number of cases depending on: i) where the guards can be placed (vertices, edges), ii) whether the polygon has holes or not and iii) whether the goal is to oversee the placed paintings (every point of a painting is seen by at least one guard), or to watch the placed paintings (at least one point of a painting is seen by at least one guard). We prove that the problem is NP-hard in all the above cases and we present polynomial time approximation algorithms for all cases, achieving constant ratios

Gathering asynchronous oblivious mobile robots in a ring

International audienc

Gathering asynchronous oblivious mobile robots in a ring

Gathering asynchronous oblivious mobile robots in a ring