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Treasure hunt is finding a hidden inert target by a mobile agent. We consider deterministic algorithms for treasure hunt in trees. Our goal is to establish the impact of different kinds of initial knowledge given to the agent on the cost of treasure hunt, defined as the total number of edge traversals until the agent reaches the treasure. The agent can be initially given either a complete map of the tree rooted at its starting node, with all port numbers marked, or a blind map of the tree rooted at its starting node but without port numbers. It may also be given, or not, the distance from the root to the treasure. This yields four different knowledge types that are partially ordered by their precision. The penalty of a less precise knowledge type 풯2 over a more precise knowledge type 풯1 measures intuitively the worst‐case ratio of the cost of an algorithm supplied with knowledge of type 풯2 over the cost of an algorithm supplied with knowledge of type 풯1. Our main results establish penalties for comparable knowledge types in this partial order. For knowledge types with known distance, the penalty for having a blind map over a complete map turns out to be very large. By contrast, for unknown distance, the penalty of having a blind map over having a complete map is small. When a map is provided (either complete or blind), the penalty of not knowing the distance over knowing it is medium. [ABSTRACT FROM AUTHOR]

Gathering is a key task in distributed and mobile systems, which becomes significantly harder if some agents are subject to Byzantine faults, known as being the worst ones. We propose here to study the task of Byzantine gathering in an arbitrary graph: despite the presence of Byzantine agents, the goal is to ensure that all the other (good) agents, executing the same algorithm, eventually meet at the same node and stop. Initially, each agent gets as input a different label and some global knowledge that is common to all agents. The agents move in synchronous rounds and communicate with each other only when located at the same node. There are f Byzantine agents. These agents act in an unpredictable way, e.g., they may convey arbitrary informations or forge any label. In the literature, the gathering algorithms working in such a context all have an exponential time complexity in the number n of nodes and the labels of the good agents. In this paper, we design a deterministic algorithm to solve Byzantine gathering in time polynomial in n and the logarithm ℓ of the smallest label of a good agent, provided the agents are a strong team i.e., a team where the number of good agents is at least some quadratic polynomial in f. Our algorithm requires global knowledge that can be coded in O (log log log n) bits: we prove this size is of optimal order of magnitude to obtain a polynomial time complexity in n and ℓ with strong teams. [ABSTRACT FROM AUTHOR]

A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most D > 0 from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than 2 π whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is Θ (D 2) . We show that if all angles given as hints are at most π , then the cost can be lowered to O(D), which is the optimal complexity. If all angles are at most β , where β < 2 π is a constant unknown to the agent, then the cost is at most O (D 2 - ϵ) , for some ϵ > 0 . For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than 2 π , then we show that cost complexity Θ (D 2) cannot be beaten. [ABSTRACT FROM AUTHOR]

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other's range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. Roughly speaking, the goal of the adversary is to prevent the agents from solving the task, or at least to ensure that the agents have covered as much distance as possible before seeing each other. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let Δ and l be the initial distance separating the agents and the length of (the binary representation of) the shortest label, respectively. Assuming that Δ andlare unknown to both agents, does there exist a deterministic approach algorithm always working at a cost that is polynomial in Δ andl? Actually the problem of approach in the plane reduces to the network problem of rendezvous in an infinite oriented grid, which consists in ensuring that both agents end up meeting at the same time at a node or on an edge of the grid. By designing such a rendezvous algorithm with appropriate properties, as we do in this paper, we provide a positive answer to the above question. Our result turns out to be an important step forward from a computational point of view, as the other algorithms allowing to solve the same problem either have an exponential cost in the initial separating distance and in the labels of the agents, or require each agent to know its starting position in a global system of coordinates, or only work under a much less powerful adversary. [ABSTRACT FROM AUTHOR]

This paper investigates an open problem introduced in Dieudonné et al. (ACM Trans Algorithms 11(1):1, 2014). Two or more mobile agents start from nodes of a network and have to accomplish the task of gathering which consists in getting all together at the same node at the same time. An adversary chooses the initial nodes of the agents and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label but does not know the labels of the other agents or their positions relative to its own. Agents move in synchronous rounds and can communicate with each other only when located at the same node. Up to f of the agents are Byzantine. A Byzantine agent can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. What is the minimum number $${\mathcal {M}}$$ of good agents that guarantees deterministic gathering of all of them, with termination? We provide exact answers to this open problem by considering the case when the agents initially know the size of the network and the case when they do not. In the former case, we prove $${\mathcal {M}}=f+1$$ while in the latter, we prove $${\mathcal {M}}=f+2$$ . More precisely, for networks of known size, we design a deterministic algorithm gathering all good agents in any network provided that the number of good agents is at least $$f+1$$ . For networks of unknown size, we also design a deterministic algorithm ensuring the gathering of all good agents in any network but provided that the number of good agents is at least $$f+2$$ . Both of our algorithms are optimal in terms of required number of good agents, as each of them perfectly matches the respective lower bound on $${\mathcal {M}}$$ shown in Dieudonné et al. (2014), which is of $$f+1$$ when the size of the network is known and of $$f+2$$ when it is unknown. Perhaps surprisingly, our results highlight an interesting feature when put in perspective with known results concerning a relaxed variant of this problem in which the Byzantine agents cannot change their initial labels. Indeed under this variant $${\mathcal {M}}=1$$ for networks of known size and $${\mathcal {M}}=f+2$$ for networks of unknown size. Following this perspective, it turns out that when the size of the network is known, the ability for the Byzantine agents to change their labels significantly impacts the value of $${\mathcal {M}}$$ . However, the relevance for $${\mathcal {M}}$$ of such an ability completely disappears in the most general case where the size of the network is unknown, as $${\mathcal {M}}=f+2$$ regardless of whether Byzantine agents can change their labels or not. [ABSTRACT FROM AUTHOR]