Showing papers by "Andrzej Pelc published in 2014"

Gathering Despite Mischief

Abstract: A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds. Each agent has a different label. Up to f of the agents are Byzantine. We consider two levels of Byzantine behavior. A strongly Byzantine agent can choose an arbitrary port when it moves and it can convey arbitrary information to other agents, while a weakly Byzantine agent can do the same, except changing its label. What is the minimum number of good agents that guarantees deterministic gathering of all of them, with terminationq We solve exactly this Byzantine gathering problem in arbitrary networks for weakly Byzantine agents and give approximate solutions for strongly Byzantine agents, both when the size of the network is known and when it is unknown. It turns out that both the strength versus the weakness of Byzantine behavior and the knowledge of network size significantly impact the results. For weakly Byzantine agents, we show that any number of good agents permits solving the problem for networks of known size. If the size is unknown, then this minimum number is fp2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least fp2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most fp1, then they are not able to gather deterministically with termination in some networks. For strongly Byzantine agents, we give a lower bound of fp1, even when the graph is known: we show that f good agents cannot gather deterministically in the presence of f Byzantine agents even in a ring of known size. On the positive side, we give deterministic gathering algorithms for at least 2fp1 good agents when the size of the network is known and for at least 4fp2 good agents when it is unknown.

Leader election for anonymous asynchronous agents in arbitrary networks

Abstract: We consider the problem of leader election among mobile agents operating in an arbitrary network modeled as an undirected graph. Nodes of the network are unlabeled and all agents are identical. Hence the only way to elect a leader among agents is by exploiting asymmetries in their initial positions in the graph. Agents do not know the graph or their positions in it, hence they must gain this knowledge by navigating in the graph and share it with other agents to accomplish leader election. This can be done using meetings of agents, which is difficult because of their asynchronous nature: an adversary has total control over the speed of agents. When can a leader be elected in this adversarial scenario and how to do it? We give a complete answer to this question by characterizing all initial configurations for which leader election is possible and by constructing an algorithm that accomplishes leader election for all configurations for which this can be done.

Deterministic Network Exploration by Anonymous Silent Agents with Local Traffic Reports

Abstract: A team consisting of an unknown number of mobile agents starting from different nodes of an unknown network, possibly at different times, have to explore the network: Every node must be visited by at least one agent, and all agents must eventually stop. Agents are anonymous (identical), execute the same deterministic algorithm, and move in synchronous rounds along links of the network. They are silent: They cannot send any messages to other agents or mark visited nodes in any way. In the absence of any additional information, exploration with termination of an arbitrary network in this model, devoid of any means of communication between agents, is impossible. Our aim is to solve the exploration problem by giving to agents very restricted local traffic reports. Specifically, an agent that is at a node v in a given round is provided with three bits of information answering the following questions: Am I alone at vq Did any agent enter v in this roundq Did any agent exit v in this roundq We show that this small amount of information permits us to solve the exploration problem in arbitrary networks. More precisely, we give a deterministic terminating exploration algorithm working in arbitrary networks for all initial configurations that are not perfectly symmetric; that is, in which there are agents with different views of the network. The algorithm works in polynomial time in the (unknown) size of the network. A deterministic terminating exploration algorithm working for all initial configurations in arbitrary networks does not exist.

Why Do We Believe Theorems

Abstract: We investigate the reasons of having confidence in mathematical theorems. The formalist point of view maintains that formal derivations underlying proofs, although usually not carried out in practice, contribute to this confidence. Opposing this opinion, the main claim of the present paper is that such a gain of confidence obtained from any link between proofs and formal derivations is, even in principle, impossible in the present state of knowledge. Our argument is based on considerations concerning length of formal derivations. We also discuss psychological and social factors that contribute to building belief in theorems.

Time versus space trade-offs for rendezvous in trees

Abstract: 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. The main result of this paper is 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 $$k \ge c\log n$$ , 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.

Time versus cost tradeoffs for deterministic rendezvous in networks

Abstract: Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds using a deterministic algorithm. In each round, an agent decides to either remain idle or to move to one of the adjacent nodes. Each agent has a distinct integer label from the set {1,...,L}, which it can use in the execution of the algorithm, but it does not know the label of the other agent.Two main efficiency measures of a rendezvous algorithm's performance are its time (the number of rounds until the meeting) and its cost (the combined number of edge traversals by both agents). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Indeed, this is a lower bound on both the time and the cost of rendezvous. Hence we express the time and cost of rendezvous as functions of an upper bound E on the time of exploration (known to both agents) and the size L of the label space. We present two natural rendezvous algorithms. Algorithm Cheap has cost O(E) (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly E) and time O(EL). Algorithm Fast has both time and cost O(E log L). Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any rendezvous algorithm of cost asymptotically E (i.e., of cost E+o(E)) must have time Ω(EL). On the other hand, we show that any rendezvous algorithm with time complexity O(E log L) must have cost Ω (E log L). Moreover, while our algorithms work for arbitrary connected graphs and arbitrary starting times of the agents, these lower bounds hold even in a scenario that is very favorable for potential rendezvous algorithms, i.e., for oriented rings of known size and with simultaneous start.

Fault-Tolerant Rendezvous in Networks

Abstract: Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adversarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals.

Election vs. Selection: Two Ways of Finding the Largest Node in a Graph.

Abstract: Finding the node with the largest label in a network, modeled as an undirected connected graph, is one of the fundamental problems in distributed computing. This is the way in which $\textit{leader election}$ is usually solved. We consider two distinct tasks in which the largest-labeled node is found deterministically. In $\textit{selection}$, this node must output 1 and all other nodes must output 0. In $\textit{election}$, the other nodes must additionally learn the largest label. Our aim is to compare the difficulty of these two tasks executed under stringent running time constraints. The measure of difficulty is the amount of information that nodes of the network must initially possess in order to solve the given task in an imposed amount of time. Following the standard framework of $\textit{algorithms with advice}$, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire graph. The length of this string is called the $\textit{size of advice}$. Consider the class of $n$-node graphs with any diameter $diam \leq D$. If time is larger than $diam$, then both tasks can be solved without advice. For the task of $\textit{election}$, we show that if time is smaller than $diam$, then the optimal size of advice is $\Theta(\log n)$, and if time is exactly $diam$, then the optimal size of advice is $\Theta(\log D)$. For the task of $\textit{selection}$, the situation changes dramatically, even within the class of rings. Indeed, for the class of rings, we show that, if time is $O(diam^{\epsilon})$, for any $\epsilon <1$, then the optimal size of advice is $\Theta(\log D)$, and, if time is $\Theta(diam)$ (and at most $diam$) then this optimal size is $\Theta(\log \log D)$.

Efficient grid exploration with a stationary token

Abstract: A mobile agent starting at an arbitrary node of an m × k grid, for 1 < m ≤ k, has to explore the grid by visiting all its nodes and traversing all edges. The cost of an exploration algorithm is the number of edge traversals by the agent. Nodes of the grid are unlabeled and ports at each node v have distinct numbers in {0,…, d − 1}, where d = 2, 3, 4 is the degree of v. Port numbering is local, i.e., there is no relation between port numbers at different nodes. When visiting a node the agent sees its degree. It also sees the port number by which it enters a node and can choose the port number by which it leaves a visited node. We are interested in deterministic exploration algorithms working at low cost. We consider the scenario in which the agent is equipped with a stationary token situated at its starting node. The agent sees the token whenever it visits this node. We give an exploration algorithm working at cost O(k2) for 2 × k grids, and at cost O(m2k), for m × k grids, when 2 < m ≤ k.

Tradeoffs between Cost and Information for Rendezvous and Treasure Hunt

Abstract: Rendezvous and treasure hunt are two basic tasks performed by mobile agents in networks. In rendezvous, two agents, initially located at distinct nodes of the network, traverse edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target (treasure) situated at an unknown node. The network is modeled as an undirected connected graph whose nodes have distinct identities. The cost of a rendezvous algorithm is the worst-case total number of edge traversals performed by both agents until meeting. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until the treasure is found. If the agents have no information about the network, the cost of both rendezvous and treasure hunt can be as large as Θ(e) for networks with e edges.

Fast Rendezvous with Advice

Abstract: Two mobile agents, starting from different nodes of an $n$-node network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds using a deterministic algorithm. In each round, an agent decides to either remain idle or to move to one of the adjacent nodes. Each agent has a distinct integer label from the set $\{1,...,L\}$, which it can use in the execution of the algorithm, but it does not know the label of the other agent. If $D$ is the distance between the initial positions of the agents, then $\Omega(D)$ is an obvious lower bound on the time of rendezvous. However, if each agent has no initial knowledge other than its label, time $O(D)$ is usually impossible to achieve. We study the minimum amount of information that has to be available a priori to the agents to achieve rendezvous in optimal time $\Theta(D)$. This information is provided to the agents at the start by an oracle knowing the entire instance of the problem, i.e., the network, the starting positions of the agents, their wake-up rounds, and both of their labels. The oracle helps the agents by providing them with the same binary string called advice, which can be used by the agents during their navigation. The length of this string is called the size of advice. Our goal is to find the smallest size of advice which enables the agents to meet in time $\Theta(D)$. We show that this optimal size of advice is $\Theta(D\log(n/D)+\log\log L)$. The upper bound is proved by constructing an advice string of this size, and providing a natural rendezvous algorithm using this advice that works in time $\Theta(D)$ for all networks. The matching lower bound, which is the main contribution of this paper, is proved by exhibiting classes of networks for which it is impossible to achieve rendezvous in time $\Theta(D)$ with smaller advice.

Fast Rendezvous with Advice

Abstract: Two mobile agents, starting from different nodes of an \(n\)-node network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds using a deterministic algorithm. In each round, an agent decides to either remain idle or to move to one of the adjacent nodes. Each agent has a distinct integer label from the set \(\{1,\ldots ,L\}\), which it can use in the execution of the algorithm, but it does not know the label of the other agent.

Fault-Tolerant Rendezvous in Networks

Abstract: Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adver- sarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not feasible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost O(nl), where l is the smaller label, working in arbitrary trees, and we show that \Omega(l) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L.