Avery Miller

Time versus cost tradeoffs for deterministic rendezvous in networks

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 algorithms 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.

Fast Rendezvous with Advice

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.

Election vs. Selection: How Much Advice is Needed to Find the Largest Node in a Graph?

Finding the node with the largest label in a labeled network, modeled as an undirected connected graph, is one of the fundamental problems in distributed computing. This is the way in which leader election is usually solved. We consider two distinct tasks in which the largest-labeled node is found deterministically. In selection, this node has to output 1 and all other nodes have to output 0. In election, the other nodes must additionally learn the largest label (everybody has to know who is the elected leader). Our aim is to compare the difficulty of these two seemingly similar 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 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 size of advice. The paradigm of algorithms with advice has a far-reaching importance in the realm of network algorithms. Lower bounds on the size of advice give us impossibility results based strictly on the amount of initial knowledge outlined in a models description. This more general approach should be contrasted with traditional results that focus on specific kinds of information available to nodes, such as the size, diameter, or maximum node degree. Consider the class of n-node graphs with any diameter diam ≤ D, for some integer D. If time is larger than diam, then both tasks can be solved without advice. For the task of election, we show that if time is smaller than

Time vs. Information Tradeoffs for Leader Election in Anonymous Trees

diam

Global Synchronization and Consensus Using Beeps in a Fault-Prone MAC

, then the optimal size of advice is Θ(log n), and if time is exactly diam, then the optimal size of advice is Θ(log D). For the task of selection, the situation changes dramatically, even within the class of rings. Indeed, for the class of rings, we show that, if time is O(diamε), for any ε < 1, then the optimal size of advice is Θ(log D), and, if time is Θ(diam) (and at most diam) then this optimal size is Θ(log log D). Thus there is an exponential increase of difficulty (measured by the size of advice) between selection in time O(diamε), for any ε < 1, and selection in time Θ(diam). As for the comparison between election and selection, our results show that, perhaps surprisingly, while for small time, the difficulty of these two tasks on rings is similar, for time Θ(diam) the difficulty of election (measured by the size of advice) is exponentially larger than that of selection.

Fast rendezvous with advice

Leader election is one of the fundamental problems in distributed computing. It calls for all nodes of a network to agree on a single node, called the leader. If the nodes of the network have distinct labels, then agreeing on a single node means that all nodes have to output the label of the elected leader. If the nodes of the network are anonymous, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this article, we study deterministic leader election in anonymous trees. Our aim is to establish tradeoffs between the allocated time τ and the amount of information that has to be given a priori to the nodes to enable leader election in time τ in all trees for which leader election in this time is at all possible. Following the framework of algorithms with advice, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire tree. The length of this string is called the size of advice. For a given time τ allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time τ. For most values of τ, our upper and lower bounds are either tight up to multiplicative constants, or they differ only by a logarithmic factor. Let T be an n-node tree of diameter diam ⩽ D. While leader election in time diam can be performed without any advice, for time diam − 1 we give tight upper and lower bounds of Θ(log D). For time diam − 2 we give tight upper and lower bounds of Θ(log D) for even values of diam, and tight upper and lower bounds of Θ(log n) for odd values of diam. Moving to shorter time, in the interval [β · diam, diam − 3] for constant β > 1/2, we prove an upper bound of O(nlog n/D) and a lower bound of Ω(n/D), the latter being valid whenever diam is odd or when the time is at most diam − 4. Hence, with the exception of the special case when diam is even and time is exactly diam − 3, our bounds leave only a logarithmic gap in this time interval. Finally, for time α · diam for any constant α < 1/2 (except for the case of very small diameters), we again give tight upper and lower bounds, this time Θ(n).

Tradeoffs between Cost and Information for Rendezvous and Treasure Hunt

Global synchronization is an important prerequisite to many distributed tasks. Communication between processors proceeds in synchronous rounds. Processors are woken up in possibly different rounds. The clock of each processor starts in its wakeup round showing local round 0, and ticks once per round, incrementing the value of the local clock by one. The global round 0, unknown to processors, is the wakeup round of the earliest processor. Global synchronization (or establishing a global clock) means that each processor chooses a local clock round such that their chosen rounds all correspond to the same global round t.

On the Complexity of Fixed-Schedule Neighbourhood Learning in Wireless Ad Hoc Radio Networks

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.The main efficiency measure of a rendezvous algorithms performance is its time, i.e., the number of rounds from the start of the earlier agent until the meeting. If D is the distance between the initial positions of the agents, then ? ( 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 ? ( D ) . Following the standard paradigm of algorithms with advice, 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 ? ( D ) . We show that this optimal size of advice is ? ( 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 ? ( 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 ? ( D ) with smaller advice.

Gossiping in Jail

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.

Deterministic distributed construction of T-dominating sets in time T

Consider a synchronous static radio network of \(n\) nodes represented by an undirected graph with maximum degree \(\varDelta \). Suppose that each node has a unique ID from \(\{1,\ldots ,N\}\), where \(N \gg n\). In the complete neighbourhood learning task, each node \(p\) must produce a set \(L_p\) of IDs such that ID \(i \in L_p\) if and only if \(p\) has a neighbour with ID \(i\). We study the complexity of this task when it is assumed that each node fixes its entire transmission schedule at the start of the algorithm. We prove a \(\varOmega (\frac{\varDelta ^2}{\log \varDelta }\log {N})\)-slot lower bound on schedule length that holds in very general models, e.g., when nodes possess collision detectors, messages can be of arbitrary size, and nodes know the schedules being followed by all other nodes. We also prove a similar result for the SINR model of radio networks. To prove these results, we introduce a new generalization of cover-free families of sets, which may be of independent interest. We also show a separation between the class of fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out some transmissions from their schedule.