18 research outputs found
Computing Boolean Functions on Anonymous Networks
AbstractWe study the bit-complexity of computing Boolean functions on anonymous networks. Let N be the number of nodes, δ the diameter, and d the maximal node degree of the network. For arbitrary, anonymous networks we give a general algorithm of polynomial bit complexity O(N3 · δ · d2 · log N) for computing any Boolean function which is computable on the network. This improves upon the previous best known algorithm, which was of exponential bit complexity O(dN2). For symmetric functions on arbitrary networks we give an algorithm with bit complexity O(N3· δ · d2 · log2N). This same algorithm is shown to have even lower bit complexity for a number of specific networks, for example tori, hypercubes, and random regular graphs. We also consider the class of distance regular unlabeled networks and show that on such networks symmetric functions can be computed efficiently in O(N · δ · d · log N) bits
Distributed anonymous function computation in information fusion and multiagent systems
We propose a model for deterministic distributed function computation by a
network of identical and anonymous nodes, with bounded computation and storage
capabilities that do not scale with the network size. Our goal is to
characterize the class of functions that can be computed within this model. In
our main result, we exhibit a class of non-computable functions, and prove that
every function outside this class can at least be approximated. The problem of
computing averages in a distributed manner plays a central role in our
development
Topology recognition with advice
In topology recognition, each node of an anonymous network has to
deterministically produce an isomorphic copy of the underlying graph, with all
ports correctly marked. This task is usually unfeasible without any a priori
information. Such information can be provided to nodes as advice. An oracle
knowing the network can give a (possibly different) string of bits to each
node, and all nodes must reconstruct the network using this advice, after a
given number of rounds of communication. During each round each node can
exchange arbitrary messages with all its neighbors and perform arbitrary local
computations. The time of completing topology recognition is the number of
rounds it takes, and the size of advice is the maximum length of a string given
to nodes.
We investigate tradeoffs between the time in which topology recognition is
accomplished and the minimum size of advice that has to be given to nodes. We
provide upper and lower bounds on the minimum size of advice that is sufficient
to perform topology recognition in a given time, in the class of all graphs of
size and diameter , for any constant . In most
cases, our bounds are asymptotically tight
Leader Election for Anonymous Asynchronous Agents in Arbitrary Networks
We study 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
Symmetries and sense of direction in labeled graphs
AbstractWe consider edge-labeled graphs which model distributed systems, focus on properties of edge-labelings, and study their impact on graph classes. In particular, we investigate the relation between symmetries, topologies and sense of direction. We study symmetries based on the notion of view and of surrounding, and characterize the corresponding graph classes. Among other results, we show that the completely surrounding symmetric labeled graphs coincides with the class of Cayley graphs with Cayley labelings. We then focus on the relationship between symmetries and sense of direction in regular graphs. We characterize the class of regular labeled graphs with minimal symmetric sense of direction, as well as the class of those with group-based sense of direction