16 research outputs found
Playing With Population Protocols
Population protocols have been introduced as a model of sensor networks
consisting of very limited mobile agents with no control over their own
movement: A collection of anonymous agents, modeled by finite automata,
interact in pairs according to some rules.
Predicates on the initial configurations that can be computed by such
protocols have been characterized under several hypotheses.
We discuss here whether and when the rules of interactions between agents can
be seen as a game from game theory. We do so by discussing several basic
protocols
Generalized solution for the Herman Protocol Conjecture
We have a cycle of nodes and there is a token on an odd number of nodes.
At each step, each token independently moves to its clockwise neighbor or stays
at its position with probability . If two tokens arrive to the
same node, then we remove both of them. The process ends when only one token
remains. The question is that for a fixed , which is the initial
configuration that maximizes the expected number of steps . The Herman
Protocol Conjecture says that the -token configuration with distances
and maximizes . We
present a proof of this conjecture not only for but also for
for some function
which method applies for different generalizations of the problem
Playing with population protocols
Abstract. Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement: A collection of anonymous agents, modeled by finite automata, interact in pairs according to some rules. Predicates on the initial configurations that can be computed by such protocols have been characterized under several hypotheses. We discuss here whether and when the rules of interactions between agents can be seen as a game from game theory. We do so by discussing several basic protocols.
A nearly optimal upper bound for the self-stabilization time in Herman's algorithm
Self-stabilization algorithms are very important in designing fault-tolerant distributed systems. In this paper we consider Herman's self-stabilization algorithm and study its expected self-stabilization time. McIver and Morgan have conjectured the optimal upper bound being 0.148N 2, where N denotes the number of processors. We present an elementary proof showing a bound of 0.167N2, a sharp improvement compared with the best known bound 0.521N2. Our proof is inspired by McIver and Morgan's approach: we find a nearly optimal closed form of the expected stabilization time for any initial configuration, and apply the Lagrange multipliers method to give an upper bound of it. © 2014 Springer-Verlag
Proving the Herman-Protocol Conjecture
Herman's self-stabilisation algorithm, introduced 25 years ago, is a well-studied synchronous randomised protocol for enabling a ring of N processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilisation is the central outstanding open problem about this protocol. It is known that there is a constant h such that any initial configuration has expected stabilisation time at most hN2. Ten years ago, McIver and Morgan established a lower bound of 4/27?0.148 for h, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of h. A series of papers over the last decade gradually reduced the upper bound on h, with the present record (achieved in 2014) standing at approximately 0.156. In this paper, we prove McIver and Morgan's conjecture and establish that h=4/27 is indeed optimal