# Optimal coteries for rings and related networks

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## Summary

Let a distributed system be represented by a graph*G=(V, E)*, where*V* is the set of nodes and*E* is the set of communication links. A coterie is defined as a family,*C*, of subsets of*V* such that any pair of subsets in*C* has at least one node in common and no subset in*C* contains any other subset in*C*. Assuming that each node*v*_{ i }*∈V* (resp. link*e*_{ j }*∈E*) is operational with probability*p*_{ i } (resp.*r*_{ j }), the availability of a coterie is defined as the probability that the operational nodes and links of*G* connect all nodes in at least one subset in the coterie. Although it is computationally intractable to find an optimal coterie that maximizes availability for general graph*G*, we show in this paper that, if*G* is a ring, either a singleton coterie or a 3-majority coterie is optimal. Therefore, for any ring, an optimal coterie can be computed in polynomial time. This result is extended to the more general graphs, in which each biconnected component is either an edge or a ring.

### Key words

Coterie Availability Network Distributed system Mutual exclusion Quorum Ring Fault tolerance## Preview

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