How to Be an Efficient Snoop, or the Probe Complexity of Quorum Systems

A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication, and dissemination of information. When the elements may fail, a user of a distributed protocol needs to quickly find a quorum all of whose elements are alive or evidence that no such quorum exists. This is done by probing the system elements, one at a time, to determine if they are alive or dead. This paper studies the probe complexity $\cal{PC(S)}$ of a quorum system $\cal{S}$, defined as the worst case number of probes required to find a live quorum or to show its nonexistence in $\cal{S}$, using the best probing strategy. We show that for large classes of quorum systems, all n elements must be probed in the worst case. Such systems are called evasive. However, not all quorum systems are evasive; we demonstrate a system where O(log n) probes always suffice. Then we prove two lower bounds on the probe complexity in terms of the minimal quorum cardinality $c \cal{(S)}$ and the number of minimal quorums $m\cal{(S)}$. Finally, we show a universal probe strategy which never makes more than $c {\cal(S)}^2 - c{\cal(S)}+ 1$ probes; thus any system with $c{\cal(S)}\le\sqrt n $ is nonevasive.