Optimizing the costs of hierarchical quorum consensus

We study the problem of how to minimize the cost of maintaining consistency among at least N copies of an object in an enviroment where the mix of read and write operations can vary. Quorum consensus requires that read and write operations must assemble appropriate quorums before an operation can succeed. The cost of an operation is proportional to the size of a quorum, and the objective is obviously to minimize the cost while still maintaining consistency. It is known that the quorum size can be reduced by organizing a number of copies into logical structures such as grids and hierarchies. In this paper, we show (a) how methods based on grids and hierarchies can be treated in a common framework, and (b) how these hierarchies can be optimized so as to minimize the cost of consensus. Of course, the optimal solution depends upon the mix of read and write operations that is present. Consequently, given N copies of an object and a ratio of write operations F, our algorithms determine the optimal values for the number of levels in the hierarchy and the read/write quorum sizes at each level. The algorithms, which run in O(N^1.63) and O(N²) time, were tested, and the results are reported and discussed.