Geographic Quorum Systems Approximations

Quorum systems are a mechanism for obtaining fault-tolerance and efficient distributed systems. We consider geographic quorum systems; a geographic quorum system is a partitioning of a set X of points in the plane (representing servers) into quorums (i.e. clusters) of size k. The distance between a point p and a cluster C is the Euclidean distance between p and the farthest point in C. We present a near linear time constant-factor approximation algorithm for partitioning X into clusters, such that, the maximal distance between a point in the underlying region and its closest cluster is minimized. Next, we describe a data-structure for answering (approximately) nearest-neighbor queries on such a clustering. Finally, we describe constant-factor approximation algorithms that associate regions of equal area to the servers in a given set of servers. Two cost measures are considered: the maximum distance of a point to its corresponding server, and the sum of average distances to the servers.