Geometric generalizations of the power of two choices

A well-known paradigm for load balancing in parallel and distributed systems is the "power of two choices," whereby an item is stored at the less loaded of two (or more) random alternative servers. We investigate the power of two choices in natural settings where items and servers reside in a geometric space and each item is associated with the server that is its nearest neighbor. This is the setting for example in the Chord distributed hash table, where the geometric space is determined by clockwise distance on a one-dimensional ring. For example, our analysis shows that when $n$ items are placed at n servers with d choices per item, the maximum load at any server is log log n/ log d + O(1) with high probability, only an additive constant more than when servers are chosen uniformly at random. Our proofs are quite general, showing that the power of two choices works under a variety of distributions, with most geometric constructions having at most an additive O(1) penalty. We also show that these techniques still work under highly unbalanced distributions, and give sharp bounds on the necessary number of choices. Finally, we provide simulation results demonstrating the load balance that results as the system size scales into the millions.