Optimal Routing in Chord

Abstract

We propose optimal routing algorithms for Chord, a popular topology for routing in peer-to-peer networks. Chord is an undirected graph on $2^b$ nodes arranged in a circle, with edges connecting pairs of nodes that are $2^k$ positions apart for any $k \geq 0$. The standard Chord routing algorithm uses edges in only one direction. Our algorithms exploit the bidirectionality of edges for optimality. At the heart of the new protocols lie algorithms for writing a positive integer $d$ as the difference of two non-negative integers $d'$ and $d''$ such that the total number of 1-bits in the binary representation of $d'$ and $d''$ is minimized. Given that Chord is a variant of the hypercube, the optimal routes possess a surprising combinatorial structure.