@inproceedings{ilprints640,
       booktitle = {ACM SIAM Symposium on Distributed Algorithms (SODA 2004)},
           title = {Optimal Routing in Chord},
          author = {Prasanna Ganesan and Gurmeet Singh Manku},
            year = {2004},
         journal = {ACM SIAM Symposium on Distributed Algorithms(SODA) 2004},
             url = {http://ilpubs.stanford.edu:8090/640/},
        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. }
}