1. Koorde: A simple degree optimal DHT
M. Frans Kaashoek and David Karger
MIT Laboratory of Computer Science

2. Introduction Koorde is a DHT based on Chord and de Bruijn graph.
O(log n) hops per lookup request with only 2 neighbors per node.
Can be generalized to O(log n/ log log n) hops per lookup request with O(log n) neighbors per node

3. Bounds
Lemma An n-node network with maximum node degree d requires at least logd(n-1) routing hops in the worst case. Why?

4. De Bruijn graph A node m has two outgoing edges to
nodes 2m mod 2b and 2m+1 mod 2b.
Call them the 0-link and the 1-link

5. Koorde
Koorde embeds a de Bruijn graph on the Chord identifier ring shown below.

6. Koorde A message from node i to node j can be routed as follows:
Shift the bits of j so that its leading r bits tally with the last r bits of i
2. Forward the query along the paths corresponding to the last (log n − r) bits of j:
Each 0 bit = a hop along the 0-link Each 1 bit = a hop along the 1-link.
Route via 0-link, 0-link
Routing takes at most log n hops (optimal)

7. Generalized version of Koorde
For k > 2, the earlier construction can easily be generalized. From each node i, there will be k routing fingers pointing to the nodes k⋅i, k⋅i + 1, k⋅i + 2,..., k⋅i + k −1 (additions mod n)
The path length will be at most logkn i.e. log n/log k between any pair of nodes
When k=log n, the path length is log n/log log n