De Bruijn's Graphs
Koorde is based on Chord but also on De Bruijn graph (De Bruijn sequence). In a d-dimensional de Bruijn graph, there are 2d nodes, each of which has a unique d-bit ID. The node with ID i is connected to nodes 2i modulo 2d and 2i+1 modulo 2d. Thanks to this property, the routing algorithm can route to any destination in d hops by successively "shifting in" the bits of the destination ID but only if the dimensions of the distance between modulo 1d and 3d are equal.
Routing a message from node m to node k is accomplished by taking the number m and shifting in the bits of k one at a time until the number has been replaced by k. Each shift corresponds to a routing hop to the next intermediate address; the hop is valid because each node's neighbors are the two possible outcomes of shifting a 0 or 1 onto its own address. Because of the structure of de Bruijn graphs, when the last bit of k has been shifted, the query will be at node k. Node k responds whether key k exists.
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