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We relate the definition of an ultrametric space to the topological distance algorithm—an algorithm defined in the context of peer-to-peer network applications. Although (greedy) algorithms for constructing minimum spanning trees such as Prim’s or Kruskal’s algorithm have been known for a long time, they require the complete graph to be specified and the weights of all edges to be known upfront in order to construct a minimum spanning tree. However, if the weights of the underlying graph stem from an ultrametric, the minimum spanning tree can be constructed incrementally and it is not necessary to know the full graph in advance. This is possible, because the join algorithm responsible for joining new nodes on behalf of the topological distance algorithm is independent of the order in which the nodes are added due to the property of an ultrametric. Apart from the mathematical elegance which some readers might find interesting in itself, this provides not only proofs (and clearer ones in the opinion of the author) for optimality theorems (i.e., proof of the minimum spanning tree construction) but a simple proof for the optimality of the reconstruction algorithm omitted in previous publications too. Furthermore, we define a new algorithm by extending the join algorithm to minimize the topological distance and (network) latency together and provide a correctness proof.

A Note on Ultrametric Spaces, Minimum Spanning Trees and the Topological Distance Algorithm