Abstract
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.
Highlights
Our motivation stems from algorithms for peer-to-peer network applications, as considered in [1,2]
In these papers a simple distance called topological distance based on IP-addresses has been introduced along with a greedy topological distance algorithm to compute trees with minimum weight measured by this distance
It is important to note that—differently to, e.g., Prim’s algorithm—the topological distance algorithm computes spanning trees incrementally which is important for the applications considered
Summary
Our motivation stems from algorithms for peer-to-peer network applications, as considered in [1,2] In these papers a simple distance called topological distance based on IP-addresses has been introduced along with a greedy topological distance algorithm to compute (spanning) trees with minimum weight measured by this distance. It is important to note that—differently to, e.g., Prim’s algorithm—the topological distance algorithm computes spanning trees incrementally (see Remark 2 below) which is important for the applications considered. In this short note we relate the topological distance and the greedy algorithm to the theory of greedy algorithms in the context of matroids and greedoids, where the objective function stems from an ultrametric instead of a usual metric.
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