On Finding Optimal (Dynamic) Arborescences

Abstract

Let G=(V,E) be a directed and weighted graph with a vertex set V of size n and an edge set E of size m such that each edge (u,v)∈E has a real-valued weight w(u,c). An arborescence in G is a subgraph T=(V,E′) such that, for a vertex u∈V, which is the root, there is a unique path in T from u to any other vertex v∈V. The weight of T is the sum of the weights of its edges. In this paper, given G, we are interested in finding an arborescence in G with a minimum weight, i.e., an optimal arborescence. Furthermore, when G is subject to changes, namely, edge insertions and deletions, we are interested in efficiently maintaining a dynamic arborescence in G. This is a well-known problem with applications in several domains such as network design optimization and phylogenetic inference. In this paper, we revisit the algorithmic ideas proposed by several authors for this problem. We provide detailed pseudocode, as well as implementation details, and we present experimental results regarding large scale-free networks and phylogenetic inference. Our implementation is publicly available.

Keywords

• optimal arborescences
• Edmonds’ algorithm
• dynamic algorithm
• algorithm engineering