Lecture notes on “ Analysis of Algorithms ” : Directed Minimum Spanning Trees ( More complete but still unfinished ) Lecturer : Uri Zwick

We describe an efficient implementation of Edmonds’ algorithm for finding minimum directed spanning trees in directed graphs. 1 Minimum Directed Spanning Trees Let G = (V,E,w) be a weighted directed graph, where w : E → R is a cost (or weight) function defined on its edges. Let r ∈ V . A directed spanning tree (DST) of G rooted at r, is a subgraph T of G such that the undirected version of T is a tree and T contains a directed path from r to any other vertex in V . The cost w(T ) of a directed spanning tree T is the sum of the costs of its edges, i.e., w(T ) = ∑ e∈T w(e). A minimum directed spanning tree (MDST) rooted at r is a directed spanning tree rooted at r of minimum cost. A directed graph contains a directed spanning tree rooted at r if and only if all vertices in G are reachable from r. This condition can be easily tested in linear time. The proof of the following lemma is trivial as is left as an exercise. Lemma 1.1 The following conditions are equivalent: (i) T is a directed spanning tree of G rooted at r. (ii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and T is acyclic, i.e., contains no directed cycles. (iii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and there are directed paths in T from r to all other vertices. ∗School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E–mail: [email protected]