Description edit

The algorithm takes as input a directed graph ${\displaystyle D=\langle V,E\rangle }$  where ${\displaystyle V}$  is the set of nodes and ${\displaystyle E}$  is the set of directed edges, a distinguished vertex ${\displaystyle r\in V}$  called the root, and a real-valued weight ${\displaystyle w(e)}$  for each edge ${\displaystyle e\in E}$ . It returns a spanning arborescence ${\displaystyle A}$  rooted at ${\displaystyle r}$  of minimum weight, where the weight of an arborescence is defined to be the sum of its edge weights, ${\displaystyle w(A)=\sum _{e\in A}{w(e)}}$ .

The algorithm has a recursive description. Let ${\displaystyle f(D,r,w)}$  denote the function which returns a spanning arborescence rooted at ${\displaystyle r}$  of minimum weight. We first remove any edge from ${\displaystyle E}$  whose destination is ${\displaystyle r}$ . We may also replace any set of parallel edges (edges between the same pair of vertices in the same direction) by a single edge with weight equal to the minimum of the weights of these parallel edges.

Now, for each node ${\displaystyle v}$  other than the root, find the edge incoming to ${\displaystyle v}$  of lowest weight (with ties broken arbitrarily). Denote the source of this edge by ${\displaystyle \pi (v)}$ . If the set of edges ${\displaystyle P=\{(\pi (v),v)\mid v\in V\setminus \{r\}\}}$  does not contain any cycles, then ${\displaystyle f(D,r,w)=P}$ .

Otherwise, ${\displaystyle P}$  contains at least one cycle. Arbitrarily choose one of these cycles and call it ${\displaystyle C}$ . We now define a new weighted directed graph ${\displaystyle D^{\prime }=\langle V^{\prime },E^{\prime }\rangle }$  in which the cycle ${\displaystyle C}$  is "contracted" into one node as follows:

The nodes of ${\displaystyle V^{\prime }}$  are the nodes of ${\displaystyle V}$  not in ${\displaystyle C}$  plus a new node denoted ${\displaystyle v_{C}}$ .

• If ${\displaystyle (u,v)}$  is an edge in ${\displaystyle E}$  with ${\displaystyle u\notin C}$  and ${\displaystyle v\in C}$  (an edge coming into the cycle), then include in ${\displaystyle E^{\prime }}$  a new edge ${\displaystyle e=(u,v_{C})}$ , and define ${\displaystyle w^{\prime }(e)=w(u,v)-w(\pi (v),v)}$ .
• If ${\displaystyle (u,v)}$  is an edge in ${\displaystyle E}$  with ${\displaystyle u\in C}$  and ${\displaystyle v\notin C}$  (an edge going away from the cycle), then include in ${\displaystyle E^{\prime }}$  a new edge ${\displaystyle e=(v_{C},v)}$ , and define ${\displaystyle w^{\prime }(e)=w(u,v)}$ .
• If ${\displaystyle (u,v)}$  is an edge in ${\displaystyle E}$  with ${\displaystyle u\notin C}$  and ${\displaystyle v\notin C}$  (an edge unrelated to the cycle), then include in ${\displaystyle E^{\prime }}$  a new edge ${\displaystyle e=(u,v)}$ , and define ${\displaystyle w^{\prime }(e)=w(u,v)}$ .

For each edge in ${\displaystyle E^{\prime }}$ , we remember which edge in ${\displaystyle E}$  it corresponds to.

Now find a minimum spanning arborescence ${\displaystyle A^{\prime }}$  of ${\displaystyle D^{\prime }}$  using a call to ${\displaystyle f(D^{\prime },r,w^{\prime })}$ . Since ${\displaystyle A^{\prime }}$  is a spanning arborescence, each vertex has exactly one incoming edge. Let ${\displaystyle (u,v_{C})}$  be the unique incoming edge to ${\displaystyle v_{C}}$  in ${\displaystyle A^{\prime }}$ . This edge corresponds to an edge ${\displaystyle (u,v)\in E}$  with ${\displaystyle v\in C}$ . Remove the edge ${\displaystyle (\pi (v),v)}$  from ${\displaystyle C}$ , breaking the cycle. Mark each remaining edge in ${\displaystyle C}$ . For each edge in ${\displaystyle A^{\prime }}$ , mark its corresponding edge in ${\displaystyle E}$ . Now we define ${\displaystyle f(D,r,w)}$  to be the set of marked edges, which form a minimum spanning arborescence.

Observe that ${\displaystyle f(D,r,w)}$  is defined in terms of ${\displaystyle f(D^{\prime },r,w^{\prime })}$ , with ${\displaystyle D^{\prime }}$  having strictly fewer vertices than ${\displaystyle D}$ . Finding ${\displaystyle f(D,r,w)}$  for a single-vertex graph is trivial (it is just ${\displaystyle D}$  itself), so the recursive algorithm is guaranteed to terminate.

The running time of this algorithm is ${\displaystyle O(EV)}$ . A faster implementation of the algorithm due to Robert Tarjan runs in time ${\displaystyle O(E\log V)}$  for sparse graphs and ${\displaystyle O(V^{2})}$  for dense graphs. This is as fast as Prim's algorithm for an undirected minimum spanning tree. In 1986, Gabow, Galil, Spencer, and Tarjan produced a faster implementation, with running time ${\displaystyle O(E+V\log V)}$ .

• Chu, Yeong-Jin; Liu, Tseng-Hong (1965), "On the Shortest Arborescence of a Directed Graph" (PDF), Scientia Sinica: Chung-kuo k`o hsueh, XIV (10): 1396–1400
• Edmonds, J. (1967), "Optimum Branchings", Journal of Research of the National Bureau of Standards Section B, 71B (4): 233–240, doi:10.6028/jres.071b.032
• Tarjan, R. E. (1977), "Finding Optimum Branchings", Networks, 7: 25–35, doi:10.1002/net.3230070103
• Camerini, P.M.; Fratta, L.; Maffioli, F. (1979), "A note on finding optimum branchings", Networks, 9 (4): 309–312, doi:10.1002/net.3230090403
• Gibbons, Alan (1985), Algorithmic Graph Theory, Cambridge University press, ISBN 0-521-28881-9
• Gabow, H. N.; Galil, Z.; Spencer, T.; Tarjan, R. E. (1986), "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs", Combinatorica, 6 (2): 109–122, doi:10.1007/bf02579168, S2CID 35618095

• Edmonds's algorithm ( edmonds-alg ) – An implementation of Edmonds's algorithm written in C++ and licensed under the MIT License. This source is using Tarjan's implementation for the dense graph.
• NetworkX, a python library distributed under BSD, has an implementation of Edmonds' Algorithm.