Edmonds' algorithm

In graph theory, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a spanning arborescence of minimum weight (sometimes called an optimum branching). It is the directed analog of the minimum spanning tree problem. The algorithm was proposed independently first by Yoeng-Jin Chu and Tseng-Hong Liu (1965) and then by Jack Edmonds (1967).

Algorithm

Description

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

The algorithm has a recursive description. Let $f(D,r,w)$ denote the function which returns a spanning arborescence rooted at $r$ of minimum weight. We first remove any edge from $E$ whose destination is $r$ . 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 $v$ other than the root, find the edge incoming to $v$ of lowest weight (with ties broken arbitrarily). Denote the source of this edge by $\pi (v)$ . If the set of edges $P=\{(\pi (v),v)\mid v\in V\setminus \{r\}\}$ does not contain any cycles, then $f(D,r,w)=P$ .

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

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

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

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

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

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

Running time

The running time of this algorithm is $O(EV)$ . A faster implementation of the algorithm due to Robert Tarjan runs in time $O(E\log V)$ for sparse graphs and $O(V^{2})$ 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 $O(E+V\log V)$ .

References

Chu, Yeong-Jin; Liu, Tseng-Hong (1965), "On the Shortest Arborescence of a Directed Graph" (PDF), Scientia Sinica, 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

External links

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.