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 $v\in C$, 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 $v\notin C$, 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 $v\notin C$, 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, Y. J.; Liu, T. H. (1965), "On the Shortest Arborescence of a Directed Graph", Science Sinica 14: 1396–1400
• Edmonds, J. (1967), "Optimum Branchings", J. Res. Nat. Bur. Standards 71B: 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: 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: 109–122, doi:10.1007/bf02579168

External links

• Edmonds's algorithm ( edmonds-alg ) – An open source 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.