Edmonds' algorithm

In graph theory, a branch of mathematics, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a maximum or minimum optimum branchings. This is similar to the minimum spanning tree problem which concerns undirected graphs. However, when nodes are connected by weighted edges that are directed, a minimum spanning tree algorithm cannot be used.

The optimum branching algorithm was proposed independently first by Yoeng-jin Chu and Tseng-hong Liu (1965) and then by Edmonds (1967). To find a maximum path length, the largest edge value is found and connected between the two nodes, then the next largest value, and so on. If an edge creates a loop, it is erased. A minimum path length is found by starting from the smallest value.

Order

The order of this algorithm is $O(EV)$ . There is a faster implementation of the algorithm by Robert Tarjan. The order is $O(E\log V)$ for a sparse graph and $O(V^{2})$ for a dense graph. This is as fast as Prim's algorithm for an undirected minimum spanning tree. In 1986, Gabow, Galil, Spencer, and Tarjan made a faster implementation, and its order is $O(E+V\log V)$ .

Algorithm

Description

The algorithm has a conceptual recursive description. We will denote by $f$ the function which, given a weighted directed graph $D$ with a distinguished vertex $r$ called the root, returns a spanning tree rooted at $r$ of minimal cost.

The precise description is as follows. Given a weighted directed graph $D$ with root $r$ we first 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, mark an (arbitrarily chosen) incoming edge of lowest cost. Denote the other endpoint of this edge by $\pi (v)$ . The edge is now denoted as $(\pi (v),v)$ with associated cost $w(\pi (v),v)$ . If the marked edges form an SRT (Shortest Route Tree), $f(D)$ is defined to be this SRT. Otherwise, the set of marked edges form at least one cycle. Call (an arbitrarily chosen) one of these cycles $C$ . We now define a weighted directed graph $D^{\prime }$ having a root $r^{\prime }$ as follows. The nodes of $D^{\prime }$ are the nodes of $D$ not in $C$ plus a new node denoted $v_{C}$ .

If $(u,v)$ is an edge in $D$ with $u\notin C$ and $v\in C$ , then include in $D^{\prime }$ the edge $e=(u,v_{C})$ , and define $w(e)=w(u,v)-w(\pi (v),v)$ .

If $(u,v)$ is an edge in $D$ with $u\in C$ and $v\notin C$ , then include in $D^{\prime }$ the edge $e=(v_{C},v)$ , and define $w(e)=w(u,v)$ .

If $(u,v)$ is an edge in $D$ with $u\notin C$ and $v\notin C$ , then include in $D^{\prime }$ the edge $e=(u,v)$ , and define $w(e)=w(u,v)$ .

We include no other edges in $D^{\prime }$ .

The root $r^{\prime }$ of $D^{\prime }$ is simply the root $r$ in $D$ .

Using a call to $f(D^{\prime })$ , find an SRT of $D^{\prime }$ . First, mark in $D$ all shared edges with $D^{\prime }$ that are marked in the SRT of $D^{\prime }$ . Also, mark in $D$ all the edges in $C$ . Now, suppose that in the SRT of $D^{\prime }$ , the (unique) incoming edge at $v_{C}$ is $(u,v_{C})$ . This edge comes from some pair $(u,v)$ with $u\notin C$ and $v\in C$ . Unmark $(\pi (v),v)$ and mark $(u,v)$ . Also, for each marked $(v_{C},v)$ in the SRT of $D^{\prime }$ and comming from an edge $(u,v)$ in $D$ with $u\in C$ and $v\notin C$ , mark the edge $(u,v)$ . Now the set of marked edges do form an SRT, which we define to be the value of $f(D)$ .

Observe that $f(D)$ is defined in terms of $f(D^{\prime })$ for weighted directed rooted graphs $D^{\prime }$ having strictly fewer vertices than $D$ , and finding $f(D)$ for a single-vertex graph is trivial.

Implementation

Let BV be a vertex bucket and BE be an edge bucket. Let v be a vertex and e be an edge of maximum positive weight that is incident to v. C_i is a circuit. G₀ = (V₀,E₀) is the original digraph. u_i is a replacement vertex for C_i.

 $BV\leftarrow BE\leftarrow \varnothing$ 
i=0


A:
if  $BV=V_{i}$  then goto B
for some vertex  $v\notin BV$  and  $v\in V_{i}$  {
    $BV\leftarrow BV\cup \lbrace v\rbrace$ 
   find an edge  $e=(x,v)$  such that w(e) = max{ w(y,v)|(y,v)  $\in$  E_i}
   if w(e) ≤ 0 then goto A
}
if  $BE\cup \lbrace e\rbrace$  contains a circuit {
   i=i+1
   construct  $G_{i}$  by shrinking  $C_{i}$  to  $u_{i}$ 
   modify BE, BV and some edge weights
}
 $BE\leftarrow BE\cup {e}$ 
goto A


B:
while i ≠ 0 {
   reconstruct  $G_{i-1}$  and rename some edges in BE
   if  $u_{i}$  was a root of an out-tree in BE {
        $BE\leftarrow BE\cup \lbrace e|e\in C_{i}$  and  $e\neq e_{0}^{i}\rbrace$ 
   }else{
        $BE\leftarrow BE\cup \lbrace e|e\in C_{i}$  and  $e\neq {\tilde {e}}_{i}\rbrace$ 
   }
   i=i-1
}
Maximum branching weight =  $\sum _{e\in BE}w(e)$

References

Y. J. Chu and T. H. Liu, "On the Shortest Arborescence of a Directed Graph", Science Sinica, vol. 14, 1965, pp. 1396–1400.
J. Edmonds, “Optimum Branchings”, J. Res. Nat. Bur. Standards, vol. 71B, 1967, pp. 233–240.
R. E. Tarjan, "Finding Optimum Branchings", Networks, v.7, 1977, pp. 25–35.
P.M. Camerini, L. Fratta, and F. Maffioli, "A note on finding optimum branchings", Networks, v.9, 1979, pp. 309–312.
Alan Gibbons Algorithmic Graph Theory, Cambridge University press, 1985 ISBN 0-521-28881-9
H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,” Combinatorica 6 (1986), 109-122.

External links

The Directed Minimum Spanning Tree Problem Description of the algorithm summarized by Shanchieh Jay Yang, May 2000.
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.
AlgoWiki – Edmonds's algorithm - A public-domain implementation of Edmonds's algorithm written in Java.