Edmonds' algorithm

This article is about the optimum branching algorithm. For the maximum matching algorithm, see Blossom 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).^[1] 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 ${\displaystyle D=\langle V,E\rangle }$ 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.

Running time

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 ${\displaystyle O(E+V\log V)}$.

References

1. The algorithm is applicable to finding a minimum spanning forest with given roots. However, when searching for the minimum spanning forest among all ${\displaystyle k}$-component spanning forests, a multiplier arises in the complexity of the algorithm ${\displaystyle C_{V}^{k}}$, corresponding to the choice of a subset of vertices designated as roots. This makes it unsuitable for such a task. Even when constructing a minimum spanning tree, regardless of the root, the algorithm must be used ${\displaystyle V}$ times, sequentially assigning each vertex as the root. An efficient algorithm for finding minimum spanning forests that solves the root assignment problem is presented in (https://link.springer.com/article/10.1007/s10958-023-06666-w). It builds a sequence of minimal ${\displaystyle k}$-component spanning forests for all ${\displaystyle k}$ up to the minimum spanning tree. The Chu-Liu/Edmonds algorithm is a component of it.

• 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
• Buslov, V. (2023), "Algorithm for Sequential Construction of Spanning Minimal Directed Forests", Journal of Mathematical Sciences, 275: 117–129, doi:10.1007/s10958-023-06666-w

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.
• (spanning-forest-builder 0.0.2) – Library for constructing oriented forests of minimum weight.