Enumerating All Directed Spanning Trees in Optimal Time

Abstract

We consider the problem of enumerating, for a given directed graph $G=(V,E)$ and a node $r\in V$, all directed spanning trees of $G$ rooted at $r$. For undirected graphs, the corresponding problem of enumerating all spanning trees has received considerable attention, culminating in the algorithm of Kapoor and Ramesh [SICOMP 1995] working in $\mathcal{O}(n+m+N)$ time, where $N, n, m$ denote the number of spanning trees, vertices, and edges of $G$, respectively. In the area of enumeration algorithms, this is known as Constant Amortised Time, or CAT. To achieve only constant time per each spanning tree, the algorithm outputs the relative change between the subsequent spanning trees instead of the whole spanning trees themselves. The natural generalization to enumerating all directed spanning trees has been already considered by Gabow and Myers [SICOMP 1978], who provided an $\mathcal{O}(n+m+Nm)$ time algorithm. This time complexity has been improved upon a couple of times, and in 1998 Uno introduced the framework of trimming and balancing that allowed him to obtain an $\mathcal{O}(n+m\log n+N\log^{2}n)$ time algorithm for this problem. By plugging in later results it is immediate to improve the time complexity to $\mathcal{O}(n+m+N\log n)$, but achieving the optimal bound of $\mathcal{O}(n+m+N)$ seems problematic within this framework. In this paper, we show how to enumerate all directed spanning trees in $\mathcal{O}(n+m+N)$ time and $\mathcal{O}(n+m)$ space, matching the time bound for undirected graphs. Our improvement is obtained by designing a purely graph-theoretical characterization of graphs with very few directed spanning trees, and using their structure to speed up the algorithm.