Efficiently Computing Directed Minimum Spanning Trees

TL;DR

This paper presents the first implementation and empirical evaluation of efficient algorithms for computing directed minimum spanning trees, comparing their performance on real-world and synthetic graphs.

Contribution

It provides the first practical implementation of Gabow et al.'s algorithm and five variants of Tarjan's algorithm, along with comprehensive performance analysis.

Findings

Gabow et al.'s algorithm outperforms Tarjan variants on various graph types.

Tarjan's implementations are often slower and less efficient in practice.

Empirical results guide the choice of algorithms for different graph regimes.

Abstract

Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in $O(\min(n^2, m\log n))$ by Tarjan which is optimal for very dense and very sparse graphs. Gabow et al.~gave a version of Edmonds' algorithm that runs in $O(n\log n + m)$, thus asymptotically beating the Tarjan variant in the regime between sparse and dense. Despite the attention the problem received theoretically, there exists, to the best of our knowledge, no empirical evaluation of either of these algorithms. In fact, the version by Gabow et al.~has never been implemented and, aside from coding competitions, all readily available Tarjan implementations run in $O(n^2)$. In this paper, we provide the first implementation of the version by Gabow et al.~as well as five variants of Tarjan's version with different underlying data structures. We evaluate these algorithms and existing solvers on a large set of real-world and random graphs.