Directed (Optimum Branchings), General MST (Minimum Spanning Tree (MST))

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A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected; edge-weighted undirected graph that connects all the vertices together; without any cycles and with the minimum possible total edge weight. Here, we're given a directed graph with a root, and we wish to find a spanning arborescence of minimum weight that is rooted at the root.

Related Problems

Subproblem: Directed (Optimum Branchings), Super Dense MST

Related: Undirected, General MST, Undirected, Dense MST, Undirected, Planar MST, Undirected, Integer Weights MST


$V$: number of vertices

$E$: number of edges

$U$: maximum edge weight

Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Chu-Liu-Edmonds Algorithm 1965 $O(EV)$ $O(E+V)$ Exact Deterministic Time
Tarjan (directed, general) 1987 $O(ElogV)$ $O(E)$ Exact Deterministic Time & Space
Gabow, Galil, Spencer 1984 $O(VlogV+Eloglog(logV/log(E/V + {2})$)) $O(E)$ Exact Deterministic Time
Gabow et al, Section 3 1986 $O(E+VlogV)$ $O(E+V)$ Exact Deterministic Time