Directed (Optimum Branchings), Super Dense 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 $E=\Omega(V^2)$ edges, and we wish to find a spanning arborescence of minimum weight that is rooted at the root.

Related Problems

Generalizations: Directed (Optimum Branchings), General 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
Tarjan (directed, dense) 1987 $O(V^{2})$ $O(E)$ Exact Deterministic Time & Space