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

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## Description

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

## Parameters

$V$: 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 |