Chu-Liu-Edmonds

Same as directed MST algorithm. Each vertex picks cheapest in-edge. Contract cycles + recurse.

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Complexity

O(VE) naive, O(E log V) Tarjan, O(E + V log V) Gabow.

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Applications

Dependency parsing (natural language). Broadcasting in directed networks. Cheapest way to reach every node from root.

Difference from MST

Undirected MST: symmetric, easy. Directed: cheapest in-edges can conflict via cycles → need contraction.