Why it matters

LCT enables dynamic tree operations. Understanding shapes advanced algorithms.

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The architecture

Preferred path decomposition.

Splay tree per preferred path.

LCT structurePreferred pathsdecompositionSplay tree per pathamortized O(log n)Operationslink/cut/queryComplex implementation; used in dynamic connectivity + max flow
LCT approach.
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How it works end to end

Applications: dynamic max flow, dynamic connectivity, Euler tour trees.

Access: bring node to root; splay.

Link: connect two trees.

Cut: separate.