Why it matters

Offline LCA is faster than online for batches.

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The architecture

DFS tree.

Union-find as we return from subtrees.

Answer query when both endpoints visited.

Offline LCA algorithmDFStraverse treeUnion-findas returningAnswer querywhen both seenNear-linear α(n) inverse-Ackermann; better than online O(log n)
Offline LCA.
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How it works end to end

α(n) per query amortized.

Requires all queries upfront.

Applications: batch tree queries.