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Virtual Tree Ladder

This ladder is for the first tree-query lane where the main trick is:

  • keep only the marked vertices from the query
  • add the LCAs that still matter
  • and run the real DP on that compressed tree

It is intentionally a thin lane:

  • one exact starter
  • one classic flagship rep
  • strong compare points back to LCA, Tree DP, and HLD

Who This Is For

Use this lane if:

  • LCA already feels comfortable
  • you can root the tree once and trust tin/tout
  • your current query set is small compared with the whole tree, but pairwise paths between those marked nodes still matter

Warm-Up

  • ancestor checks by tin/tout
  • why LCAs of consecutive marked vertices in DFS order are enough
  • stack-building one compressed ancestor tree

Target skill:

  • explain why one query with k marked vertices can be reduced to O(k) relevant vertices

Warm-up compare points:

Core

  • sort marked vertices by tin
  • add consecutive LCAs
  • sort/unique again
  • build the parent relation by one stack
  • attach original-path metadata such as depth difference or "can this segment be cut with one non-important vertex?"
  • flagship rep: Kingdom and its Cities

Target skill:

  • know when the hard part is not one whole-tree DP, but one query-local DP on a compressed tree

Stretch

Target skill:

  • separate "build the auxiliary tree" from "what summary/DP lives on the compressed edges and nodes"

Retrieval Layer

Repo Anchors And Compare Points

The cleanest in-repo loop here is:

  1. reopen LCA until tin/tout and lca() feel automatic
  2. trust the build step from virtual-tree.cpp
  3. solve or revisit Kingdom and its Cities
  4. only then jump to heavier marked-subset DPs like Leaf Color

Exit Criteria

You are ready to move on when:

  • you can explain why consecutive-by-tin LCAs are enough
  • you can build the compressed tree without guessing the stack logic
  • you can say what metadata a compressed edge must carry for the current query DP
  • you can distinguish virtual tree from HLD path decomposition and from whole-tree DP

External Practice