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X-Fast / Y-Fast Tries Ladder

This lane is for the first time one ordered-set lesson is really about bounded integer universe + predecessor / successor via prefix hashing instead of one ordinary O(log n) tree.

Who This Is For

Use this lane if:

  • you already understand what one ordered set does
  • Binary Trie / XOR Queries already feels separate from ordinary ordered sets
  • you now want the classical x-fast / y-fast breadth story itself

This lane is intentionally narrow:

Warm-Up

  • explain why the universe bound U = 2^w is the real parameter
  • explain why storing every prefix lets you binary-search depth
  • explain why leaf links are still needed after the prefix search stops

Target skill:

  • recognize when the real lesson is "bounded-integer predecessor structure," not "which ordinary ordered set should I use?"

Core

  • per-depth prefix hashing
  • deepest-existing-prefix binary search
  • leaf linked order
  • jump pointers on one-child internal nodes
  • exact flagship rep: X-Fast Dictionary

Target skill:

  • trust the x-fast predecessor / successor route and understand why leaf order finishes the job after prefix search

Stretch

Target skill:

  • separate x-fast mechanics from the y-fast refinement that fixes x-fast's space and update cost

Retrieval Layer

Repo Anchors And Compare Points

The cleanest in-repo loop here is:

  1. reopen Binary Trie / XOR Queries just enough to keep xor objectives separate from predecessor objectives
  2. learn the bounded-universe route in X-Fast / Y-Fast Tries
  3. solve X-Fast Dictionary
  4. compare back against PBDS / Order Statistics Tree so the "textbook breadth vs contest-first route" boundary stays sharp

Exit Criteria

You are ready to move on when:

  • you can explain why x-fast needs the leaf list even after hashing all prefixes
  • you know the difference between x-fast and y-fast in one sentence
  • you know why this family is rarely the first contest retrieval route in the repo

External Practice