Add Edit Distance and Catalan Numbers

[Hardik1106]

Description

This PR adds two new algorithm modules implemented in the project’s mapcode style and integrates them into the main document so they render with the existing mapcode-viz visualizer.

What’s Included

1. New Algorithm Modules

• algorithms/editdistance.typ

  • Formal DP recurrence definition.
  • Mapcode-style specification (rho, F, pi).
  • Concrete DP computation for a sample instance.
  • Full DP-table visualization using the project’s mapcode-viz framework.
  • Structured layout for item metadata, tensor updates, and stepwise evaluation.

• algorithms/catalannumbers.typ

  • Formal DP recurrence definition.
  • Mapcode-style specification (rho, F, pi).
  • Concrete DP computation for a sample instance.
  • Full DP-table visualization using the project’s mapcode-viz framework.
  • Structured layout for item metadata, tensor updates, and stepwise evaluation.

2. Documentation Integration

• Updated main.typ:
  • Added "#include "algorithms/editdistance.typ" and "algorithms/catalannumbers.typ" directly after the add-two-numbers section.

Why this matters

• Adds representative DP examples (1D convolution-like DP and 2D DP table) to the mapcode corpus.
• Exercises both vector and tensor visualizers and helps validate DP-history/fixed-point visualization logic.
• Three-way dependency for Edit Distance Problem: Each DP cell depends on (i-1,j), (i,j-1), (i-1,j-1) (delete, insert, substitute) a simultaneous 3‑way recursion per non‑base cell.
• Convolutional dependency: Cat(n) = Σ_{k=0..n-1} Cat(k)·Cat(n-1-k) - each new index depends on all strictly smaller indices (a full convolution), not just a fixed small window.

Files Changed

• algorithms/editdistance.typ - 141 lines added
• algorithms/catalannumbers.typ - 108 lines added
• main.typ - 4 lines added
• No existing code modified; additions are isolated and non-breaking

Implementation notes

• catalannumbers.typ:

  • Defines I = n:NN, X = [0..n] -> NN_bot, A = NN.
  • rho(n) initializes the DP vector to bot (none), F fills Cat(0)=1 and computes Cat(i) = sum_{k=0}^{i-1} Cat(k) * Cat(i-1-k) when dependencies are ready.
  • pi(n) returns x[n].
  • X_h visualizes the DP vector and highlights updated entries per iteration (consistent with fibonacci.typ / factorial.typ).

• editdistance.typ:

  • Defines I = S × T, X = [0..|S|] × [0..|T|] -> NN_bot, A = NN.
  • rho(S,T) initializes the full DP table to bot (none).
  • F implements the standard Levenshtein DP update:
    • base rows/columns: i if j=0, j if i=0
    • match: x[i-1,j-1]
    • mismatch: 1 + min(x[i-1,j], x[i,j-1], x[i-1,j-1])
  • pi returns x[|S|, |T|].
  • x_h renders the DP table with labeled row/column characters and highlights changed cells per iteration (consistent with LongestCommonSubsequence.typ).

  Traces Generated: