add edit distance

algorithms/EditDistance.typ

@@ -0,0 +1,142 @@
+#import "../lib/style.typ": *
+#import "../lib/mapcode.typ": *
+
+== Edit Distance (Levenshtein Distance, dynamic programming, mapcode)
+
+Compute the minimum number of single-character edits (insertions, deletions, or substitutions) required to transform one string into another.
+
+Formal definition:
+$
+"ED"(i, j) = cases(
+  i & "if " j = 0,
+  j & "if " i = 0,
+  "ED"(i-1, j-1) & "if " s_1[i] = s_2[j],
+  1 + min("ED"(i-1, j), "ED"(i, j-1), "ED"(i-1, j-1)) & "otherwise"
+)
+$
+
+*As mapcode:*
+
+_primitives_: `min`, `add`($+$)
+
+$
+I = s_1:[Sigma]^m times s_2:[Sigma]^n quad quad quad
+X_(m,n) & = [0..m] times [0..n] -> NN_bot quad quad quad
+A = NN\
+rho(m, n) & = { (i,j) -> bot | i in {0 dots m}, j in {0 dots n}} \
+F_(s_1, s_2)(x_(i,j)) & = cases(
+    i & "if " j = 0,
+    j & "if " i = 0,
+    x_(i-1,j-1) & "if " s_1[i-1] = s_2[j-1],
+    1 + min(x_(i-1,j), x_(i,j-1), x_(i-1,j-1)) & "otherwise"
+  )\
+pi_(m,n) (x) & = x_(m,n)
+$
+
+#let inst_s1 = "cat";
+#let inst_s2 = "dog";
+#let inst_m = inst_s1.len();
+#let inst_n = inst_s2.len();
+
+#figure(
+  caption: [Edit Distance (Levenshtein) DP table for $s_1 =$ "#inst_s1" and $s_2 =$ "#inst_s2".],
+$#{
+  let rho = ((s1, s2)) => {
+    let m = s1.len()
+    let n = s2.len()
+    let x = ()
+    for i in range(0, m + 1) {
+      let row = ()
+      for j in range(0, n + 1) {
+        row.push(none)
+      }
+      x.push(row)
+    }
+    x
+  }
+
+  let F_i = ((s1, s2)) => (x) => ((i,j)) => {
+    if j == 0 {
+      i
+    } else if i == 0 {
+      j
+    } else {
+      // Check if all dependencies are met
+      let dep1 = x.at(i - 1).at(j)      // deletion
+      let dep2 = x.at(i).at(j - 1)      // insertion
+      let dep3 = x.at(i - 1).at(j - 1)  // substitution/match
+
+      if dep1 != none and dep2 != none and dep3 != none {
+        if s1.at(i - 1) == s2.at(j - 1) {
+          dep3
+        } else {
+          1 + calc.min(dep1, calc.min(dep2, dep3))
+        }
+      } else {
+        none
+      }
+    }
+  }
+
+  let F = ((s1, s2)) => map_tensor(F_i((s1, s2)), dim: 2)
+
+  let pi = ((s1, s2)) => (x) => {
+    let m = s1.len()
+    let n = s2.len()
+    x.at(m).at(n)
+  }
+
+  let I_h((s1, s2)) = {
+    [
+      [$s_1:$ "#s1"],
+      [$s_2:$ "#s2"],
+    ]
+  }
+
+  // draw DP table with string character labels
+  let x_h(x, diff_mask:none) = {
+    set text(weight: "bold")
+    let rows = ()
+
+    // header row: show s2 characters (with an initial empty corner)
+    let header_cells = ()
+    header_cells.push(rect(stroke: none, inset: 4pt)[$bot$])
+    header_cells.push(rect(fill: orange.transparentize(70%), inset: 4pt)[$epsilon$])
+    for j in range(0, inst_n) {
+      header_cells.push(rect(fill: orange.transparentize(70%), inset: 4pt)[$#inst_s2.at(j)$])
+    }
+    rows.push(grid(columns: header_cells.len() * (14pt,), rows: 14pt, align: center + horizon, ..header_cells))
+
+    for i in range(0, x.len()) {
+      let row = ()
+      // left label: s1 character for i>0, empty for i=0
+      if i == 0 {
+        row.push(rect(fill: green.transparentize(70%), inset: 4pt)[$epsilon$])
+      } else {
+        row.push(rect(fill: green.transparentize(70%), inset: 4pt)[$#inst_s1.at(i - 1)$])
+      }
+
+      for j in range(0, x.at(i).len()) {
+        let val = if x.at(i).at(j) != none {[$#x.at(i).at(j)$]} else {[$bot$]}
+        if diff_mask != none and diff_mask.at(i).at(j) {
+          row.push(rect(stroke: gray, fill: yellow.transparentize(70%), inset: 4pt)[$#val$])
+        } else {
+          row.push(rect(stroke: gray, inset: 4pt)[$#val$])
+        }
+      }
+      rows.push(grid(columns: row.len() * (14pt,), rows: 14pt, align: center + horizon, ..row))
+    }
+    grid(align: center, ..rows)
+  }
+
+  mapcode-viz(
+    rho, F((inst_s1, inst_s2)), pi((inst_s1, inst_s2)),
+    I_h: I_h,
+    X_h: x_h,
+    F_name: [$F_(s_1, s_2)$],
+    pi_name: [$\pi_(#inst_m, #inst_n)$],
+    group-size: 3,
+    cell-size: 40mm, scale-fig: 85%
+  )((inst_s1, inst_s2))
+}$)
+

main.typ

@@ -51,4 +51,6 @@ All primitives are _strict_ meaning they do not allow for undefined values (i.e.
 #pagebreak()
 #include "algorithms/LongestCommonSubsequence.typ"
 #pagebreak()
+#include "algorithms/EditDistance.typ"
+#pagebreak()
 #include "algorithms/leetcode/P2_add-two-numbers.typ"