src/algorithms/math/karatsuba-multiplication/README.md

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+# Karatsuba Multiplication
+
+Karatsuba is a fast multiplication algorithm discovered by Anatoly Karatsuba in 1960. Given two n-digit numbers, the "grade-school" method of long multiplication has a time complexity of O(n<sup>2</sup>), whereas the karatsuba algorithm has a time complexity of O(n<sup>1.59</sup>).
+
+## Recursive Formula
+
+```
+x = 1234
+y = 5678
+
+karatsuba(x, y)
+```
+
+1. Split each number into numbers with half as many digits
+```
+a = 12
+b = 34
+
+c = 56
+d = 78
+```
+
+2. Compute 3 subexpressions from the smaller numbers
+  - `ac = a * c`
+  - `bd = b * d`
+  - `abcd = (a + b) * (c + d)`
+
+3. Combine subexpressions to calculate the product
+```
+A = ac * 10000
+B = (abcd - ac - bd) * 100
+C = bd
+
+x * y = A + B + C
+```
+
+_**Note:**_ *The karatsuba algorithm can be applied recursively to calculate each product in the subexpressions.* (`a * c = karatsuba(a, c)`*). When the numbers get smaller than some arbitrary threshold, they are multiplied in the traditional way.*
+
+## References
+[Stanford Algorithms (YouTube)](https://www.youtube.com/watch?v=JCbZayFr9RE)
+[Wikipedia](https://en.wikipedia.org/wiki/Karatsuba_algorithm)

src/algorithms/math/karatsuba-multiplication/__test__/karatsuba.test.js

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+import karatsuba from '../karatsuba';
+
+describe('karatsuba multiplication', () => {
+  it('should multiply simple numbers correctly', () => {
+    expect(karatsuba(0, 37)).toEqual(0);
+    expect(karatsuba(1, 8)).toEqual(8);
+    expect(karatsuba(5, 6)).toEqual(30);
+  });
+
+  it('should multiply larger numbers correctly', () => {
+    expect(karatsuba(1234, 5678)).toEqual(7006652);
+    expect(karatsuba(9182734, 726354172)).toEqual(6669917151266248);
+  });
+});

src/algorithms/math/karatsuba-multiplication/karatsuba.js

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+/**
+ *
+ * @param {number} x
+ * @param {number} y
+ * @return {number}
+ */
+export default function karatsuba(x, y) {
+  // BASE CASE:
+  // if numbers are sufficiently small,
+  // multiply them together in the traditional way
+  if (x < 10 || y < 10) {
+    return x * y;
+  }
+
+  // SCALE FACTOR:
+  // scaleFactor is used to split the numbers
+  // into smaller numbers for recursion.
+  // when combining the subexpressions back
+  // together, the scaleFactor is used to
+  // recreate the magnitude of the original numbers
+  const minDigits = Math.min(
+    String(x).length,
+    String(y).length,
+  );
+  const scaleFactor = 10 ** Math.floor(minDigits / 2);
+
+  // PARAMETER COMPONENTS:
+  // a b are the two components of x
+  // c d are the two components of y
+  //
+  // e.g.
+  // x = 1234 -> a = 12, b = 34
+  // y = 5678 -> c = 56, d = 78
+
+  // example of component computations:
+  // x = 1234, y = 5678
+  // scaleFactor = 100
+
+  // a = floor(1234 / 100) = floor(12.34) = 12
+  const a = Math.floor(x / scaleFactor);
+
+  // b = 1234 - (12 * 100) = 1234 - 1200 = 34
+  const b = x - (a * scaleFactor);
+
+  // c = floor(5678 / 100) = floor(56.78) = 56
+  const c = Math.floor(y / scaleFactor);
+
+  // d = 5678 - (56 * 100) = 5678 - 5600 = 78
+  const d = y - (c * scaleFactor);
+
+  // COMPUTE SUBEXPRESSIONS:
+  // since a + b is less than x, and c + d is less than y
+  // the recursion is guaranteed to reach the base case
+  const ac = karatsuba(a, c);
+  const bd = karatsuba(b, d);
+  const abcd = karatsuba(a + b, c + d);
+
+  // COMBINE SUBEXPRESSIONS:
+  // since the scaleFactor was used to
+  // reduce the size of the components,
+  // the scaleFactor must be applied in reverse
+  // to reconstruct the magnitude of the original components
+  const A = ac * (scaleFactor ** 2);
+  const B = (abcd - ac - bd) * scaleFactor;
+  const C = bd;
+
+  return A + B + C;
+}