Merge pull request #531 from rahul1995/fib-matrix-expo

Add O(log n) algorithm using matrix exponentiation to find n-th fibonacci

Author: ruppysuppy

Maths/Fibonacci.js

@@ -69,7 +69,83 @@ const FibonacciDpWithoutRecursion = (number) => {
   return table
 }
 
+// Using Matrix exponentiation to find n-th fibonacci in O(log n) time
+
+const copyMatrix = (A) => {
+  return A.map(row => row.map(cell => cell))
+}
+
+const Identity = (size) => {
+  const I = Array(size).fill(null).map(() => Array(size).fill())
+  return I.map((row, rowIdx) => row.map((_col, colIdx) => {
+    return rowIdx === colIdx ? 1 : 0
+  }))
+}
+
+// A of size (l x m) and B of size (m x n)
+// product C will be of size (l x n)
+const matrixMultiply = (A, B) => {
+  A = copyMatrix(A)
+  B = copyMatrix(B)
+  const l = A.length
+  const m = B.length
+  const n = B[0].length // Assuming non-empty matrices
+  const C = Array(l).fill(null).map(() => Array(n).fill())
+  for (let i = 0; i < l; i++) {
+    for (let j = 0; j < n; j++) {
+      C[i][j] = 0
+      for (let k = 0; k < m; k++) {
+        C[i][j] += A[i][k] * B[k][j]
+      }
+    }
+  }
+  return C
+}
+
+// A is a square matrix
+const matrixExpo = (A, n) => {
+  A = copyMatrix(A)
+  if (n === 0) return Identity(A.length) // Identity matrix
+  if (n === 1) return A
+
+  // Just like Binary exponentiation mentioned in ./BinaryExponentiationIterative.js
+  let result = Identity(A.length)
+  while (n > 0) {
+    if (n % 2 !== 0) result = matrixMultiply(result, A)
+    n = Math.floor(n / 2)
+    if (n > 0) A = matrixMultiply(A, A)
+  }
+  return result
+}
+
+const FibonacciMatrixExpo = (n) => {
+  // F(0) = 0, F(1) = 1
+  // F(n) = F(n-1) + F(n-2)
+  // Consider below matrix multiplication:
+
+  // | F(n) |   |1  1|   |F(n-1)|
+  // |      | = |    | * |      |
+  // |F(n-1)|   |1  0|   |F(n-2)|
+
+  // F(n, n-1) = pow(A, n-1) * F(1, 0)
+
+  if (n === 0) return 0
+
+  const A = [
+    [1, 1],
+    [1, 0]
+  ]
+  const poweredA = matrixExpo(A, n - 1) // A raise to the power n
+  let F = [
+    [1],
+    [0]
+  ]
+  F = matrixMultiply(poweredA, F)
+  return F[0][0]
+}
+
 export { FibonacciDpWithoutRecursion }
 export { FibonacciIterative }
 export { FibonacciRecursive }
 export { FibonacciRecursiveDP }
+export { FibonacciMatrixExpo }

Maths/test/Fibonacci.test.js

@@ -2,7 +2,8 @@ import {
   FibonacciDpWithoutRecursion,
   FibonacciRecursiveDP,
   FibonacciIterative,
-  FibonacciRecursive
+  FibonacciRecursive,
+  FibonacciMatrixExpo
 } from '../Fibonacci'
 
 describe('Fibonanci', () => {
@@ -27,4 +28,8 @@ describe('Fibonanci', () => {
       expect.arrayContaining([1, 1, 2, 3, 5])
     )
   })
+
+  it('should return number for FibonnaciMatrixExpo', () => {
+    expect(FibonacciMatrixExpo(5)).toBe(5)
+  })
 })