fibonacci: add binary lifting version (#828)

Signed-off-by: y5c4l3 <[email protected]>
Co-authored-by: Piotr Idzik <[email protected]>

Author: y5c4l3

src/dynamic_programming/fibonacci.rs

@@ -180,6 +180,33 @@ fn matrix_multiply(multiplier: &[Vec<u128>], multiplicand: &[Vec<u128>]) -> Vec<
     result
 }
 
+/// Binary lifting fibonacci
+///
+/// Following properties of F(n) could be deduced from the matrix formula above:
+///
+/// F(2n)   = F(n) * (2F(n+1) - F(n))
+/// F(2n+1) = F(n+1)^2 + F(n)^2
+///
+/// Therefore F(n) and F(n+1) can be derived from F(n>>1) and F(n>>1 + 1), which
+/// has a smaller constant in both time and space compared to matrix fibonacci.
+pub fn binary_lifting_fibonacci(n: u32) -> u128 {
+    // the state always stores F(k), F(k+1) for some k, initially F(0), F(1)
+    let mut state = (0u128, 1u128);
+
+    for i in (0..u32::BITS - n.leading_zeros()).rev() {
+        // compute F(2k), F(2k+1) from F(k), F(k+1)
+        state = (
+            state.0 * (2 * state.1 - state.0),
+            state.0 * state.0 + state.1 * state.1,
+        );
+        if n & (1 << i) != 0 {
+            state = (state.1, state.0 + state.1);
+        }
+    }
+
+    state.0
+}
+
 /// nth_fibonacci_number_modulo_m(n, m) returns the nth fibonacci number modulo the specified m
 /// i.e. F(n) % m
 pub fn nth_fibonacci_number_modulo_m(n: i64, m: i64) -> i128 {
@@ -251,6 +278,7 @@ pub fn last_digit_of_the_sum_of_nth_fibonacci_number(n: i64) -> i64 {
 
 #[cfg(test)]
 mod tests {
+    use super::binary_lifting_fibonacci;
     use super::classical_fibonacci;
     use super::fibonacci;
     use super::last_digit_of_the_sum_of_nth_fibonacci_number;
@@ -398,6 +426,24 @@ mod tests {
         );
     }
 
+    #[test]
+    fn test_binary_lifting_fibonacci() {
+        assert_eq!(binary_lifting_fibonacci(0), 0);
+        assert_eq!(binary_lifting_fibonacci(1), 1);
+        assert_eq!(binary_lifting_fibonacci(2), 1);
+        assert_eq!(binary_lifting_fibonacci(3), 2);
+        assert_eq!(binary_lifting_fibonacci(4), 3);
+        assert_eq!(binary_lifting_fibonacci(5), 5);
+        assert_eq!(binary_lifting_fibonacci(10), 55);
+        assert_eq!(binary_lifting_fibonacci(20), 6765);
+        assert_eq!(binary_lifting_fibonacci(21), 10946);
+        assert_eq!(binary_lifting_fibonacci(100), 354224848179261915075);
+        assert_eq!(
+            binary_lifting_fibonacci(184),
+            127127879743834334146972278486287885163
+        );
+    }
+
     #[test]
     fn test_nth_fibonacci_number_modulo_m() {
         assert_eq!(nth_fibonacci_number_modulo_m(5, 10), 5);

src/dynamic_programming/mod.rs

@@ -20,6 +20,7 @@ mod word_break;
 
 pub use self::coin_change::coin_change;
 pub use self::egg_dropping::egg_drop;
+pub use self::fibonacci::binary_lifting_fibonacci;
 pub use self::fibonacci::classical_fibonacci;
 pub use self::fibonacci::fibonacci;
 pub use self::fibonacci::last_digit_of_the_sum_of_nth_fibonacci_number;