num_bigint/algorithms.rs
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use std::borrow::Cow;
use std::cmp;
use std::cmp::Ordering::{self, Less, Greater, Equal};
use std::iter::repeat;
use std::mem;
use traits;
use traits::{Zero, One};
use biguint::BigUint;
use bigint::BigInt;
use bigint::Sign;
use bigint::Sign::{Minus, NoSign, Plus};
#[allow(non_snake_case)]
pub mod big_digit {
/// A `BigDigit` is a `BigUint`'s composing element.
pub type BigDigit = u32;
/// A `DoubleBigDigit` is the internal type used to do the computations. Its
/// size is the double of the size of `BigDigit`.
pub type DoubleBigDigit = u64;
pub const ZERO_BIG_DIGIT: BigDigit = 0;
// `DoubleBigDigit` size dependent
pub const BITS: usize = 32;
pub const BASE: DoubleBigDigit = 1 << BITS;
const LO_MASK: DoubleBigDigit = (-1i32 as DoubleBigDigit) >> BITS;
#[inline]
fn get_hi(n: DoubleBigDigit) -> BigDigit {
(n >> BITS) as BigDigit
}
#[inline]
fn get_lo(n: DoubleBigDigit) -> BigDigit {
(n & LO_MASK) as BigDigit
}
/// Split one `DoubleBigDigit` into two `BigDigit`s.
#[inline]
pub fn from_doublebigdigit(n: DoubleBigDigit) -> (BigDigit, BigDigit) {
(get_hi(n), get_lo(n))
}
/// Join two `BigDigit`s into one `DoubleBigDigit`
#[inline]
pub fn to_doublebigdigit(hi: BigDigit, lo: BigDigit) -> DoubleBigDigit {
(lo as DoubleBigDigit) | ((hi as DoubleBigDigit) << BITS)
}
}
use big_digit::{BigDigit, DoubleBigDigit};
// Generic functions for add/subtract/multiply with carry/borrow:
// Add with carry:
#[inline]
fn adc(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) + (b as DoubleBigDigit) +
(*carry as DoubleBigDigit));
*carry = hi;
lo
}
// Subtract with borrow:
#[inline]
fn sbb(a: BigDigit, b: BigDigit, borrow: &mut BigDigit) -> BigDigit {
let (hi, lo) = big_digit::from_doublebigdigit(big_digit::BASE + (a as DoubleBigDigit) -
(b as DoubleBigDigit) -
(*borrow as DoubleBigDigit));
// hi * (base) + lo == 1*(base) + ai - bi - borrow
// => ai - bi - borrow < 0 <=> hi == 0
*borrow = (hi == 0) as BigDigit;
lo
}
#[inline]
pub fn mac_with_carry(a: BigDigit, b: BigDigit, c: BigDigit, carry: &mut BigDigit) -> BigDigit {
let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) +
(b as DoubleBigDigit) * (c as DoubleBigDigit) +
(*carry as DoubleBigDigit));
*carry = hi;
lo
}
#[inline]
pub fn mul_with_carry(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) * (b as DoubleBigDigit) +
(*carry as DoubleBigDigit));
*carry = hi;
lo
}
/// Divide a two digit numerator by a one digit divisor, returns quotient and remainder:
///
/// Note: the caller must ensure that both the quotient and remainder will fit into a single digit.
/// This is _not_ true for an arbitrary numerator/denominator.
///
/// (This function also matches what the x86 divide instruction does).
#[inline]
fn div_wide(hi: BigDigit, lo: BigDigit, divisor: BigDigit) -> (BigDigit, BigDigit) {
debug_assert!(hi < divisor);
let lhs = big_digit::to_doublebigdigit(hi, lo);
let rhs = divisor as DoubleBigDigit;
((lhs / rhs) as BigDigit, (lhs % rhs) as BigDigit)
}
pub fn div_rem_digit(mut a: BigUint, b: BigDigit) -> (BigUint, BigDigit) {
let mut rem = 0;
for d in a.data.iter_mut().rev() {
let (q, r) = div_wide(rem, *d, b);
*d = q;
rem = r;
}
(a.normalized(), rem)
}
// Only for the Add impl:
#[inline]
pub fn __add2(a: &mut [BigDigit], b: &[BigDigit]) -> BigDigit {
debug_assert!(a.len() >= b.len());
let mut carry = 0;
let (a_lo, a_hi) = a.split_at_mut(b.len());
for (a, b) in a_lo.iter_mut().zip(b) {
*a = adc(*a, *b, &mut carry);
}
if carry != 0 {
for a in a_hi {
*a = adc(*a, 0, &mut carry);
if carry == 0 { break }
}
}
carry
}
/// /Two argument addition of raw slices:
/// a += b
///
/// The caller _must_ ensure that a is big enough to store the result - typically this means
/// resizing a to max(a.len(), b.len()) + 1, to fit a possible carry.
pub fn add2(a: &mut [BigDigit], b: &[BigDigit]) {
let carry = __add2(a, b);
debug_assert!(carry == 0);
}
pub fn sub2(a: &mut [BigDigit], b: &[BigDigit]) {
let mut borrow = 0;
let len = cmp::min(a.len(), b.len());
let (a_lo, a_hi) = a.split_at_mut(len);
let (b_lo, b_hi) = b.split_at(len);
for (a, b) in a_lo.iter_mut().zip(b_lo) {
*a = sbb(*a, *b, &mut borrow);
}
if borrow != 0 {
for a in a_hi {
*a = sbb(*a, 0, &mut borrow);
if borrow == 0 { break }
}
}
// note: we're _required_ to fail on underflow
assert!(borrow == 0 && b_hi.iter().all(|x| *x == 0),
"Cannot subtract b from a because b is larger than a.");
}
pub fn sub2rev(a: &[BigDigit], b: &mut [BigDigit]) {
debug_assert!(b.len() >= a.len());
let mut borrow = 0;
let len = cmp::min(a.len(), b.len());
let (a_lo, a_hi) = a.split_at(len);
let (b_lo, b_hi) = b.split_at_mut(len);
for (a, b) in a_lo.iter().zip(b_lo) {
*b = sbb(*a, *b, &mut borrow);
}
assert!(a_hi.is_empty());
// note: we're _required_ to fail on underflow
assert!(borrow == 0 && b_hi.iter().all(|x| *x == 0),
"Cannot subtract b from a because b is larger than a.");
}
pub fn sub_sign(a: &[BigDigit], b: &[BigDigit]) -> (Sign, BigUint) {
// Normalize:
let a = &a[..a.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
let b = &b[..b.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
match cmp_slice(a, b) {
Greater => {
let mut a = a.to_vec();
sub2(&mut a, b);
(Plus, BigUint::new(a))
}
Less => {
let mut b = b.to_vec();
sub2(&mut b, a);
(Minus, BigUint::new(b))
}
_ => (NoSign, Zero::zero()),
}
}
/// Three argument multiply accumulate:
/// acc += b * c
pub fn mac_digit(acc: &mut [BigDigit], b: &[BigDigit], c: BigDigit) {
if c == 0 {
return;
}
let mut carry = 0;
let (a_lo, a_hi) = acc.split_at_mut(b.len());
for (a, &b) in a_lo.iter_mut().zip(b) {
*a = mac_with_carry(*a, b, c, &mut carry);
}
let mut a = a_hi.iter_mut();
while carry != 0 {
let a = a.next().expect("carry overflow during multiplication!");
*a = adc(*a, 0, &mut carry);
}
}
/// Three argument multiply accumulate:
/// acc += b * c
fn mac3(acc: &mut [BigDigit], b: &[BigDigit], c: &[BigDigit]) {
let (x, y) = if b.len() < c.len() {
(b, c)
} else {
(c, b)
};
// We use three algorithms for different input sizes.
//
// - For small inputs, long multiplication is fastest.
// - Next we use Karatsuba multiplication (Toom-2), which we have optimized
// to avoid unnecessary allocations for intermediate values.
// - For the largest inputs we use Toom-3, which better optimizes the
// number of operations, but uses more temporary allocations.
//
// The thresholds are somewhat arbitrary, chosen by evaluating the results
// of `cargo bench --bench bigint multiply`.
if x.len() <= 32 {
// Long multiplication:
for (i, xi) in x.iter().enumerate() {
mac_digit(&mut acc[i..], y, *xi);
}
} else if x.len() <= 256 {
/*
* Karatsuba multiplication:
*
* The idea is that we break x and y up into two smaller numbers that each have about half
* as many digits, like so (note that multiplying by b is just a shift):
*
* x = x0 + x1 * b
* y = y0 + y1 * b
*
* With some algebra, we can compute x * y with three smaller products, where the inputs to
* each of the smaller products have only about half as many digits as x and y:
*
* x * y = (x0 + x1 * b) * (y0 + y1 * b)
*
* x * y = x0 * y0
* + x0 * y1 * b
* + x1 * y0 * b
* + x1 * y1 * b^2
*
* Let p0 = x0 * y0 and p2 = x1 * y1:
*
* x * y = p0
* + (x0 * y1 + x1 * y0) * b
* + p2 * b^2
*
* The real trick is that middle term:
*
* x0 * y1 + x1 * y0
*
* = x0 * y1 + x1 * y0 - p0 + p0 - p2 + p2
*
* = x0 * y1 + x1 * y0 - x0 * y0 - x1 * y1 + p0 + p2
*
* Now we complete the square:
*
* = -(x0 * y0 - x0 * y1 - x1 * y0 + x1 * y1) + p0 + p2
*
* = -((x1 - x0) * (y1 - y0)) + p0 + p2
*
* Let p1 = (x1 - x0) * (y1 - y0), and substitute back into our original formula:
*
* x * y = p0
* + (p0 + p2 - p1) * b
* + p2 * b^2
*
* Where the three intermediate products are:
*
* p0 = x0 * y0
* p1 = (x1 - x0) * (y1 - y0)
* p2 = x1 * y1
*
* In doing the computation, we take great care to avoid unnecessary temporary variables
* (since creating a BigUint requires a heap allocation): thus, we rearrange the formula a
* bit so we can use the same temporary variable for all the intermediate products:
*
* x * y = p2 * b^2 + p2 * b
* + p0 * b + p0
* - p1 * b
*
* The other trick we use is instead of doing explicit shifts, we slice acc at the
* appropriate offset when doing the add.
*/
/*
* When x is smaller than y, it's significantly faster to pick b such that x is split in
* half, not y:
*/
let b = x.len() / 2;
let (x0, x1) = x.split_at(b);
let (y0, y1) = y.split_at(b);
/*
* We reuse the same BigUint for all the intermediate multiplies and have to size p
* appropriately here: x1.len() >= x0.len and y1.len() >= y0.len():
*/
let len = x1.len() + y1.len() + 1;
let mut p = BigUint { data: vec![0; len] };
// p2 = x1 * y1
mac3(&mut p.data[..], x1, y1);
// Not required, but the adds go faster if we drop any unneeded 0s from the end:
p.normalize();
add2(&mut acc[b..], &p.data[..]);
add2(&mut acc[b * 2..], &p.data[..]);
// Zero out p before the next multiply:
p.data.truncate(0);
p.data.extend(repeat(0).take(len));
// p0 = x0 * y0
mac3(&mut p.data[..], x0, y0);
p.normalize();
add2(&mut acc[..], &p.data[..]);
add2(&mut acc[b..], &p.data[..]);
// p1 = (x1 - x0) * (y1 - y0)
// We do this one last, since it may be negative and acc can't ever be negative:
let (j0_sign, j0) = sub_sign(x1, x0);
let (j1_sign, j1) = sub_sign(y1, y0);
match j0_sign * j1_sign {
Plus => {
p.data.truncate(0);
p.data.extend(repeat(0).take(len));
mac3(&mut p.data[..], &j0.data[..], &j1.data[..]);
p.normalize();
sub2(&mut acc[b..], &p.data[..]);
},
Minus => {
mac3(&mut acc[b..], &j0.data[..], &j1.data[..]);
},
NoSign => (),
}
} else {
// Toom-3 multiplication:
//
// Toom-3 is like Karatsuba above, but dividing the inputs into three parts.
// Both are instances of Toom-Cook, using `k=3` and `k=2` respectively.
//
// FIXME: It would be nice to have comments breaking down the operations below.
let i = y.len()/3 + 1;
let x0_len = cmp::min(x.len(), i);
let x1_len = cmp::min(x.len() - x0_len, i);
let y0_len = i;
let y1_len = cmp::min(y.len() - y0_len, i);
let x0 = BigInt::from_slice(Plus, &x[..x0_len]);
let x1 = BigInt::from_slice(Plus, &x[x0_len..x0_len + x1_len]);
let x2 = BigInt::from_slice(Plus, &x[x0_len + x1_len..]);
let y0 = BigInt::from_slice(Plus, &y[..y0_len]);
let y1 = BigInt::from_slice(Plus, &y[y0_len..y0_len + y1_len]);
let y2 = BigInt::from_slice(Plus, &y[y0_len + y1_len..]);
let p = &x0 + &x2;
let q = &y0 + &y2;
let p2 = &p - &x1;
let q2 = &q - &y1;
let r0 = &x0 * &y0;
let r4 = &x2 * &y2;
let r1 = (p + x1) * (q + y1);
let r2 = &p2 * &q2;
let r3 = ((p2 + x2)*2 - x0) * ((q2 + y2)*2 - y0);
let mut comp3: BigInt = (r3 - &r1) / 3;
let mut comp1: BigInt = (r1 - &r2) / 2;
let mut comp2: BigInt = r2 - &r0;
comp3 = (&comp2 - comp3)/2 + &r4*2;
comp2 = comp2 + &comp1 - &r4;
comp1 = comp1 - &comp3;
let result = r0 + (comp1 << 32*i) + (comp2 << 2*32*i) + (comp3 << 3*32*i) + (r4 << 4*32*i);
let result_pos = result.to_biguint().unwrap();
add2(&mut acc[..], &result_pos.data);
}
}
pub fn mul3(x: &[BigDigit], y: &[BigDigit]) -> BigUint {
let len = x.len() + y.len() + 1;
let mut prod = BigUint { data: vec![0; len] };
mac3(&mut prod.data[..], x, y);
prod.normalized()
}
pub fn scalar_mul(a: &mut [BigDigit], b: BigDigit) -> BigDigit {
let mut carry = 0;
for a in a.iter_mut() {
*a = mul_with_carry(*a, b, &mut carry);
}
carry
}
pub fn div_rem(u: &BigUint, d: &BigUint) -> (BigUint, BigUint) {
if d.is_zero() {
panic!()
}
if u.is_zero() {
return (Zero::zero(), Zero::zero());
}
if d.data == [1] {
return (u.clone(), Zero::zero());
}
if d.data.len() == 1 {
let (div, rem) = div_rem_digit(u.clone(), d.data[0]);
return (div, rem.into());
}
// Required or the q_len calculation below can underflow:
match u.cmp(d) {
Less => return (Zero::zero(), u.clone()),
Equal => return (One::one(), Zero::zero()),
Greater => {} // Do nothing
}
// This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:
//
// First, normalize the arguments so the highest bit in the highest digit of the divisor is
// set: the main loop uses the highest digit of the divisor for generating guesses, so we
// want it to be the largest number we can efficiently divide by.
//
let shift = d.data.last().unwrap().leading_zeros() as usize;
let mut a = u << shift;
let b = d << shift;
// The algorithm works by incrementally calculating "guesses", q0, for part of the
// remainder. Once we have any number q0 such that q0 * b <= a, we can set
//
// q += q0
// a -= q0 * b
//
// and then iterate until a < b. Then, (q, a) will be our desired quotient and remainder.
//
// q0, our guess, is calculated by dividing the last few digits of a by the last digit of b
// - this should give us a guess that is "close" to the actual quotient, but is possibly
// greater than the actual quotient. If q0 * b > a, we simply use iterated subtraction
// until we have a guess such that q0 * b <= a.
//
let bn = *b.data.last().unwrap();
let q_len = a.data.len() - b.data.len() + 1;
let mut q = BigUint { data: vec![0; q_len] };
// We reuse the same temporary to avoid hitting the allocator in our inner loop - this is
// sized to hold a0 (in the common case; if a particular digit of the quotient is zero a0
// can be bigger).
//
let mut tmp = BigUint { data: Vec::with_capacity(2) };
for j in (0..q_len).rev() {
/*
* When calculating our next guess q0, we don't need to consider the digits below j
* + b.data.len() - 1: we're guessing digit j of the quotient (i.e. q0 << j) from
* digit bn of the divisor (i.e. bn << (b.data.len() - 1) - so the product of those
* two numbers will be zero in all digits up to (j + b.data.len() - 1).
*/
let offset = j + b.data.len() - 1;
if offset >= a.data.len() {
continue;
}
/* just avoiding a heap allocation: */
let mut a0 = tmp;
a0.data.truncate(0);
a0.data.extend(a.data[offset..].iter().cloned());
/*
* q0 << j * big_digit::BITS is our actual quotient estimate - we do the shifts
* implicitly at the end, when adding and subtracting to a and q. Not only do we
* save the cost of the shifts, the rest of the arithmetic gets to work with
* smaller numbers.
*/
let (mut q0, _) = div_rem_digit(a0, bn);
let mut prod = &b * &q0;
while cmp_slice(&prod.data[..], &a.data[j..]) == Greater {
let one: BigUint = One::one();
q0 = q0 - one;
prod = prod - &b;
}
add2(&mut q.data[j..], &q0.data[..]);
sub2(&mut a.data[j..], &prod.data[..]);
a.normalize();
tmp = q0;
}
debug_assert!(a < b);
(q.normalized(), a >> shift)
}
/// Find last set bit
/// fls(0) == 0, fls(u32::MAX) == 32
pub fn fls<T: traits::PrimInt>(v: T) -> usize {
mem::size_of::<T>() * 8 - v.leading_zeros() as usize
}
pub fn ilog2<T: traits::PrimInt>(v: T) -> usize {
fls(v) - 1
}
#[inline]
pub fn biguint_shl(n: Cow<BigUint>, bits: usize) -> BigUint {
let n_unit = bits / big_digit::BITS;
let mut data = match n_unit {
0 => n.into_owned().data,
_ => {
let len = n_unit + n.data.len() + 1;
let mut data = Vec::with_capacity(len);
data.extend(repeat(0).take(n_unit));
data.extend(n.data.iter().cloned());
data
}
};
let n_bits = bits % big_digit::BITS;
if n_bits > 0 {
let mut carry = 0;
for elem in data[n_unit..].iter_mut() {
let new_carry = *elem >> (big_digit::BITS - n_bits);
*elem = (*elem << n_bits) | carry;
carry = new_carry;
}
if carry != 0 {
data.push(carry);
}
}
BigUint::new(data)
}
#[inline]
pub fn biguint_shr(n: Cow<BigUint>, bits: usize) -> BigUint {
let n_unit = bits / big_digit::BITS;
if n_unit >= n.data.len() {
return Zero::zero();
}
let mut data = match n {
Cow::Borrowed(n) => n.data[n_unit..].to_vec(),
Cow::Owned(mut n) => {
n.data.drain(..n_unit);
n.data
}
};
let n_bits = bits % big_digit::BITS;
if n_bits > 0 {
let mut borrow = 0;
for elem in data.iter_mut().rev() {
let new_borrow = *elem << (big_digit::BITS - n_bits);
*elem = (*elem >> n_bits) | borrow;
borrow = new_borrow;
}
}
BigUint::new(data)
}
pub fn cmp_slice(a: &[BigDigit], b: &[BigDigit]) -> Ordering {
debug_assert!(a.last() != Some(&0));
debug_assert!(b.last() != Some(&0));
let (a_len, b_len) = (a.len(), b.len());
if a_len < b_len {
return Less;
}
if a_len > b_len {
return Greater;
}
for (&ai, &bi) in a.iter().rev().zip(b.iter().rev()) {
if ai < bi {
return Less;
}
if ai > bi {
return Greater;
}
}
return Equal;
}
#[cfg(test)]
mod algorithm_tests {
use {BigDigit, BigUint, BigInt};
use Sign::Plus;
use traits::Num;
#[test]
fn test_sub_sign() {
use super::sub_sign;
fn sub_sign_i(a: &[BigDigit], b: &[BigDigit]) -> BigInt {
let (sign, val) = sub_sign(a, b);
BigInt::from_biguint(sign, val)
}
let a = BigUint::from_str_radix("265252859812191058636308480000000", 10).unwrap();
let b = BigUint::from_str_radix("26525285981219105863630848000000", 10).unwrap();
let a_i = BigInt::from_biguint(Plus, a.clone());
let b_i = BigInt::from_biguint(Plus, b.clone());
assert_eq!(sub_sign_i(&a.data[..], &b.data[..]), &a_i - &b_i);
assert_eq!(sub_sign_i(&b.data[..], &a.data[..]), &b_i - &a_i);
}
}