Merge pull request #1525 from nbraud/factor/faster

Performance improvements for `factor`

src/uu/factor/build.rs

@@ -26,7 +26,7 @@ use std::path::Path;
 use std::u64::MAX as MAX_U64;
 
 #[cfg(test)]
-use numeric::is_prime;
+use miller_rabin::is_prime;
 
 #[cfg(test)]
 #[path = "src/numeric.rs"]

src/uu/factor/src/factor.rs

@@ -18,139 +18,95 @@ extern crate rand;
 #[macro_use]
 extern crate uucore;
 
-use numeric::*;
-use rand::distributions::{Distribution, Uniform};
-use rand::rngs::SmallRng;
-use rand::{thread_rng, SeedableRng};
-use std::cmp::{max, min};
+use std::collections::HashMap;
+use std::fmt;
 use std::io::{stdin, BufRead};
-use std::mem::swap;
-use std::num::Wrapping;
+use std::ops;
 
+mod miller_rabin;
 mod numeric;
-
-include!(concat!(env!("OUT_DIR"), "/prime_table.rs"));
+mod rho;
+mod table;
 
 static SYNTAX: &str = "[OPTION] [NUMBER]...";
 static SUMMARY: &str = "Print the prime factors of the given number(s).
  If none are specified, read from standard input.";
 static LONG_HELP: &str = "";
 
-fn rho_pollard_pseudorandom_function(x: u64, a: u64, b: u64, num: u64) -> u64 {
-    if num < 1 << 63 {
-        (sm_mul(a, sm_mul(x, x, num), num) + b) % num
-    } else {
-        big_add(big_mul(a, big_mul(x, x, num), num), b, num)
-    }
+struct Factors {
+    f: HashMap<u64, u8>,
 }
 
-fn gcd(mut a: u64, mut b: u64) -> u64 {
-    while b > 0 {
-        a %= b;
-        swap(&mut a, &mut b);
+impl Factors {
+    fn new() -> Factors {
+        Factors { f: HashMap::new() }
+    }
+
+    fn add(&mut self, prime: u64, exp: u8) {
+        assert!(exp > 0);
+        let n = *self.f.get(&prime).unwrap_or(&0);
+        self.f.insert(prime, exp + n);
+    }
+
+    fn push(&mut self, prime: u64) {
+        self.add(prime, 1)
     }
-    a
 }
 
-fn rho_pollard_find_divisor(num: u64) -> u64 {
-    #![allow(clippy::many_single_char_names)]
-    let range = Uniform::new(1, num);
-    let mut rng = SmallRng::from_rng(&mut thread_rng()).unwrap();
-    let mut x = range.sample(&mut rng);
-    let mut y = x;
-    let mut a = range.sample(&mut rng);
-    let mut b = range.sample(&mut rng);
-
-    loop {
-        x = rho_pollard_pseudorandom_function(x, a, b, num);
-        y = rho_pollard_pseudorandom_function(y, a, b, num);
-        y = rho_pollard_pseudorandom_function(y, a, b, num);
-        let d = gcd(num, max(x, y) - min(x, y));
-        if d == num {
-            // Failure, retry with different function
-            x = range.sample(&mut rng);
-            y = x;
-            a = range.sample(&mut rng);
-            b = range.sample(&mut rng);
-        } else if d > 1 {
-            return d;
+impl ops::MulAssign<Factors> for Factors {
+    fn mul_assign(&mut self, other: Factors) {
+        for (prime, exp) in &other.f {
+            self.add(*prime, *exp);
         }
     }
 }
 
-fn rho_pollard_factor(num: u64, factors: &mut Vec<u64>) {
-    if is_prime(num) {
-        factors.push(num);
-        return;
+impl fmt::Display for Factors {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        // TODO: Use a representation with efficient in-order iteration
+        let mut primes: Vec<&u64> = self.f.keys().collect();
+        primes.sort();
+
+        for p in primes {
+            for _ in 0..self.f[&p] {
+                write!(f, " {}", p)?
+            }
+        }
+
+        Ok(())
     }
-    let divisor = rho_pollard_find_divisor(num);
-    rho_pollard_factor(divisor, factors);
-    rho_pollard_factor(num / divisor, factors);
 }
 
-fn table_division(mut num: u64, factors: &mut Vec<u64>) {
-    if num < 2 {
-        return;
-    }
-    while num % 2 == 0 {
-        num /= 2;
-        factors.push(2);
+fn factor(mut n: u64) -> Factors {
+    let mut factors = Factors::new();
+
+    if n < 2 {
+        factors.push(n);
+        return factors;
     }
-    if num == 1 {
-        return;
+
+    let z = n.trailing_zeros();
+    if z > 0 {
+        factors.add(2, z as u8);
+        n >>= z;
     }
-    if is_prime(num) {
-        factors.push(num);
-        return;
+
+    if n == 1 {
+        return factors;
     }
-    for &(prime, inv, ceil) in P_INVS_U64 {
-        if num == 1 {
-            break;
-        }
 
-        // inv = prime^-1 mod 2^64
-        // ceil = floor((2^64-1) / prime)
-        // if (num * inv) mod 2^64 <= ceil, then prime divides num
-        // See http://math.stackexchange.com/questions/1251327/
-        // for a nice explanation.
-        loop {
-            let Wrapping(x) = Wrapping(num) * Wrapping(inv); // x = num * inv mod 2^64
-            if x <= ceil {
-                num = x;
-                factors.push(prime);
-                if is_prime(num) {
-                    factors.push(num);
-                    return;
-                }
-            } else {
-                break;
-            }
-        }
+    let (f, n) = table::factor(n);
+    factors *= f;
+
+    if n >= table::NEXT_PRIME {
+        factors *= rho::factor(n);
     }
 
-    // do we still have more factoring to do?
-    // Decide whether to use Pollard Rho or slow divisibility based on
-    // number's size:
-    //if num >= 1 << 63 {
-    // number is too big to use rho pollard without overflowing
-    //trial_division_slow(num, factors);
-    //} else if num > 1 {
-    // number is still greater than 1, but not so big that we have to worry
-    rho_pollard_factor(num, factors);
-    //}
+    factors
 }
 
 fn print_factors(num: u64) {
-    print!("{}:", num);
-
-    let mut factors = Vec::new();
-    // we always start with table division, and go from there
-    table_division(num, &mut factors);
-    factors.sort();
-
-    for fac in &factors {
-        print!(" {}", fac);
-    }
+    print!("{}:{}", num, factor(num));
     println!();
 }
 

src/uu/factor/src/miller_rabin.rs

@@ -0,0 +1,88 @@
+use crate::numeric::*;
+
+// Small set of bases for the Miller-Rabin prime test, valid for all 64b integers;
+//  discovered by Jim Sinclair on 2011-04-20, see miller-rabin.appspot.com
+#[allow(clippy::unreadable_literal)]
+const BASIS: [u64; 7] = [2, 325, 9375, 28178, 450775, 9780504, 1795265022];
+
+#[derive(Eq, PartialEq)]
+pub(crate) enum Result {
+    Prime,
+    Pseudoprime,
+    Composite(u64),
+}
+
+impl Result {
+    pub(crate) fn is_prime(&self) -> bool {
+        *self == Result::Prime
+    }
+}
+
+// Deterministic Miller-Rabin primality-checking algorithm, adapted to extract
+// (some) dividers; it will fail to factor strong pseudoprimes.
+#[allow(clippy::many_single_char_names)]
+pub(crate) fn test<A: Arithmetic>(n: u64) -> Result {
+    use self::Result::*;
+
+    if n < 2 {
+        return Pseudoprime;
+    }
+    if n % 2 == 0 {
+        return if n == 2 { Prime } else { Composite(2) };
+    }
+
+    // n-1 = r 2ⁱ
+    let i = (n - 1).trailing_zeros();
+    let r = (n - 1) >> i;
+
+    for a in BASIS.iter() {
+        let a = a % n;
+        if a == 0 {
+            break;
+        }
+
+        // x = a^r mod n
+        let mut x = A::pow(a, r, n);
+
+        {
+            // y = ((x²)²...)² i times = x ^ (2ⁱ) = a ^ (r 2ⁱ) = x ^ (n - 1)
+            let mut y = x;
+            for _ in 0..i {
+                y = A::mul(y, y, n)
+            }
+            if y != 1 {
+                return Pseudoprime;
+            };
+        }
+
+        if x == 1 || x == n - 1 {
+            break;
+        }
+
+        loop {
+            let y = A::mul(x, x, n);
+            if y == 1 {
+                return Composite(gcd(x - 1, n));
+            }
+            if y == n - 1 {
+                // This basis element is not a witness of `n` being composite.
+                // Keep looking.
+                break;
+            }
+            x = y;
+        }
+    }
+
+    Prime
+}
+
+// Used by build.rs' tests
+#[allow(dead_code)]
+pub(crate) fn is_prime(n: u64) -> bool {
+    if n < 1 << 63 {
+        test::<Small>(n)
+    } else {
+        test::<Big>(n)
+    }
+    .is_prime()
+}