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To: coreutils@HIDDEN, bug-coreutils@HIDDEN
From: Denys Vlasenko <dvlasenk@HIDDEN>
Subject: factor: missed optimizations
Message-ID: <9436ddae-b31b-7e1f-30a2-b56746344d85@HIDDEN>
Date: Sun, 20 Dec 2020 17:16:49 +0100
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--------------CF1065E22818613C93660811
Content-Type: text/plain; charset=utf-8; format=flowed
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factor ---debug output is rather rudimentary.
In order to understand the logic, I added more printouts
(try attached debug patch)
and immediately noticed some obvious snafus:
$ src/factor ---debug 340282366920938461286658806734041124249
[using arbitrary-precision arithmetic] 340282366920938461286658806734041124249:[trial division 340282366920938461286658806734041124249] [trial stopped 4999(max)] [is number 34028236692093846128665880673404112424
9 prime?] [no] [pollard-rho (1)] 1:1:2:4:8:16:32:64:128:256:512:1024:2048:4096:8192:16384:32768:65536:131072:262144:524288:1048576:2097152:4194304:8388608:16777216:33554432:67108864:134217728:268435456:536870912
[factor found][rho factor 18446744073709551557] [trial division 18446744073709551556] [factor 2] [factor 2] [factor 11] [factor 137] [factor 547] [trial stopped 4999(max)] [is number 5594472617641 prime?] [trial
division 5594472617640] [factor 2] [factor 2] [factor 2] [factor 3] [factor 5] [factor 1427] [factor 2131] [trial stopped 2131] [is number 15331 prime?] [yes] [factor 15331] [yes] ...
Look at the ens of the last line.
We don't need to check whether 15331 is prime. We know it is: we just stopped
trial division because next prime divisor to try is larger than
sqrt(number_we_try_to_factor) - therefore number_we_try_to_factor
has no more divisors, ergo it's prime. The fix:
for (unsigned int i = 1; i <= PRIMES_PTAB_ENTRIES;)
{
if (! mpz_divisible_ui_p (t, p))
{
p += primes_diff[i++];
if (mpz_cmp_ui (t, p * p) < 0)
+ mp_factor_insert (factors, t);
+ mpz_set_ui (t, 1);
break;
}
else
... [factor 5594472617641] [factor
18446744073709551557] [trial division 18446744073709551556] [factor 2] [factor 2] [factor 11] [factor 137] [factor 547] [trial stopped 4999(max)] [is number 5594472617641 prime?] [trial division 5594472617640]
[factor 2] [factor 2] [factor 2] [factor 3] [factor 5] [factor 1427] [factor 2131] [trial stopped 2131] [is number 15331 prime?] [yes] [factor 15331] [yes] [factor 5594472617641] [factor 18446744073709551557] [p
ollard-rho end]
18446744073709551557 18446744073709551557
Second optimization is this: as you see, I deliberately fed it a square
(the factorization result is 18446744073709551557^2).
It happily did not notice it and spent an awful lot of time in pollard rho:
1:1:2:4:8:16:32:64:128:256:512:1024:2048:4096:8192:16384:32768:65536:131072:262144:524288:1048576:2097152:4194304:8388608:16777216:33554432:67108864:134217728:268435456:536870912
A quick check for square drastically cuts down execution time:
/* Use Pollard-rho to compute the prime factors of
arbitrary-precision T, and put the results in FACTORS. */
static void
+mp_factor_big (mpz_t t, struct mp_factors *factors)
+{
+ /* the _big() function assumes trial division was already tried */
+ devmsg ("[is number prime?] ");
+ if (mp_prime_p (t))
+ mp_factor_insert (factors, t);
+ else
+ if (mpz_perfect_square_p (t))
+ {
+ struct mp_factors f2;
+ devmsg ("[it is square] ");
+ mpz_sqrt (t, t);
+ mp_factor_init (&f2);
+ mp_factor_big (t, &f2);
+ for (unsigned long int i = 0; i < f2.nfactors; i++)
+ {
+ unsigned long int e = f2.e[i];
+ while (e--) { //
+ mp_factor_insert (factors, f2.p[i]); //FIXME: obvious optimization:
+ mp_factor_insert (factors, f2.p[i]); // need to introduce/use mp_factor_insert_multiplicity(..., e*2);
+ }
+ }
+ mp_factor_clear (&f2);
+ }
+ else
+ mp_factor_using_pollard_rho (t, 1, factors);
+}
+
+static void
mp_factor (mpz_t t, struct mp_factors *factors)
{
mp_factor_init (factors);
@@ -2265,11 +2309,7 @@ mp_factor (mpz_t t, struct mp_factors *f
.
if (mpz_cmp_ui (t, 1) != 0)
{
- devmsg ("[is number prime?] ");
- if (mp_prime_p (t))
- mp_factor_insert (factors, t);
- else
- mp_factor_using_pollard_rho (t, 1, factors);
+ mp_factor_big (t, factors);
}
}
}
--------------CF1065E22818613C93660811
Content-Type: text/x-patch;
name="zz.diff"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment;
filename="zz.diff"
diff -urp coreutils-8.32/src/factor.c coreutils-8.32-TT/src/factor.c
--- coreutils-8.32/src/factor.c 2020-01-01 15:13:12.000000000 +0100
+++ coreutils-8.32-TT/src/factor.c 2020-12-20 17:08:33.022081567 +0100
@@ -620,6 +620,8 @@ mp_factor_insert (struct mp_factors *fac
unsigned long int *e = factors->e;
long i;
+gmp_printf ("[factor %Zd] ", prime);
+
/* Locate position for insert new or increment e. */
for (i = nfactors - 1; i >= 0; i--)
{
@@ -842,7 +844,8 @@ mp_factor_using_division (mpz_t t, struc
mpz_t q;
unsigned long int p;
- devmsg ("[trial division] ");
+gmp_printf ("[trial division %Zd] ", t);
+// devmsg ("[trial division] ");
mpz_init (q);
@@ -861,6 +864,8 @@ mp_factor_using_division (mpz_t t, struc
{
p += primes_diff[i++];
if (mpz_cmp_ui (t, p * p) < 0)
+ mp_factor_insert (factors, t);
+ mpz_set_ui (t, 1);
break;
}
else
@@ -1407,6 +1412,7 @@ mp_prime_p (mpz_t n)
if (!mp_millerrabin (n, nm1, a, tmp, q, k))
{
is_prime = false;
+gmp_printf ("[ML1:no] ");
goto ret2;
}
@@ -1414,6 +1420,7 @@ mp_prime_p (mpz_t n)
{
/* Factor n-1 for Lucas. */
mpz_set (tmp, nm1);
+gmp_printf ("[ML does not know, trying Lucas] ");
mp_factor (tmp, &factors);
}
@@ -1438,13 +1445,16 @@ mp_prime_p (mpz_t n)
}
if (is_prime)
+{gmp_printf ("[Lucas:yes] ");
goto ret1;
+}
mpz_add_ui (a, a, primes_diff[r]); /* Establish new base. */
if (!mp_millerrabin (n, nm1, a, tmp, q, k))
{
is_prime = false;
+gmp_printf ("[ML2:no] ");
goto ret1;
}
}
@@ -1692,6 +1702,7 @@ mp_factor_using_pollard_rho (mpz_t n, un
{
for (;;)
{
+printf("%llu", k);
do
{
mpz_mul (t, x, x);
@@ -1715,6 +1726,7 @@ mp_factor_using_pollard_rho (mpz_t n, un
mpz_set (z, x);
k = l;
l = 2 * l;
+devmsg(":");
for (unsigned long long int i = 0; i < k; i++)
{
mpz_mul (t, x, x);
@@ -1725,6 +1737,7 @@ mp_factor_using_pollard_rho (mpz_t n, un
}
factor_found:
+gmp_printf("[factor found]");
do
{
mpz_mul (t, y, y);
@@ -1738,6 +1751,7 @@ mp_factor_using_pollard_rho (mpz_t n, un
mpz_divexact (n, n, t); /* divide by t, before t is overwritten */
+gmp_printf("[rho factor %Zd] ", t);
if (!mp_prime_p (t))
{
devmsg ("[composite factor--restarting pollard-rho] ");
@@ -1759,6 +1773,7 @@ mp_factor_using_pollard_rho (mpz_t n, un
mpz_mod (y, y, n);
}
+ devmsg ("[pollard-rho end] ");
mpz_clears (P, t2, t, z, x, y, NULL);
}
#endif
@@ -2255,6 +2270,35 @@ factor (uintmax_t t1, uintmax_t t0, stru
/* Use Pollard-rho to compute the prime factors of
arbitrary-precision T, and put the results in FACTORS. */
static void
+mp_factor_big (mpz_t t, struct mp_factors *factors)
+{
+ /* the _big() function assumes trial division was already tried */
+ devmsg ("[is number prime?] ");
+ if (mp_prime_p (t))
+ mp_factor_insert (factors, t);
+ else
+ if (mpz_perfect_square_p (t))
+ {
+ struct mp_factors f2;
+ devmsg ("[it is square] ");
+ mpz_sqrt (t, t);
+ mp_factor_init (&f2);
+ mp_factor_big (t, &f2);
+ for (unsigned long int i = 0; i < f2.nfactors; i++)
+ {
+ unsigned long int e = f2.e[i];
+ while (e--) {
+ mp_factor_insert (factors, f2.p[i]);
+ mp_factor_insert (factors, f2.p[i]);
+ }
+ }
+ mp_factor_clear (&f2);
+ }
+ else
+ mp_factor_using_pollard_rho (t, 1, factors);
+}
+
+static void
mp_factor (mpz_t t, struct mp_factors *factors)
{
mp_factor_init (factors);
@@ -2265,11 +2309,7 @@ mp_factor (mpz_t t, struct mp_factors *f
if (mpz_cmp_ui (t, 1) != 0)
{
- devmsg ("[is number prime?] ");
- if (mp_prime_p (t))
- mp_factor_insert (factors, t);
- else
- mp_factor_using_pollard_rho (t, 1, factors);
+ mp_factor_big (t, factors);
}
}
}
@@ -2599,6 +2639,8 @@ do_stdin (void)
int
main (int argc, char **argv)
{
+ setvbuf(stdout, NULL, _IONBF, 0);
+
initialize_main (&argc, &argv);
set_program_name (argv[0]);
setlocale (LC_ALL, "");
--------------CF1065E22818613C93660811--
Denys Vlasenko <dvlasenk@HIDDEN>:bug-coreutils@HIDDEN.
Full text available.bug-coreutils@HIDDEN:bug#45343; Package coreutils.
Full text available.
GNU bug tracking system
Copyright (C) 1999 Darren O. Benham,
1997 nCipher Corporation Ltd,
1994-97 Ian Jackson.