-
Notifications
You must be signed in to change notification settings - Fork 1
/
expand_kara.c
630 lines (577 loc) · 16.7 KB
/
expand_kara.c
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
/* This file is part of the MAYLIB libray.
Copyright 2007-2018 Patrick Pelissier
This Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
This Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with th Library; see the file COPYING.LESSER.txt.
If not, write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */
#include "may-impl.h"
/* Karatsuba multiplication for multivariate and univariate polynomials */
/* dump */
#if 0
static void
dump (may_linked_term_t a, int n)
{
for ( ;a!= NULL; a = a->next) {
printf ("[");
for (int i = 0; i < n ;i ++)
printf ("%lu%c", (unsigned long) a->expo[i], (i==(n-1)) ? ']' : ',');
may_dump (a->coeff);
}
}
#endif
/* Allocate a new uninitialized linked term */
MAY_INLINE may_linked_term_t
new_term (int n)
{
may_linked_term_t t;
if (may_g.kara.freeterm == NULL) {
t = MAY_ALLOC(sizeof (struct may_linked_term_s) + n*sizeof (may_karatsuba_unsigned_long));
t->coeff = MAY_DUMMY;
} else {
t = may_g.kara.freeterm;
may_g.kara.freeterm = may_g.kara.freeterm->next;
}
MAY_ASSERT ((t->coeff == MAY_DUMMY) || !MAY_EVAL_P (t->coeff));
return t;
}
/* Free and next */
MAY_INLINE may_linked_term_t
free_and_next (may_linked_term_t a)
{
may_linked_term_t b = a->next;
a->next = may_g.kara.freeterm;
may_g.kara.freeterm = a;
return b;
}
/* Compare two linked_term_t */
MAY_INLINE int
compare (may_linked_term_t a, may_linked_term_t b, int n)
{
MAY_ASSERT (n > 0);
for (int i = 0; i < n; i ++)
if (a->expo[i] != b->expo[i])
return a->expo[i] < b->expo[i] ? -1 : 1;
return 0;
}
/* Sort the linked list.
Use a merge sort */
static may_linked_term_t
sort (may_linked_term_t a, int n)
{
may_linked_term_t b, c, d;
/* If there is no element, one element */
if (a == NULL || a->next == NULL)
return a;
/* If there is 2 elements */
if (a->next->next == NULL) {
if (compare (a, a->next, n) >= 0)
return a;
b = a->next;
b->next = a;
a->next = NULL;
return b;
}
/* Split */
b = c = NULL;
while (a != NULL) {
d = a->next;
a->next = b;
b = a;
if (d == NULL)
break;
a = d->next;
d->next = c;
c = d;
}
/* Sort recursive */
b = sort (b, n);
c = sort (c, n);
/* Merge */
may_linked_term_t *ap = &a;
while (b != NULL && c != NULL) {
if (compare (b, c, n) >= 0) {
*ap = b;
ap = &(b->next);
b = b->next;
} else {
*ap = c;
ap = &(c->next);
c = c->next;
}
}
*ap = (b != NULL) ? b : c;
return a;
}
/* Convertion of an expression composed of n variables varlist[n]
to a linked list of term.
Return NULL if the convertion failed */
static may_linked_term_t
convert_to_linked_term (may_t x, int n, may_t varlist[n])
{
MAY_ASSERT ((MAY_FLAGS(x) & MAY_EXPAND_F) == MAY_EXPAND_F);
may_linked_term_t list = NULL;
may_iterator_t it;
may_t num, cx, bx;
for (num = may_sum_iterator_init (it, x) ;
may_sum_iterator_end (&cx, &bx, it) ;
may_sum_iterator_next (it)) {
may_iterator_t it2;
may_t bx2, pw2;
may_linked_term_t tmp = new_term (n);
tmp->coeff = may_num_set (MAY_DUMMY, cx);
for (int i = 0; i < n; i++)
tmp->expo[i] = 0;
for (may_product_iterator_init (it2, bx) ;
may_product_iterator_end (&pw2, &bx2, it2) ;
may_product_iterator_next (it2)) {
unsigned long p;
/* If we can't convert the exponent into an integer
between 0 and ULONG_MAX/2, abort karatsuba */
if (may_get_ui (&p, pw2) && p < (((may_karatsuba_unsigned_long)-1)/2) )
return NULL;
for (int i = 0; i < n; i++) {
if (may_identical (varlist[i], bx2) == 0) {
tmp->expo[i] = p;
goto convertion_done;
}
} /* for i */
/* Fail to find a correspondance between the factor and the list
Can not convert */
return NULL;
convertion_done:
(void) 0;
} /* for product iterator */
/* Add the new term into the list */
tmp->next = list;
list = tmp;
} /* for sum iterator */
/* The numerical constant term */
if (!MAY_ZERO_P (num)) {
may_linked_term_t tmp = new_term (n);
tmp->coeff = may_num_set (MAY_DUMMY, num);
for (int i = 0; i < n; i++)
tmp->expo[i] = 0;
tmp->next = list;
list = tmp;
}
/* Sort list */
list = sort (list, n);
return list;
}
/* Convertion from an expression composed of n variables varlist[n]
to a linked list of term */
static may_t
convert_from_linked_term (may_linked_term_t term, int n, may_t varlist[n])
{
may_t s = MAY_ZERO;
for (; term != NULL; term = term->next) {
if (may_num_zero_p (term->coeff))
continue;
may_mark ();
may_t t = MAY_NODE_C (MAY_PRODUCT_T, n+1);
for (int i = 0; i < n; i++)
MAY_SET_AT (t, i, may_pow_c (varlist[i], may_set_ui (term->expo[i])));
MAY_SET_AT (t, n, term->coeff);
t = may_compact (may_eval (t));
MAY_SET_FLAG (t, MAY_EXPAND_F);
s = may_addinc_c (s, t);
}
s = may_eval (s);
MAY_SET_FLAG (s, MAY_EXPAND_F);
return s;
}
/* Split a linked list of term in two (reusing the memory).
i is the index of the variable to split (in 0..n-1)
It will separate according to the least significant bit of the exponent,
so that A[X_i^2]*X_i+B(X_i^2) = org(X_i) */
static void
split (may_linked_term_t *a, may_linked_term_t *b,
may_linked_term_t org, int i)
{
may_linked_term_t *ap = a;
may_linked_term_t *bp = b;
unsigned int as = 0, bs = 0;
MAY_ASSERT (org != NULL);
while (org != NULL) {
/* Check the least significant bit */
if ((org->expo[i] & 1) == 0) {
/* Add it in B keeping the order */
*bp = org;
bp = &(org->next);
bs++;
} else {
/* Add it in A keeping the order */
*ap = org;
ap = &(org->next);
as++;
}
/* Reduce the exponent */
org->expo[i] = (org->expo[i] / 2);
/* Next */
org = org->next;
}
/* Terminate the lists */
*ap = NULL;
*bp = NULL;
/* Let's see if the split was a success or not.
If not, let's increase the threshold so that we will use the base case method
for such sizes */
if (!(as <= 2*bs && bs <= 2*as)) {
if (!(as <= 3*bs && bs <= 3*as))
may_g.kara.threshold = 3*(as+bs+1);
else if (!(as <= 4*bs && bs <= 4*as))
may_g.kara.threshold = 3*(as+bs+1);
else
may_g.kara.threshold = as+bs+1;
}
}
/* Merge two list so that the returned multivariate poly is
A[X_i^2]*X_i^2+B[X_i^2]*X_i+C(X_i^2) (Reusing memory)
A & B may overlap. C doesn't overlap with B and C */
static may_linked_term_t
merge (may_linked_term_t a, may_linked_term_t b, may_linked_term_t c, int i, int n)
{
may_linked_term_t *sp;
/* Fix the degree of A for X_i */
may_linked_term_t tmp = a;
while (tmp != NULL) {
tmp->expo[i] ++;
tmp = tmp->next;
}
/* Fix the degree of B for X_i */
tmp = b;
while (tmp != NULL) {
tmp->expo[i] = 2 * tmp->expo[i] + 1;
tmp = tmp->next;
}
/* First sum A and C and fix the degree for X_i */
sp = &tmp;
while (a != NULL && c != NULL) {
for (int j = 0; j < n ; j++) {
if (a->expo[j] > c->expo[j]) {
/* A is bigger than C */
*sp = a;
a->expo[i] = 2*a->expo[i];
sp = &(a->next);
a = a->next;
goto next_term;
}else if (a->expo[j] < c->expo[j]) {
/* C is bigger than A */
*sp = c;
c->expo[i] = 2*c->expo[i];
sp = &(c->next);
c = c->next;
goto next_term;
}
}
/* Both exponents are equals: sum coeff */
MAY_ASSERT (!MAY_EVAL_P (a->coeff) && !MAY_EVAL_P (c->coeff));
*sp = a;
a->expo[i] = 2*a->expo[i];
sp = &(a->next);
a->coeff = may_num_add (a->coeff, a->coeff, c->coeff);
a = a->next;
c = free_and_next (c);
next_term:
(void) 0;
} /* while a and c != NULL */
while (a != NULL) {
*sp = a;
a->expo[i] = 2*a->expo[i];
sp = &(a->next);
a = a->next;
}
while (c != NULL) {
*sp = c;
c->expo[i] = 2*c->expo[i];
sp = &(c->next);
c = c->next;
}
*sp = NULL;
/* Merge tmp and b */
a = tmp;
sp = &tmp;
while (a != NULL && b != NULL) {
for (int j = 0; j < n ; j++) {
if (a->expo[j] > b->expo[j]) {
*sp = a;
sp = &(a->next);
a = a->next;
break;
}else if (a->expo[j] < b->expo[j]) {
*sp = b;
sp = &(b->next);
b = b->next;
break;
}
}
/* Both exponents are equals. This case is not possible */
} /* while (a!= NULL && b != NULL) */
*sp = (a!=NULL) ? a : b;
return tmp;
}
/* Sum two linked list of n variables
Allocating a new area of memory
*/
static may_linked_term_t
add (may_linked_term_t a, may_linked_term_t b, int n)
{
may_linked_term_t *sp;
may_linked_term_t tmp, ret;
int i;
sp = &ret;
while (a != NULL && b != NULL) {
tmp = new_term (n);
*sp = tmp;
sp = &(tmp->next);
MAY_ASSERT (n > 0);
for (i = 0; i < n ; i++) {
if (a->expo[i] > b->expo[i]) {
tmp->coeff = may_num_set (tmp->coeff, a->coeff);
memcpy (tmp->expo, a->expo, n*sizeof (may_karatsuba_unsigned_long));
a = a->next;
goto next_term;
}else if (a->expo[i] < b->expo[i]) {
tmp->coeff = may_num_set (tmp->coeff, b->coeff);
memcpy (tmp->expo, b->expo, n*sizeof (may_karatsuba_unsigned_long));
b = b->next;
goto next_term;
}
}
/* Both exponents are equals */
memcpy (tmp->expo, a->expo, n*sizeof (may_karatsuba_unsigned_long));
tmp->coeff = may_num_add (tmp->coeff, a->coeff, b->coeff);
a = a->next;
b = b->next;
next_term:
(void) 0;
}
/* Add the latest terms */
for ( ; a!=NULL ; a = a->next) {
tmp = new_term (n);
tmp->coeff = may_num_set (tmp->coeff, a->coeff);
memcpy (tmp->expo, a->expo, n*sizeof (may_karatsuba_unsigned_long));
*sp = tmp;
sp = &(tmp->next);
}
for ( ; b!=NULL ; b = b->next) {
tmp = new_term (n);
tmp->coeff = may_num_set (tmp->coeff, b->coeff);
memcpy (tmp->expo, b->expo, n*sizeof (may_karatsuba_unsigned_long));
*sp = tmp;
sp = &(tmp->next);
}
/* Terminate the list */
*sp = NULL;
return ret;
}
/* Substract two linked list of n variables
Reusing the memory if possible (even the coefficient) */
static may_linked_term_t
sub (may_linked_term_t a, may_linked_term_t b, int n)
{
may_linked_term_t *sp;
may_linked_term_t ret;
int i;
sp = &ret;
while (a != NULL && b != NULL) {
MAY_ASSERT (n > 0);
for (i = 0; i < n ; i++) {
if (a->expo[i] > b->expo[i]) {
/* Reuse a */
*sp = a;
sp = &(a->next);
a = a->next;
goto next_term;
}else if (a->expo[i] < b->expo[i]) {
/* Reuse b */
*sp = b;
sp = &(b->next);
b->coeff = may_num_neg (b->coeff, b->coeff);
b = b->next;
goto next_term;
}
}
/* Both exponents are equals: reuse a or b and free the other */
a->coeff = may_num_sub (a->coeff, a->coeff, b->coeff);
if (may_num_zero_p (a->coeff)) {
/* Result is zero. Don't add this term. Free A and B */
a = free_and_next (a);
b = free_and_next (b);
} else {
*sp = a;
sp = &(a->next);
b = free_and_next (b);
a = a->next;
}
next_term:
(void) 0;
}
/* Add the latest terms */
if (a!=NULL) {
*sp = a; /* Auto terminate the list */
} else {
for ( ; b!=NULL ; b = b->next) {
*sp = b;
sp = &(b->next);
b->coeff = may_num_neg (b->coeff, b->coeff);
b = b->next;
}
/* Terminate the list */
*sp = NULL;
}
return ret;
}
/* Perform basecase multiplication */
static may_linked_term_t
basecase (may_linked_term_t a2, may_linked_term_t b2, int n)
{
may_linked_term_t product = NULL;
while (a2 != NULL) {
/* Multiply A2[] by B2 and add it in product */
/* Try to reuse memory if possible for a but not for b */
may_linked_term_t *sp;
may_linked_term_t a, b, tmp;
int i;
a = product;
b = b2;
sp = &product;
while (a != NULL && b != NULL) {
MAY_ASSERT (n > 0);
for (i = 0; i < n ; i++) {
if (a->expo[i] > b->expo[i]+a2->expo[i] ) {
/* a exponent is bigger: reuse a */
*sp = a;
sp = &(a->next);
a = a->next;
goto next_term;
}else if (a->expo[i] < b->expo[i]+a2->expo[i]) {
/* b+a2 is bigger. Allocate a new term */
tmp = new_term (n);
tmp->coeff = may_num_mul (tmp->coeff,
b->coeff, a2->coeff);
*sp = tmp;
sp = &(tmp->next);
for (i = 0; i < n; i++)
tmp->expo[i] = b->expo[i] + a2->expo[i];
b = b->next;
goto next_term;
}
}
/* Both exponents are equals: reuse a */
*sp = a;
sp = &(a->next);
a->coeff = may_num_add (a->coeff,
a->coeff,
may_g.kara.tmpnum = may_num_mul (may_g.kara.tmpnum,
b->coeff, a2->coeff) );
a = a->next;
b = b->next;
next_term:
(void) 0;
}
/* Add the latest terms */
if (a!=NULL) {
*sp = a; /* Auto terminate the list */
} else {
for ( ; b!=NULL ; b = b->next) {
tmp = new_term (n);
tmp->coeff = may_num_mul (tmp->coeff, b->coeff, a2->coeff);
*sp = tmp;
sp = &(tmp->next);
for (i = 0; i < n; i++)
tmp->expo[i] = b->expo[i] + a2->expo[i];
}
/* Terminate the list */
*sp = NULL;
}
/* Add A2 in the free term list for later reallocating use */
a2 = free_and_next (a2);
} /* while a2 */
/* B2 wasn't used to create A. Add them in the free term list */
while (b2 != NULL)
b2 = free_and_next (b2);
return product;
}
/* Test if size is small than n */
static int
size_smaller_than_p (may_linked_term_t a, unsigned long n)
{
unsigned long s = 0;
for ( ;a != NULL; a = a->next)
if (++s > n)
return 0;
return 1;
}
/* Perform Karatsuba multiplication */
static may_linked_term_t
karatsuba (may_linked_term_t a, may_linked_term_t b, int i, int n, may_mark_t mark)
{
may_linked_term_t a1, a2, b1, b2;
may_linked_term_t a1pa2,b1pb2,a1pa2fb1pb2,a1fb1,a2fb2;
/* If a or b is too small, use basecase */
if (size_smaller_than_p (a, may_g.kara.threshold)
|| size_smaller_than_p (b, may_g.kara.threshold))
return basecase (a, b, n);
/* Select another variable to split: j = (i+1) % n */
int j = (i+1) % n;
MAY_SPAWN_BLOCK(block, mark);
MAY_SPAWN(block, (a,i,n), {
/* A(X_i) = A1(X_i^2)*X_i+A2(X_i^2) */
split (&a1, &a2, a, i);
/* (A1+A2)*(B1+B2) with j */
a1pa2 = add (a1, a2, n);
}, (a1, a2, a1pa2));
/* B(X_i) = B1(X_i^2)*X_i+B2(X_i^2) */
split (&b1, &b2, b, i);
/* B(X_i) = B1(X_i^2)*X_i+B2(X_i^2) */
b1pb2 = add (b1, b2, n);
MAY_SPAWN_SYNC(block);
MAY_SPAWN(block, (a1pa2, b1pb2, j, n), {
a1pa2fb1pb2 = karatsuba(a1pa2, b1pb2, j, n, mark);
}, (a1pa2fb1pb2));
MAY_SPAWN(block, (a1, b1, j, n), {
/* A1*B1 with j */
a1fb1 = karatsuba (a1, b1, j, n, mark);
}, (a1fb1));
/* A2*B2 with j */
a2fb2 = karatsuba (a2, b2, j, n, mark);
MAY_SPAWN_SYNC(block);
/* Result(X_i) = A1*B1(X_i^2)*X_i^2+(((A1+A2)*(B1+B2)-A1*B1-A2*B2)(X_i^2)*X_i+A2*B2(X_i^2) */
a1pa2fb1pb2 = sub (a1pa2fb1pb2, add (a1fb1, a2fb2, n), n);
return merge (a1fb1, a1pa2fb1pb2, a2fb2, i, n);
}
may_t
may_karatsuba (may_t a, may_t b, may_t varlist)
{
MAY_LOG_FUNC (("a='%Y' b='%Y' varlist='%Y", a, b, varlist));
may_mark_t mark;
may_mark (mark);
/* Define the threshold for which below we call the basecase.
It is a dynamic threshold and may be increase if the function failed for some
part of the input */
may_g.kara.threshold = 10;
/* Define the cache used by the base case */
may_g.kara.tmpnum = MAY_DUMMY;
/* Define the list of free terms */
may_g.kara.freeterm = NULL;
int n = may_nops (varlist);
may_linked_term_t ac = convert_to_linked_term (a, n, MAY_AT_PTR (varlist, 0));
if (ac == NULL)
return NULL;
may_linked_term_t bc = convert_to_linked_term (b, n, MAY_AT_PTR (varlist, 0));
if (bc == NULL)
return NULL;
ac = karatsuba (ac, bc, 0, n, mark);
MAY_LOG_MSG (("kara.threshold =%lu", may_g.kara.threshold));
return may_keep (mark, convert_from_linked_term (ac, n, MAY_AT_PTR (varlist, 0)));
}