| blob | 2bb06d49ca344324b9ff56928cb48812209d2290 |
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2012-2013 Ecole Normale Superieure
4 * Copyright 2014-2015 INRIA Rocquencourt
5 * Copyright 2016 Sven Verdoolaege
6 *
7 * Use of this software is governed by the MIT license
8 *
9 * Written by Sven Verdoolaege, K.U.Leuven, Departement
10 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
11 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
12 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
13 * B.P. 105 - 78153 Le Chesnay, France
14 */
16 #include <isl_ctx_private.h>
17 #include <isl_map_private.h>
19 #include <isl/map.h>
20 #include <isl_seq.h>
22 #include <isl_space_private.h>
23 #include <isl_mat_private.h>
24 #include <isl_vec_private.h>
26 #include <bset_to_bmap.c>
27 #include <bset_from_bmap.c>
28 #include <set_to_map.c>
29 #include <set_from_map.c>
32 {
36 }
39 {
44 }
45 }
48 {
51 }
53 /* Drop n dimensions starting at first.
54 *
55 * In principle, this frees up some extra variables as the number
56 * of columns remains constant, but we would have to extend
57 * the div array too as the number of rows in this array is assumed
58 * to be equal to extra.
59 */
62 {
64 }
66 /* Move "n" divs starting at "first" to the end of the list of divs.
67 */
70 {
88 error:
91 }
93 /* Drop "n" dimensions of type "type" starting at "first".
94 *
95 * In principle, this frees up some extra variables as the number
96 * of columns remains constant, but we would have to extend
97 * the div array too as the number of rows in this array is assumed
98 * to be equal to extra.
99 */
102 {
146 error:
149 }
153 {
156 }
160 {
181 }
185 error:
188 }
192 {
194 }
196 /*
197 * We don't cow, as the div is assumed to be redundant.
198 */
201 {
220 }
222 }
235 }
241 error:
244 }
248 {
250 isl_int gcd;
263 }
266 }
274 }
276 }
284 }
287 }
294 }
298 }
302 {
305 }
307 /* Reduce the coefficient of the variable at position "pos"
308 * in integer division "div", such that it lies in the half-open
309 * interval (1/2,1/2], extracting any excess value from this integer division.
310 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
311 * corresponds to the constant term.
312 *
313 * That is, the integer division is of the form
314 *
315 * floor((... + (c * d + r) * x_pos + ...)/d)
316 *
317 * with -d < 2 * r <= d.
318 * Replace it by
319 *
320 * floor((... + r * x_pos + ...)/d) + c * x_pos
321 *
322 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
323 * Otherwise, c = floor((c * d + r)/d) + 1.
324 *
325 * This is the same normalization that is performed by isl_aff_floor.
326 */
329 {
330 isl_int shift;
345 }
347 /* Does the coefficient of the variable at position "pos"
348 * in integer division "div" need to be reduced?
349 * That is, does it lie outside the half-open interval (1/2,1/2]?
350 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
351 * 2 * c != d.
352 */
355 {
356 isl_bool r;
368 }
370 /* Reduce the coefficients (including the constant term) of
371 * integer division "div", if needed.
372 * In particular, make sure all coefficients lie in
373 * the half-open interval (1/2,1/2].
374 */
377 {
382 isl_bool reduce;
392 }
395 }
397 /* Reduce the coefficients (including the constant term) of
398 * the known integer divisions, if needed
399 * In particular, make sure all coefficients lie in
400 * the half-open interval (1/2,1/2].
401 */
404 {
418 }
421 }
423 /* Remove any common factor in numerator and denominator of the div expression,
424 * not taking into account the constant term.
425 * That is, if the div is of the form
426 *
427 * floor((a + m f(x))/(m d))
428 *
429 * then replace it by
430 *
431 * floor((floor(a/m) + f(x))/d)
432 *
433 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
434 * and can therefore not influence the result of the floor.
435 */
437 {
453 }
455 /* Remove any common factor in numerator and denominator of a div expression,
456 * not taking into account the constant term.
457 * That is, look for any div of the form
458 *
459 * floor((a + m f(x))/(m d))
460 *
461 * and replace it by
462 *
463 * floor((floor(a/m) + f(x))/d)
464 *
465 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
466 * and can therefore not influence the result of the floor.
467 */
470 {
482 }
484 /* Assumes divs have been ordered if keep_divs is set.
485 */
488 {
506 }
516 }
525 /* We need to be careful about circular definitions,
526 * so for now we just remove the definition of div k
527 * if the equality contains any divs.
528 * If keep_divs is set, then the divs have been ordered
529 * and we can keep the definition as long as the result
530 * is still ordered.
531 */
539 }
540 }
542 /* Assumes divs have been ordered if keep_divs is set.
543 */
546 {
554 }
556 /* Check if elimination of div "div" using equality "eq" would not
557 * result in a div depending on a later div.
558 */
561 {
576 }
579 }
581 /* Eliminate divs based on equalities
582 */
585 {
600 isl_bool ok;
616 }
617 }
621 }
623 /* Elimininate divs based on inequalities
624 */
627 {
657 }
659 }
661 /* Does the equality constraint at position "eq" in "bmap" involve
662 * any local variables in the range [first, first + n)
663 * that are not marked as having an explicit representation?
664 */
667 {
676 isl_bool unknown;
685 }
688 }
690 /* The last local variable involved in the equality constraint
691 * at position "eq" in "bmap" is the local variable at position "div".
692 * It can therefore be used to extract an explicit representation
693 * for that variable.
694 * Do so unless the local variable already has an explicit representation or
695 * the explicit representation would involve any other local variables
696 * that in turn do not have an explicit representation.
697 * An equality constraint involving local variables without an explicit
698 * representation can be used in isl_basic_map_drop_redundant_divs
699 * to separate out an independent local variable. Introducing
700 * an explicit representation here would block this transformation,
701 * while the partial explicit representation in itself is not very useful.
702 * Set *progress if anything is changed.
703 *
704 * The equality constraint is of the form
705 *
706 * f(x) + n e >= 0
707 *
708 * with n a positive number. The explicit representation derived from
709 * this constraint is
710 *
711 * floor((-f(x))/n)
712 */
715 {
717 isl_bool involves;
741 }
745 {
768 }
777 progress);
784 }
791 }
794 }
798 {
800 progress));
801 }
805 {
811 }
813 }
815 /* Hash table of inequalities in a basic map.
816 * "index" is an array of addresses of inequalities in the basic map, some
817 * of which are NULL. The inequalities are hashed on the coefficients
818 * except the constant term.
819 * "size" is the number of elements in the array and is always a power of two
820 * "bits" is the number of bits need to represent an index into the array.
821 * "total" is the total dimension of the basic map.
822 */
828 };
830 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
831 */
834 {
851 }
853 /* Free the memory allocated by create_constraint_index.
854 */
856 {
858 }
860 /* Return the position in ci->index that contains the address of
861 * an inequality that is equal to *ineq up to the constant term,
862 * provided this address is not identical to "ineq".
863 * If there is no such inequality, then return the position where
864 * such an inequality should be inserted.
865 */
867 {
875 }
877 /* Return the position in ci->index that contains the address of
878 * an inequality that is equal to the k'th inequality of "bmap"
879 * up to the constant term, provided it does not point to the very
880 * same inequality.
881 * If there is no such inequality, then return the position where
882 * such an inequality should be inserted.
883 */
886 {
888 }
892 {
894 }
896 /* Fill in the "ci" data structure with the inequalities of "bset".
897 */
900 {
909 }
912 }
914 /* Is the inequality ineq (obviously) redundant with respect
915 * to the constraints in "ci"?
916 *
917 * Look for an inequality in "ci" with the same coefficients and then
918 * check if the contant term of "ineq" is greater than or equal
919 * to the constant term of that inequality. If so, "ineq" is clearly
920 * redundant.
921 *
922 * Note that hash_index_ineq ignores a stored constraint if it has
923 * the same address as the passed inequality. It is ok to pass
924 * the address of a local variable here since it will never be
925 * the same as the address of a constraint in "ci".
926 */
929 {
936 }
938 /* If we can eliminate more than one div, then we need to make
939 * sure we do it from last div to first div, in order not to
940 * change the position of the other divs that still need to
941 * be removed.
942 */
945 {
999 }
1001 }
1013 }
1016 out:
1020 }
1023 {
1035 }
1037 }
1039 /* Normalize divs that appear in equalities.
1040 *
1041 * In particular, we assume that bmap contains some equalities
1042 * of the form
1043 *
1044 * a x = m * e_i
1045 *
1046 * and we want to replace the set of e_i by a minimal set and
1047 * such that the new e_i have a canonical representation in terms
1048 * of the vector x.
1049 * If any of the equalities involves more than one divs, then
1050 * we currently simply bail out.
1051 *
1052 * Let us first additionally assume that all equalities involve
1053 * a div. The equalities then express modulo constraints on the
1054 * remaining variables and we can use "parameter compression"
1055 * to find a minimal set of constraints. The result is a transformation
1056 *
1057 * x = T(x') = x_0 + G x'
1058 *
1059 * with G a lower-triangular matrix with all elements below the diagonal
1060 * non-negative and smaller than the diagonal element on the same row.
1061 * We first normalize x_0 by making the same property hold in the affine
1062 * T matrix.
1063 * The rows i of G with a 1 on the diagonal do not impose any modulo
1064 * constraint and simply express x_i = x'_i.
1065 * For each of the remaining rows i, we introduce a div and a corresponding
1066 * equality. In particular
1067 *
1068 * g_ii e_j = x_i - g_i(x')
1069 *
1070 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
1071 * corresponding div (if g_kk != 1).
1072 *
1073 * If there are any equalities not involving any div, then we
1074 * first apply a variable compression on the variables x:
1075 *
1076 * x = C x'' x'' = C_2 x
1077 *
1078 * and perform the above parameter compression on A C instead of on A.
1079 * The resulting compression is then of the form
1080 *
1081 * x'' = T(x') = x_0 + G x'
1082 *
1083 * and in constructing the new divs and the corresponding equalities,
1084 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
1085 * by the corresponding row from C_2.
1086 */
1089 {
1098 isl_int v;
1130 }
1131 }
1140 }
1146 }
1156 }
1163 }
1168 /* We have to be careful because dropping equalities may reorder them */
1178 }
1179 }
1185 else
1186 needed++;
1187 }
1193 }
1203 else
1211 else
1214 }
1218 }
1225 done:
1229 error:
1235 }
1239 {
1249 }
1251 /* Check whether it is ok to define a div based on an inequality.
1252 * To avoid the introduction of circular definitions of divs, we
1253 * do not allow such a definition if the resulting expression would refer to
1254 * any other undefined divs or if any known div is defined in
1255 * terms of the unknown div.
1256 */
1259 {
1263 /* Not defined in terms of unknown divs */
1271 }
1273 /* No other div defined in terms of this one => avoid loops */
1281 }
1284 }
1286 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1287 * be a better expression than the current one?
1288 *
1289 * If we do not have any expression yet, then any expression would be better.
1290 * Otherwise we check if the last variable involved in the inequality
1291 * (disregarding the div that it would define) is in an earlier position
1292 * than the last variable involved in the current div expression.
1293 */
1296 {
1313 }
1315 /* Given two constraints "k" and "l" that are opposite to each other,
1316 * except for the constant term, check if we can use them
1317 * to obtain an expression for one of the hitherto unknown divs or
1318 * a "better" expression for a div for which we already have an expression.
1319 * "sum" is the sum of the constant terms of the constraints.
1320 * If this sum is strictly smaller than the coefficient of one
1321 * of the divs, then this pair can be used define the div.
1322 * To avoid the introduction of circular definitions of divs, we
1323 * do not use the pair if the resulting expression would refer to
1324 * any other undefined divs or if any known div is defined in
1325 * terms of the unknown div.
1326 */
1329 {
1334 isl_bool set_div;
1349 else
1354 }
1356 }
1360 {
1364 isl_int sum;
1379 }
1387 }
1402 }
1404 /* We need to break out of the loop after these
1405 * changes since the contents of the hash
1406 * will no longer be valid.
1407 * Plus, we probably we want to regauss first.
1408 */
1416 }
1421 }
1423 /* Detect all pairs of inequalities that form an equality.
1424 *
1425 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1426 * Call it repeatedly while it is making progress.
1427 */
1430 {
1442 }
1444 /* Eliminate knowns divs from constraints where they appear with
1445 * a (positive or negative) unit coefficient.
1446 *
1447 * That is, replace
1448 *
1449 * floor(e/m) + f >= 0
1450 *
1451 * by
1452 *
1453 * e + m f >= 0
1454 *
1455 * and
1456 *
1457 * -floor(e/m) + f >= 0
1458 *
1459 * by
1460 *
1461 * -e + m f + m - 1 >= 0
1462 *
1463 * The first conversion is valid because floor(e/m) >= -f is equivalent
1464 * to e/m >= -f because -f is an integral expression.
1465 * The second conversion follows from the fact that
1466 *
1467 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1468 *
1469 *
1470 * Note that one of the div constraints may have been eliminated
1471 * due to being redundant with respect to the constraint that is
1472 * being modified by this function. The modified constraint may
1473 * no longer imply this div constraint, so we add it back to make
1474 * sure we do not lose any information.
1475 *
1476 * We skip integral divs, i.e., those with denominator 1, as we would
1477 * risk eliminating the div from the div constraints. We do not need
1478 * to handle those divs here anyway since the div constraints will turn
1479 * out to form an equality and this equality can then be used to eliminate
1480 * the div from all constraints.
1481 */
1484 {
1516 else
1526 }
1531 }
1532 }
1535 }
1538 {
1543 isl_bool empty;
1559 /* requires equalities in normal form */
1565 }
1567 }
1570 {
1572 }
1577 {
1609 }
1613 {
1615 }
1618 /* If the only constraints a div d=floor(f/m)
1619 * appears in are its two defining constraints
1620 *
1621 * f - m d >=0
1622 * -(f - (m - 1)) + m d >= 0
1623 *
1624 * then it can safely be removed.
1625 */
1627 {
1636 isl_bool red;
1643 }
1650 }
1653 }
1655 /*
1656 * Remove divs that don't occur in any of the constraints or other divs.
1657 * These can arise when dropping constraints from a basic map or
1658 * when the divs of a basic map have been temporarily aligned
1659 * with the divs of another basic map.
1660 */
1662 {
1669 isl_bool redundant;
1677 }
1679 }
1681 /* Mark "bmap" as final, without checking for obviously redundant
1682 * integer divisions. This function should be used when "bmap"
1683 * is known not to involve any such integer divisions.
1684 */
1687 {
1692 }
1694 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1695 */
1697 {
1701 }
1704 {
1706 }
1708 /* Remove definition of any div that is defined in terms of the given variable.
1709 * The div itself is not removed. Functions such as
1710 * eliminate_divs_ineq depend on the other divs remaining in place.
1711 */
1714 {
1728 }
1730 }
1732 /* Eliminate the specified variables from the constraints using
1733 * Fourier-Motzkin. The variables themselves are not removed.
1734 */
1737 {
1769 }
1776 n_lower++;
1778 n_upper++;
1779 }
1803 }
1806 }
1818 }
1819 }
1824 error:
1827 }
1831 {
1834 }
1836 /* Eliminate the specified n dimensions starting at first from the
1837 * constraints, without removing the dimensions from the space.
1838 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1839 * Otherwise, they are projected out and the original space is restored.
1840 */
1844 {
1860 }
1867 error:
1870 }
1875 {
1877 }
1879 /* Remove all constraints from "bmap" that reference any unknown local
1880 * variables (directly or indirectly).
1881 *
1882 * Dropping all constraints on a local variable will make it redundant,
1883 * so it will get removed implicitly by
1884 * isl_basic_map_drop_constraints_involving_dims. Some other local
1885 * variables may also end up becoming redundant if they only appear
1886 * in constraints together with the unknown local variable.
1887 * Therefore, start over after calling
1888 * isl_basic_map_drop_constraints_involving_dims.
1889 */
1892 {
1893 isl_bool known;
1918 }
1921 }
1923 /* Remove all constraints from "map" that reference any unknown local
1924 * variables (directly or indirectly).
1925 *
1926 * Since constraints may get dropped from the basic maps,
1927 * they may no longer be disjoint from each other.
1928 */
1931 {
1933 isl_bool known;
1951 }
1957 }
1959 /* Don't assume equalities are in order, because align_divs
1960 * may have changed the order of the divs.
1961 */
1963 {
1976 }
1977 }
1978 }
1981 {
1983 }
1987 {
2001 }
2003 }
2005 }
2009 {
2012 }
2016 {
2026 }
2047 error:
2051 }
2053 /* For each inequality in "ineq" that is a shifted (more relaxed)
2054 * copy of an inequality in "context", mark the corresponding entry
2055 * in "row" with -1.
2056 * If an inequality only has a non-negative constant term, then
2057 * mark it as well.
2058 */
2061 {
2078 isl_bool redundant;
2084 }
2091 }
2094 error:
2097 }
2101 {
2114 isl_bool redundant;
2126 }
2129 error:
2132 }
2134 /* Remove constraints from "bmap" that are identical to constraints
2135 * in "context" or that are more relaxed (greater constant term).
2136 *
2137 * We perform the test for shifted copies on the pure constraints
2138 * in remove_shifted_constraints.
2139 */
2142 {
2151 }
2164 error:
2168 }
2170 /* Does the (linear part of a) constraint "c" involve any of the "len"
2171 * "relevant" dimensions?
2172 */
2174 {
2182 }
2185 }
2187 /* Drop constraints from "bmap" that do not involve any of
2188 * the dimensions marked "relevant".
2189 */
2192 {
2207 }
2214 }
2217 }
2219 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2220 *
2221 * In particular, for any variable involved in the constraint,
2222 * find the actual group id from before and replace the group
2223 * of the corresponding variable by the minimal group of all
2224 * the variables involved in the constraint considered so far
2225 * (if this minimum is smaller) or replace the minimum by this group
2226 * (if the minimum is larger).
2227 *
2228 * At the end, all the variables in "c" will (indirectly) point
2229 * to the minimal of the groups that they referred to originally.
2230 */
2232 {
2249 }
2250 }
2252 /* Allocate an array of groups of variables, one for each variable
2253 * in "context", initialized to zero.
2254 */
2256 {
2263 }
2265 /* Drop constraints from "bmap" that only involve variables that are
2266 * not related to any of the variables marked with a "-1" in "group".
2267 *
2268 * We construct groups of variables that collect variables that
2269 * (indirectly) appear in some common constraint of "bmap".
2270 * Each group is identified by the first variable in the group,
2271 * except for the special group of variables that was already identified
2272 * in the input as -1 (or are related to those variables).
2273 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2274 * otherwise the group of i is the group of group[i].
2275 *
2276 * We first initialize groups for the remaining variables.
2277 * Then we iterate over the constraints of "bmap" and update the
2278 * group of the variables in the constraint by the smallest group.
2279 * Finally, we resolve indirect references to groups by running over
2280 * the variables.
2281 *
2282 * After computing the groups, we drop constraints that do not involve
2283 * any variables in the -1 group.
2284 */
2287 {
2304 }
2322 }
2324 /* Drop constraints from "context" that are irrelevant for computing
2325 * the gist of "bset".
2326 *
2327 * In particular, drop constraints in variables that are not related
2328 * to any of the variables involved in the constraints of "bset"
2329 * in the sense that there is no sequence of constraints that connects them.
2330 *
2331 * We first mark all variables that appear in "bset" as belonging
2332 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2333 */
2336 {
2357 }
2363 }
2366 }
2368 /* Drop constraints from "context" that are irrelevant for computing
2369 * the gist of the inequalities "ineq".
2370 * Inequalities in "ineq" for which the corresponding element of row
2371 * is set to -1 have already been marked for removal and should be ignored.
2372 *
2373 * In particular, drop constraints in variables that are not related
2374 * to any of the variables involved in "ineq"
2375 * in the sense that there is no sequence of constraints that connects them.
2376 *
2377 * We first mark all variables that appear in "bset" as belonging
2378 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2379 */
2382 {
2403 }
2406 }
2409 }
2411 /* Do all "n" entries of "row" contain a negative value?
2412 */
2414 {
2422 }
2424 /* Update the inequalities in "bset" based on the information in "row"
2425 * and "tab".
2426 *
2427 * In particular, the array "row" contains either -1, meaning that
2428 * the corresponding inequality of "bset" is redundant, or the index
2429 * of an inequality in "tab".
2430 *
2431 * If the row entry is -1, then drop the inequality.
2432 * Otherwise, if the constraint is marked redundant in the tableau,
2433 * then drop the inequality. Similarly, if it is marked as an equality
2434 * in the tableau, then turn the inequality into an equality and
2435 * perform Gaussian elimination.
2436 */
2439 {
2456 }
2466 }
2467 }
2473 }
2475 /* Update the inequalities in "bset" based on the information in "row"
2476 * and "tab" and free all arguments (other than "bset").
2477 */
2482 {
2491 }
2493 /* Remove all information from bset that is redundant in the context
2494 * of context.
2495 * "ineq" contains the (possibly transformed) inequalities of "bset",
2496 * in the same order.
2497 * The (explicit) equalities of "bset" are assumed to have been taken
2498 * into account by the transformation such that only the inequalities
2499 * are relevant.
2500 * "context" is assumed not to be empty.
2501 *
2502 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2503 * A value of -1 means that the inequality is obviously redundant and may
2504 * not even appear in "tab".
2505 *
2506 * We first mark the inequalities of "bset"
2507 * that are obviously redundant with respect to some inequality in "context".
2508 * Then we remove those constraints from "context" that have become
2509 * irrelevant for computing the gist of "bset".
2510 * Note that this removal of constraints cannot be replaced by
2511 * a factorization because factors in "bset" may still be connected
2512 * to each other through constraints in "context".
2513 *
2514 * If there are any inequalities left, we construct a tableau for
2515 * the context and then add the inequalities of "bset".
2516 * Before adding these inequalities, we freeze all constraints such that
2517 * they won't be considered redundant in terms of the constraints of "bset".
2518 * Then we detect all redundant constraints (among the
2519 * constraints that weren't frozen), first by checking for redundancy in the
2520 * the tableau and then by checking if replacing a constraint by its negation
2521 * would lead to an empty set. This last step is fairly expensive
2522 * and could be optimized by more reuse of the tableau.
2523 * Finally, we update bset according to the results.
2524 */
2527 {
2542 }
2578 }
2602 }
2607 }
2611 error:
2619 }
2621 /* Extract the inequalities of "bset" as an isl_mat.
2622 */
2624 {
2638 }
2640 /* Remove all information from "bset" that is redundant in the context
2641 * of "context", for the case where both "bset" and "context" are
2642 * full-dimensional.
2643 */
2646 {
2651 }
2653 /* Remove all information from "bset" that is redundant in the context
2654 * of "context", for the case where the combined equalities of
2655 * "bset" and "context" allow for a compression that can be obtained
2656 * by preapplication of "T".
2657 *
2658 * "bset" itself is not transformed by "T". Instead, the inequalities
2659 * are extracted from "bset" and those are transformed by "T".
2660 * uset_gist_full then determines which of the transformed inequalities
2661 * are redundant with respect to the transformed "context" and removes
2662 * the corresponding inequalities from "bset".
2663 *
2664 * After preapplying "T" to the inequalities, any common factor is
2665 * removed from the coefficients. If this results in a tightening
2666 * of the constant term, then the same tightening is applied to
2667 * the corresponding untransformed inequality in "bset".
2668 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2669 *
2670 * g f'(x) + r >= 0
2671 *
2672 * with 0 <= r < g, then it is equivalent to
2673 *
2674 * f'(x) >= 0
2675 *
2676 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2677 * subspace compressed by T since the latter would be transformed to
2678 *
2679 * g f'(x) >= 0
2680 */
2684 {
2688 isl_int rem;
2700 }
2723 }
2727 error:
2732 }
2734 /* Project "bset" onto the variables that are involved in "template".
2735 */
2738 {
2747 isl_bool involved;
2756 }
2759 }
2761 /* Remove all information from bset that is redundant in the context
2762 * of context. In particular, equalities that are linear combinations
2763 * of those in context are removed. Then the inequalities that are
2764 * redundant in the context of the equalities and inequalities of
2765 * context are removed.
2766 *
2767 * First of all, we drop those constraints from "context"
2768 * that are irrelevant for computing the gist of "bset".
2769 * Alternatively, we could factorize the intersection of "context" and "bset".
2770 *
2771 * We first compute the intersection of the integer affine hulls
2772 * of "bset" and "context",
2773 * compute the gist inside this intersection and then reduce
2774 * the constraints with respect to the equalities of the context
2775 * that only involve variables already involved in the input.
2776 *
2777 * If two constraints are mutually redundant, then uset_gist_full
2778 * will remove the second of those constraints. We therefore first
2779 * sort the constraints so that constraints not involving existentially
2780 * quantified variables are given precedence over those that do.
2781 * We have to perform this sorting before the variable compression,
2782 * because that may effect the order of the variables.
2783 */
2786 {
2811 }
2816 }
2826 }
2837 }
2840 error:
2844 }
2846 /* Return the number of equality constraints in "bmap" that involve
2847 * local variables. This function assumes that Gaussian elimination
2848 * has been applied to the equality constraints.
2849 */
2851 {
2871 }
2873 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2874 * The constraints are assumed not to involve any local variables.
2875 */
2878 {
2896 }
2901 error:
2906 }
2908 /* Construct and return a variable compression based on the equality
2909 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2910 * "n1" is the number of (initial) equality constraints in "bmap1"
2911 * that do involve local variables.
2912 * "n2" is the number of (initial) equality constraints in "bmap2"
2913 * that do involve local variables.
2914 * "total" is the total number of other variables.
2915 * This function assumes that Gaussian elimination
2916 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2917 * such that the equality constraints not involving local variables
2918 * are those that start at "n1" or "n2".
2919 *
2920 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2921 * then simply compute the compression based on the equality constraints
2922 * in the other basic map.
2923 * Otherwise, combine the equality constraints from both into a new
2924 * basic map such that Gaussian elimination can be applied to this combination
2925 * and then construct a variable compression from the resulting
2926 * equality constraints.
2927 */
2931 {
2941 }
2946 }
2961 }
2963 /* Extract the stride constraints from "bmap", compressed
2964 * with respect to both the stride constraints in "context" and
2965 * the remaining equality constraints in both "bmap" and "context".
2966 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2967 * "context_n_eq" is the number of (initial) stride constraints in "context".
2968 *
2969 * Let x be all variables in "bmap" (and "context") other than the local
2970 * variables. First compute a variable compression
2971 *
2972 * x = V x'
2973 *
2974 * based on the non-stride equality constraints in "bmap" and "context".
2975 * Consider the stride constraints of "context",
2976 *
2977 * A(x) + B(y) = 0
2978 *
2979 * with y the local variables and plug in the variable compression,
2980 * resulting in
2981 *
2982 * A(V x') + B(y) = 0
2983 *
2984 * Use these constraints to compute a parameter compression on x'
2985 *
2986 * x' = T x''
2987 *
2988 * Now consider the stride constraints of "bmap"
2989 *
2990 * C(x) + D(y) = 0
2991 *
2992 * and plug in x = V*T x''.
2993 * That is, return A = [C*V*T D].
2994 */
2998 {
3027 }
3029 /* Remove the prime factors from *g that have an exponent that
3030 * is strictly smaller than the exponent in "c".
3031 * All exponents in *g are known to be smaller than or equal
3032 * to those in "c".
3033 *
3034 * That is, if *g is equal to
3035 *
3036 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
3037 *
3038 * and "c" is equal to
3039 *
3040 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
3041 *
3042 * then update *g to
3043 *
3044 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
3045 * p_n^{e_n * (e_n = f_n)}
3046 *
3047 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
3048 * neither does the gcd of *g and c / *g.
3049 * If e_i < f_i, then the gcd of *g and c / *g has a positive
3050 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
3051 * Dividing *g by this gcd therefore strictly reduces the exponent
3052 * of the prime factors that need to be removed, while leaving the
3053 * other prime factors untouched.
3054 * Repeating this process until gcd(*g, c / *g) = 1 therefore
3055 * removes all undesired factors, without removing any others.
3056 */
3058 {
3059 isl_int t;
3068 }
3070 }
3072 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3073 * of the same stride constraints in a compressed space that exploits
3074 * all equalities in the context and the other equalities in "bmap".
3075 *
3076 * If the stride constraints of "bmap" are of the form
3077 *
3078 * C(x) + D(y) = 0
3079 *
3080 * then A is of the form
3081 *
3082 * B(x') + D(y) = 0
3083 *
3084 * If any of these constraints involves only a single local variable y,
3085 * then the constraint appears as
3086 *
3087 * f(x) + m y_i = 0
3088 *
3089 * in "bmap" and as
3090 *
3091 * h(x') + m y_i = 0
3092 *
3093 * in "A".
3094 *
3095 * Let g be the gcd of m and the coefficients of h.
3096 * Then, in particular, g is a divisor of the coefficients of h and
3097 *
3098 * f(x) = h(x')
3099 *
3100 * is known to be a multiple of g.
3101 * If some prime factor in m appears with the same exponent in g,
3102 * then it can be removed from m because f(x) is already known
3103 * to be a multiple of g and therefore in particular of this power
3104 * of the prime factors.
3105 * Prime factors that appear with a smaller exponent in g cannot
3106 * be removed from m.
3107 * Let g' be the divisor of g containing all prime factors that
3108 * appear with the same exponent in m and g, then
3109 *
3110 * f(x) + m y_i = 0
3111 *
3112 * can be replaced by
3113 *
3114 * f(x) + m/g' y_i' = 0
3115 *
3116 * Note that (if g' != 1) this changes the explicit representation
3117 * of y_i to that of y_i', so the integer division at position i
3118 * is marked unknown and later recomputed by a call to
3119 * isl_basic_map_gauss.
3120 */
3123 {
3127 isl_int gcd;
3161 }
3168 error:
3172 }
3174 /* Simplify the stride constraints in "bmap" based on
3175 * the remaining equality constraints in "bmap" and all equality
3176 * constraints in "context".
3177 * Only do this if both "bmap" and "context" have stride constraints.
3178 *
3179 * First extract a copy of the stride constraints in "bmap" in a compressed
3180 * space exploiting all the other equality constraints and then
3181 * use this compressed copy to simplify the original stride constraints.
3182 */
3185 {
3207 }
3209 /* Return a basic map that has the same intersection with "context" as "bmap"
3210 * and that is as "simple" as possible.
3211 *
3212 * The core computation is performed on the pure constraints.
3213 * When we add back the meaning of the integer divisions, we need
3214 * to (re)introduce the div constraints. If we happen to have
3215 * discovered that some of these integer divisions are equal to
3216 * some affine combination of other variables, then these div
3217 * constraints may end up getting simplified in terms of the equalities,
3218 * resulting in extra inequalities on the other variables that
3219 * may have been removed already or that may not even have been
3220 * part of the input. We try and remove those constraints of
3221 * this form that are most obviously redundant with respect to
3222 * the context. We also remove those div constraints that are
3223 * redundant with respect to the other constraints in the result.
3224 *
3225 * The stride constraints among the equality constraints in "bmap" are
3226 * also simplified with respecting to the other equality constraints
3227 * in "bmap" and with respect to all equality constraints in "context".
3228 */
3231 {
3242 }
3248 }
3252 }
3274 }
3293 error:
3297 }
3299 /*
3300 * Assumes context has no implicit divs.
3301 */
3304 {
3315 }
3335 }
3336 }
3340 error:
3344 }
3346 /* Drop all inequalities from "bmap" that also appear in "context".
3347 * "context" is assumed to have only known local variables and
3348 * the initial local variables of "bmap" are assumed to be the same
3349 * as those of "context".
3350 * The constraints of both "bmap" and "context" are assumed
3351 * to have been sorted using isl_basic_map_sort_constraints.
3352 *
3353 * Run through the inequality constraints of "bmap" and "context"
3354 * in sorted order.
3355 * If a constraint of "bmap" involves variables not in "context",
3356 * then it cannot appear in "context".
3357 * If a matching constraint is found, it is removed from "bmap".
3358 */
3361 {
3380 }
3386 }
3390 }
3395 }
3398 }
3401 }
3403 /* Drop all equalities from "bmap" that also appear in "context".
3404 * "context" is assumed to have only known local variables and
3405 * the initial local variables of "bmap" are assumed to be the same
3406 * as those of "context".
3407 *
3408 * Run through the equality constraints of "bmap" and "context"
3409 * in sorted order.
3410 * If a constraint of "bmap" involves variables not in "context",
3411 * then it cannot appear in "context".
3412 * If a matching constraint is found, it is removed from "bmap".
3413 */
3416 {
3440 }
3444 }
3449 }
3452 }
3455 }
3457 /* Remove the constraints in "context" from "bmap".
3458 * "context" is assumed to have explicit representations
3459 * for all local variables.
3460 *
3461 * First align the divs of "bmap" to those of "context" and
3462 * sort the constraints. Then drop all constraints from "bmap"
3463 * that appear in "context".
3464 */
3467 {
3482 }
3501 error:
3505 }
3507 /* Replace "map" by the disjunct at position "pos" and free "context".
3508 */
3511 {
3518 }
3520 /* Remove the constraints in "context" from "map".
3521 * If any of the disjuncts in the result turns out to be the universe,
3522 * then return this universe.
3523 * "context" is assumed to have explicit representations
3524 * for all local variables.
3525 */
3528 {
3538 }
3557 }
3564 error:
3568 }
3570 /* Replace "map" by a universe map in the same space and free "drop".
3571 */
3574 {
3581 }
3583 /* Return a map that has the same intersection with "context" as "map"
3584 * and that is as "simple" as possible.
3585 *
3586 * If "map" is already the universe, then we cannot make it any simpler.
3587 * Similarly, if "context" is the universe, then we cannot exploit it
3588 * to simplify "map"
3589 * If "map" and "context" are identical to each other, then we can
3590 * return the corresponding universe.
3591 *
3592 * If either "map" or "context" consists of multiple disjuncts,
3593 * then check if "context" happens to be a subset of "map",
3594 * in which case all constraints can be removed.
3595 * In case of multiple disjuncts, the standard procedure
3596 * may not be able to detect that all constraints can be removed.
3597 *
3598 * If none of these cases apply, we have to work a bit harder.
3599 * During this computation, we make use of a single disjunct context,
3600 * so if the original context consists of more than one disjunct
3601 * then we need to approximate the context by a single disjunct set.
3602 * Simply taking the simple hull may drop constraints that are
3603 * only implicitly available in each disjunct. We therefore also
3604 * look for constraints among those defining "map" that are valid
3605 * for the context. These can then be used to simplify away
3606 * the corresponding constraints in "map".
3607 */
3610 {
3614 isl_bool subset;
3625 }
3641 }
3657 list);
3658 }
3660 error:
3664 }
3668 {
3670 }
3674 {
3677 }
3681 {
3684 }
3688 {
3693 }
3697 {
3699 }
3701 /* Compute the gist of "bmap" with respect to the constraints "context"
3702 * on the domain.
3703 */
3706 {
3712 }
3716 {
3720 }
3724 {
3728 }
3732 {
3736 }
3740 {
3742 }
3744 /* Quick check to see if two basic maps are disjoint.
3745 * In particular, we reduce the equalities and inequalities of
3746 * one basic map in the context of the equalities of the other
3747 * basic map and check if we get a contradiction.
3748 */
3751 {
3783 }
3791 }
3800 }
3804 disjoint:
3808 error:
3812 }
3816 {
3819 }
3821 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3822 */
3826 {
3837 }
3838 }
3841 }
3843 /* Are "map1" and "map2" obviously disjoint, based on information
3844 * that can be derived without looking at the individual basic maps?
3845 *
3846 * In particular, if one of them is empty or if they live in different spaces
3847 * (ignoring parameters), then they are clearly disjoint.
3848 */
3851 {
3852 isl_bool disjoint;
3853 isl_bool match;
3877 }
3879 /* Are "map1" and "map2" obviously disjoint?
3880 *
3881 * If one of them is empty or if they live in different spaces (ignoring
3882 * parameters), then they are clearly disjoint.
3883 * This is checked by isl_map_plain_is_disjoint_global.
3884 *
3885 * If they have different parameters, then we skip any further tests.
3886 *
3887 * If they are obviously equal, but not obviously empty, then we will
3888 * not be able to detect if they are disjoint.
3889 *
3890 * Otherwise we check if each basic map in "map1" is obviously disjoint
3891 * from each basic map in "map2".
3892 */
3895 {
3896 isl_bool disjoint;
3897 isl_bool intersect;
3898 isl_bool match;
3913 }
3915 /* Are "map1" and "map2" disjoint?
3916 *
3917 * They are disjoint if they are "obviously disjoint" or if one of them
3918 * is empty. Otherwise, they are not disjoint if one of them is universal.
3919 * If the two inputs are (obviously) equal and not empty, then they are
3920 * not disjoint.
3921 * If none of these cases apply, then check if all pairs of basic maps
3922 * are disjoint.
3923 */
3925 {
3926 isl_bool disjoint;
3927 isl_bool intersect;
3954 }
3956 /* Are "bmap1" and "bmap2" disjoint?
3957 *
3958 * They are disjoint if they are "obviously disjoint" or if one of them
3959 * is empty. Otherwise, they are not disjoint if one of them is universal.
3960 * If none of these cases apply, we compute the intersection and see if
3961 * the result is empty.
3962 */
3965 {
3966 isl_bool disjoint;
3967 isl_bool intersect;
3996 }
3998 /* Are "bset1" and "bset2" disjoint?
3999 */
4002 {
4004 }
4008 {
4010 }
4012 /* Are "set1" and "set2" disjoint?
4013 */
4015 {
4017 }
4019 /* Is "v" equal to 0, 1 or -1?
4020 */
4022 {
4024 }
4026 /* Check if we can combine a given div with lower bound l and upper
4027 * bound u with some other div and if so return that other div.
4028 * Otherwise return -1.
4029 *
4030 * We first check that
4031 * - the bounds are opposites of each other (except for the constant
4032 * term)
4033 * - the bounds do not reference any other div
4034 * - no div is defined in terms of this div
4035 *
4036 * Let m be the size of the range allowed on the div by the bounds.
4037 * That is, the bounds are of the form
4038 *
4039 * e <= a <= e + m - 1
4040 *
4041 * with e some expression in the other variables.
4042 * We look for another div b such that no third div is defined in terms
4043 * of this second div b and such that in any constraint that contains
4044 * a (except for the given lower and upper bound), also contains b
4045 * with a coefficient that is m times that of b.
4046 * That is, all constraints (execpt for the lower and upper bound)
4047 * are of the form
4048 *
4049 * e + f (a + m b) >= 0
4050 *
4051 * Furthermore, in the constraints that only contain b, the coefficient
4052 * of b should be equal to 1 or -1.
4053 * If so, we return b so that "a + m b" can be replaced by
4054 * a single div "c = a + m b".
4055 */
4058 {
4080 }
4090 }
4102 }
4113 }
4126 }
4131 }
4135 }
4137 /* Internal data structure used during the construction and/or evaluation of
4138 * an inequality that ensures that a pair of bounds always allows
4139 * for an integer value.
4140 *
4141 * "tab" is the tableau in which the inequality is evaluated. It may
4142 * be NULL until it is actually needed.
4143 * "v" contains the inequality coefficients.
4144 * "g", "fl" and "fu" are temporary scalars used during the construction and
4145 * evaluation.
4146 */
4150 isl_int g;
4151 isl_int fl;
4152 isl_int fu;
4153 };
4155 /* Free all the memory allocated by the fields of "data".
4156 */
4158 {
4164 }
4166 /* Is the inequality stored in data->v satisfied by "bmap"?
4167 * That is, does it only attain non-negative values?
4168 * data->tab is a tableau corresponding to "bmap".
4169 */
4172 {
4183 }
4185 /* Given a lower and an upper bound on div i, do they always allow
4186 * for an integer value of the given div?
4187 * Determine this property by constructing an inequality
4188 * such that the property is guaranteed when the inequality is nonnegative.
4189 * The lower bound is inequality l, while the upper bound is inequality u.
4190 * The constructed inequality is stored in data->v.
4191 *
4192 * Let the upper bound be
4193 *
4194 * -n_u a + e_u >= 0
4195 *
4196 * and the lower bound
4197 *
4198 * n_l a + e_l >= 0
4199 *
4200 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4201 * We have
4202 *
4203 * - f_u e_l <= f_u f_l g a <= f_l e_u
4204 *
4205 * Since all variables are integer valued, this is equivalent to
4206 *
4207 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4208 *
4209 * If this interval is at least f_u f_l g, then it contains at least
4210 * one integer value for a.
4211 * That is, the test constraint is
4212 *
4213 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4214 *
4215 * or
4216 *
4217 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4218 *
4219 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4220 * then the constraint can be scaled down by a factor g',
4221 * with the constant term replaced by
4222 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4223 * Note that the result of applying Fourier-Motzkin to this pair
4224 * of constraints is
4225 *
4226 * f_l e_u + f_u e_l >= 0
4227 *
4228 * If the constant term of the scaled down version of this constraint,
4229 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4230 * term of the scaled down test constraint, then the test constraint
4231 * is known to hold and no explicit evaluation is required.
4232 * This is essentially the Omega test.
4233 *
4234 * If the test constraint consists of only a constant term, then
4235 * it is sufficient to look at the sign of this constant term.
4236 */
4239 {
4264 }
4274 }
4276 /* Remove more kinds of divs that are not strictly needed.
4277 * In particular, if all pairs of lower and upper bounds on a div
4278 * are such that they allow at least one integer value of the div,
4279 * then we can eliminate the div using Fourier-Motzkin without
4280 * introducing any spurious solutions.
4281 *
4282 * If at least one of the two constraints has a unit coefficient for the div,
4283 * then the presence of such a value is guaranteed so there is no need to check.
4284 * In particular, the value attained by the bound with unit coefficient
4285 * can serve as this intermediate value.
4286 */
4289 {
4312 isl_bool has_int;
4320 }
4341 }
4344 }
4348 }
4352 }
4355 }
4366 error:
4371 }
4373 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4374 * and the upper bound u, div1 always occurs together with div2 in the form
4375 * (div1 + m div2), where m is the constant range on the variable div1
4376 * allowed by l and u, replace the pair div1 and div2 by a single
4377 * div that is equal to div1 + m div2.
4378 *
4379 * The new div will appear in the location that contains div2.
4380 * We need to modify all constraints that contain
4381 * div2 = (div - div1) / m
4382 * The coefficient of div2 is known to be equal to 1 or -1.
4383 * (If a constraint does not contain div2, it will also not contain div1.)
4384 * If the constraint also contains div1, then we know they appear
4385 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4386 * i.e., the coefficient of div is f.
4387 *
4388 * Otherwise, we first need to introduce div1 into the constraint.
4389 * Let the l be
4390 *
4391 * div1 + f >=0
4392 *
4393 * and u
4394 *
4395 * -div1 + f' >= 0
4396 *
4397 * A lower bound on div2
4398 *
4399 * div2 + t >= 0
4400 *
4401 * can be replaced by
4402 *
4403 * m div2 + div1 + m t + f >= 0
4404 *
4405 * An upper bound
4406 *
4407 * -div2 + t >= 0
4408 *
4409 * can be replaced by
4410 *
4411 * -(m div2 + div1) + m t + f' >= 0
4412 *
4413 * These constraint are those that we would obtain from eliminating
4414 * div1 using Fourier-Motzkin.
4415 *
4416 * After all constraints have been modified, we drop the lower and upper
4417 * bound and then drop div1.
4418 */
4421 {
4423 isl_int m;
4445 else
4448 }
4452 }
4461 }
4464 }
4466 /* First check if we can coalesce any pair of divs and
4467 * then continue with dropping more redundant divs.
4468 *
4469 * We loop over all pairs of lower and upper bounds on a div
4470 * with coefficient 1 and -1, respectively, check if there
4471 * is any other div "c" with which we can coalesce the div
4472 * and if so, perform the coalescing.
4473 */
4476 {
4499 }
4500 }
4501 }
4506 }
4509 }
4511 /* Are the "n" coefficients starting at "first" of inequality constraints
4512 * "i" and "j" of "bmap" equal to each other?
4513 */
4516 {
4518 }
4520 /* Are the "n" coefficients starting at "first" of inequality constraints
4521 * "i" and "j" of "bmap" opposite to each other?
4522 */
4525 {
4527 }
4529 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4530 * apart from the constant term?
4531 */
4533 {
4538 }
4540 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4541 * apart from the constant term and the coefficient at position "pos"?
4542 */
4545 {
4551 }
4553 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4554 * apart from the constant term and the coefficient at position "pos"?
4555 */
4558 {
4564 }
4566 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4567 * been modified, simplying it if "simplify" is set.
4568 * Free the temporary data structure "pairs" that was associated
4569 * to the old version of "bmap".
4570 */
4573 {
4578 }
4580 /* Is "div" the single unknown existentially quantified variable
4581 * in inequality constraint "ineq" of "bmap"?
4582 * "div" is known to have a non-zero coefficient in "ineq".
4583 */
4586 {
4589 isl_bool known;
4599 isl_bool known;
4608 }
4611 }
4613 /* Does integer division "div" have coefficient 1 in inequality constraint
4614 * "ineq" of "map"?
4615 */
4617 {
4625 }
4627 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4628 * then try and drop redundant divs again,
4629 * freeing the temporary data structure "pairs" that was associated
4630 * to the old version of "bmap".
4631 */
4634 {
4638 }
4640 /* Drop the integer division at position "div", along with the two
4641 * inequality constraints "ineq1" and "ineq2" in which it appears
4642 * from "bmap" and then try and drop redundant divs again,
4643 * freeing the temporary data structure "pairs" that was associated
4644 * to the old version of "bmap".
4645 */
4649 {
4656 }
4659 }
4661 /* Given two inequality constraints
4662 *
4663 * f(x) + n d + c >= 0, (ineq)
4664 *
4665 * with d the variable at position "pos", and
4666 *
4667 * f(x) + c0 >= 0, (lower)
4668 *
4669 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4670 * determined by the first constraint.
4671 * That is, store
4672 *
4673 * ceil((c0 - c)/n)
4674 *
4675 * in *l.
4676 */
4679 {
4683 }
4685 /* Given two inequality constraints
4686 *
4687 * f(x) + n d + c >= 0, (ineq)
4688 *
4689 * with d the variable at position "pos", and
4690 *
4691 * -f(x) - c0 >= 0, (upper)
4692 *
4693 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4694 * determined by the first constraint.
4695 * That is, store
4696 *
4697 * ceil((-c1 - c)/n)
4698 *
4699 * in *u.
4700 */
4703 {
4707 }
4709 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4710 * does the corresponding lower bound have a fixed value in "bmap"?
4711 *
4712 * In particular, "ineq" is of the form
4713 *
4714 * f(x) + n d + c >= 0
4715 *
4716 * with n > 0, c the constant term and
4717 * d the existentially quantified variable "div".
4718 * That is, the lower bound is
4719 *
4720 * ceil((-f(x) - c)/n)
4721 *
4722 * Look for a pair of constraints
4723 *
4724 * f(x) + c0 >= 0
4725 * -f(x) + c1 >= 0
4726 *
4727 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4728 * That is, check that
4729 *
4730 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4731 *
4732 * If so, return the index of inequality f(x) + c0 >= 0.
4733 * Otherwise, return -1.
4734 */
4736 {
4753 }
4757 }
4758 }
4775 }
4777 /* Given a lower bound constraint "ineq" on the existentially quantified
4778 * variable "div", such that the corresponding lower bound has
4779 * a fixed value in "bmap", assign this fixed value to the variable and
4780 * then try and drop redundant divs again,
4781 * freeing the temporary data structure "pairs" that was associated
4782 * to the old version of "bmap".
4783 * "lower" determines the constant value for the lower bound.
4784 *
4785 * In particular, "ineq" is of the form
4786 *
4787 * f(x) + n d + c >= 0,
4788 *
4789 * while "lower" is of the form
4790 *
4791 * f(x) + c0 >= 0
4792 *
4793 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4794 * is ceil((c0 - c)/n).
4795 */
4798 {
4799 isl_int c;
4812 }
4814 /* Remove divs that are not strictly needed based on the inequality
4815 * constraints.
4816 * In particular, if a div only occurs positively (or negatively)
4817 * in constraints, then it can simply be dropped.
4818 * Also, if a div occurs in only two constraints and if moreover
4819 * those two constraints are opposite to each other, except for the constant
4820 * term and if the sum of the constant terms is such that for any value
4821 * of the other values, there is always at least one integer value of the
4822 * div, i.e., if one plus this sum is greater than or equal to
4823 * the (absolute value) of the coefficient of the div in the constraints,
4824 * then we can also simply drop the div.
4825 *
4826 * If an existentially quantified variable does not have an explicit
4827 * representation, appears in only a single lower bound that does not
4828 * involve any other such existentially quantified variables and appears
4829 * in this lower bound with coefficient 1,
4830 * then fix the variable to the value of the lower bound. That is,
4831 * turn the inequality into an equality.
4832 * If for any value of the other variables, there is any value
4833 * for the existentially quantified variable satisfying the constraints,
4834 * then this lower bound also satisfies the constraints.
4835 * It is therefore safe to pick this lower bound.
4836 *
4837 * The same reasoning holds even if the coefficient is not one.
4838 * However, fixing the variable to the value of the lower bound may
4839 * in general introduce an extra integer division, in which case
4840 * it may be better to pick another value.
4841 * If this integer division has a known constant value, then plugging
4842 * in this constant value removes the existentially quantified variable
4843 * completely. In particular, if the lower bound is of the form
4844 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4845 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4846 * then the existentially quantified variable can be assigned this
4847 * shared value.
4848 *
4849 * We skip divs that appear in equalities or in the definition of other divs.
4850 * Divs that appear in the definition of other divs usually occur in at least
4851 * 4 constraints, but the constraints may have been simplified.
4852 *
4853 * If any divs are left after these simple checks then we move on
4854 * to more complicated cases in drop_more_redundant_divs.
4855 */
4858 {
4898 }
4902 }
4903 }
4911 }
4914 else
4934 pairs);
4938 pairs);
4940 }
4957 else
4964 }
4967 }
4974 error:
4978 }
4980 /* Consider the coefficients at "c" as a row vector and replace
4981 * them with their product with "T". "T" is assumed to be a square matrix.
4982 */
4984 {
5006 }
5008 /* Plug in T for the variables in "bmap" starting at "pos".
5009 * T is a linear unimodular matrix, i.e., without constant term.
5010 */
5013 {
5042 }
5046 error:
5050 }
5052 /* Remove divs that are not strictly needed.
5053 *
5054 * First look for an equality constraint involving two or more
5055 * existentially quantified variables without an explicit
5056 * representation. Replace the combination that appears
5057 * in the equality constraint by a single existentially quantified
5058 * variable such that the equality can be used to derive
5059 * an explicit representation for the variable.
5060 * If there are no more such equality constraints, then continue
5061 * with isl_basic_map_drop_redundant_divs_ineq.
5062 *
5063 * In particular, if the equality constraint is of the form
5064 *
5065 * f(x) + \sum_i c_i a_i = 0
5066 *
5067 * with a_i existentially quantified variable without explicit
5068 * representation, then apply a transformation on the existentially
5069 * quantified variables to turn the constraint into
5070 *
5071 * f(x) + g a_1' = 0
5072 *
5073 * with g the gcd of the c_i.
5074 * In order to easily identify which existentially quantified variables
5075 * have a complete explicit representation, i.e., without being defined
5076 * in terms of other existentially quantified variables without
5077 * an explicit representation, the existentially quantified variables
5078 * are first sorted.
5079 *
5080 * The variable transformation is computed by extending the row
5081 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5082 *
5083 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5084 * [a_2'] [ a_2 ]
5085 * ... = U ....
5086 * [a_n'] [ a_n ]
5087 *
5088 * with [c_1/g ... c_n/g] representing the first row of U.
5089 * The inverse of U is then plugged into the original constraints.
5090 * The call to isl_basic_map_simplify makes sure the explicit
5091 * representation for a_1' is extracted from the equality constraint.
5092 */
5095 {
5130 }
5149 }
5151 /* Does "bmap" satisfy any equality that involves more than 2 variables
5152 * and/or has coefficients different from -1 and 1?
5153 */
5155 {
5184 }
5187 }
5189 /* Remove any common factor g from the constraint coefficients in "v".
5190 * The constant term is stored in the first position and is replaced
5191 * by floor(c/g). If any common factor is removed and if this results
5192 * in a tightening of the constraint, then set *tightened.
5193 */
5196 {
5216 }
5218 /* If "bmap" is an integer set that satisfies any equality involving
5219 * more than 2 variables and/or has coefficients different from -1 and 1,
5220 * then use variable compression to reduce the coefficients by removing
5221 * any (hidden) common factor.
5222 * In particular, apply the variable compression to each constraint,
5223 * factor out any common factor in the non-constant coefficients and
5224 * then apply the inverse of the compression.
5225 * At the end, we mark the basic map as having reduced constants.
5226 * If this flag is still set on the next invocation of this function,
5227 * then we skip the computation.
5228 *
5229 * Removing a common factor may result in a tightening of some of
5230 * the constraints. If this happens, then we may end up with two
5231 * opposite inequalities that can be replaced by an equality.
5232 * We therefore call isl_basic_map_detect_inequality_pairs,
5233 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5234 * and isl_basic_map_gauss if such a pair was found.
5235 */
5238 {
5272 }
5283 }
5298 }
5299 }
5302 error:
5307 }
5309 /* Shift the integer division at position "div" of "bmap"
5310 * by "shift" times the variable at position "pos".
5311 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5312 * corresponds to the constant term.
5313 *
5314 * That is, if the integer division has the form
5315 *
5316 * floor(f(x)/d)
5317 *
5318 * then replace it by
5319 *
5320 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5321 */
5324 {
5343 }
5349 }
5357 }
5360 }