| blob | d5709c07ea862a062b00d18e0ea6ad6b3016dc29 |
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 }
49 {
51 isl_int gcd;
64 }
67 }
75 }
77 }
85 }
88 }
95 }
99 }
103 {
106 }
108 /* Reduce the coefficient of the variable at position "pos"
109 * in integer division "div", such that it lies in the half-open
110 * interval (1/2,1/2], extracting any excess value from this integer division.
111 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
112 * corresponds to the constant term.
113 *
114 * That is, the integer division is of the form
115 *
116 * floor((... + (c * d + r) * x_pos + ...)/d)
117 *
118 * with -d < 2 * r <= d.
119 * Replace it by
120 *
121 * floor((... + r * x_pos + ...)/d) + c * x_pos
122 *
123 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
124 * Otherwise, c = floor((c * d + r)/d) + 1.
125 *
126 * This is the same normalization that is performed by isl_aff_floor.
127 */
130 {
131 isl_int shift;
146 }
148 /* Does the coefficient of the variable at position "pos"
149 * in integer division "div" need to be reduced?
150 * That is, does it lie outside the half-open interval (1/2,1/2]?
151 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
152 * 2 * c != d.
153 */
156 {
157 isl_bool r;
169 }
171 /* Reduce the coefficients (including the constant term) of
172 * integer division "div", if needed.
173 * In particular, make sure all coefficients lie in
174 * the half-open interval (1/2,1/2].
175 */
178 {
183 isl_bool reduce;
193 }
196 }
198 /* Reduce the coefficients (including the constant term) of
199 * the known integer divisions, if needed
200 * In particular, make sure all coefficients lie in
201 * the half-open interval (1/2,1/2].
202 */
205 {
219 }
222 }
224 /* Remove any common factor in numerator and denominator of the div expression,
225 * not taking into account the constant term.
226 * That is, if the div is of the form
227 *
228 * floor((a + m f(x))/(m d))
229 *
230 * then replace it by
231 *
232 * floor((floor(a/m) + f(x))/d)
233 *
234 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
235 * and can therefore not influence the result of the floor.
236 */
239 {
257 }
259 /* Remove any common factor in numerator and denominator of a div expression,
260 * not taking into account the constant term.
261 * That is, look for any div of the form
262 *
263 * floor((a + m f(x))/(m d))
264 *
265 * and replace it by
266 *
267 * floor((floor(a/m) + f(x))/d)
268 *
269 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
270 * and can therefore not influence the result of the floor.
271 */
274 {
286 }
288 /* Assumes divs have been ordered if keep_divs is set.
289 */
293 {
313 }
324 }
333 /* We need to be careful about circular definitions,
334 * so for now we just remove the definition of div k
335 * if the equality contains any divs.
336 * If keep_divs is set, then the divs have been ordered
337 * and we can keep the definition as long as the result
338 * is still ordered.
339 */
348 }
351 }
353 /* Assumes divs have been ordered if keep_divs is set.
354 */
357 {
370 }
372 /* Check if elimination of div "div" using equality "eq" would not
373 * result in a div depending on a later div.
374 */
377 {
397 }
400 }
402 /* Eliminate divs based on equalities
403 */
406 {
421 isl_bool ok;
437 }
438 }
442 }
444 /* Eliminate divs based on inequalities
445 */
448 {
478 }
480 }
482 /* Does the equality constraint at position "eq" in "bmap" involve
483 * any local variables in the range [first, first + n)
484 * that are not marked as having an explicit representation?
485 */
488 {
497 isl_bool unknown;
506 }
509 }
511 /* The last local variable involved in the equality constraint
512 * at position "eq" in "bmap" is the local variable at position "div".
513 * It can therefore be used to extract an explicit representation
514 * for that variable.
515 * Do so unless the local variable already has an explicit representation or
516 * the explicit representation would involve any other local variables
517 * that in turn do not have an explicit representation.
518 * An equality constraint involving local variables without an explicit
519 * representation can be used in isl_basic_map_drop_redundant_divs
520 * to separate out an independent local variable. Introducing
521 * an explicit representation here would block this transformation,
522 * while the partial explicit representation in itself is not very useful.
523 * Set *progress if anything is changed.
524 *
525 * The equality constraint is of the form
526 *
527 * f(x) + n e >= 0
528 *
529 * with n a positive number. The explicit representation derived from
530 * this constraint is
531 *
532 * floor((-f(x))/n)
533 */
536 {
538 isl_bool involves;
561 }
565 {
588 }
604 }
611 }
614 }
618 {
620 progress));
621 }
625 {
631 }
633 }
635 /* Hash table of inequalities in a basic map.
636 * "index" is an array of addresses of inequalities in the basic map, some
637 * of which are NULL. The inequalities are hashed on the coefficients
638 * except the constant term.
639 * "size" is the number of elements in the array and is always a power of two
640 * "bits" is the number of bits need to represent an index into the array.
641 * "total" is the total dimension of the basic map.
642 */
648 };
650 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
651 */
654 {
671 }
673 /* Free the memory allocated by create_constraint_index.
674 */
676 {
678 }
680 /* Return the position in ci->index that contains the address of
681 * an inequality that is equal to *ineq up to the constant term,
682 * provided this address is not identical to "ineq".
683 * If there is no such inequality, then return the position where
684 * such an inequality should be inserted.
685 */
687 {
695 }
697 /* Return the position in ci->index that contains the address of
698 * an inequality that is equal to the k'th inequality of "bmap"
699 * up to the constant term, provided it does not point to the very
700 * same inequality.
701 * If there is no such inequality, then return the position where
702 * such an inequality should be inserted.
703 */
706 {
708 }
712 {
714 }
716 /* Fill in the "ci" data structure with the inequalities of "bset".
717 */
720 {
729 }
732 }
734 /* Is the inequality ineq (obviously) redundant with respect
735 * to the constraints in "ci"?
736 *
737 * Look for an inequality in "ci" with the same coefficients and then
738 * check if the contant term of "ineq" is greater than or equal
739 * to the constant term of that inequality. If so, "ineq" is clearly
740 * redundant.
741 *
742 * Note that hash_index_ineq ignores a stored constraint if it has
743 * the same address as the passed inequality. It is ok to pass
744 * the address of a local variable here since it will never be
745 * the same as the address of a constraint in "ci".
746 */
749 {
756 }
758 /* If we can eliminate more than one div, then we need to make
759 * sure we do it from last div to first div, in order not to
760 * change the position of the other divs that still need to
761 * be removed.
762 */
765 {
821 }
823 }
835 }
838 out:
842 }
845 {
859 }
861 }
863 /* Normalize divs that appear in equalities.
864 *
865 * In particular, we assume that bmap contains some equalities
866 * of the form
867 *
868 * a x = m * e_i
869 *
870 * and we want to replace the set of e_i by a minimal set and
871 * such that the new e_i have a canonical representation in terms
872 * of the vector x.
873 * If any of the equalities involves more than one divs, then
874 * we currently simply bail out.
875 *
876 * Let us first additionally assume that all equalities involve
877 * a div. The equalities then express modulo constraints on the
878 * remaining variables and we can use "parameter compression"
879 * to find a minimal set of constraints. The result is a transformation
880 *
881 * x = T(x') = x_0 + G x'
882 *
883 * with G a lower-triangular matrix with all elements below the diagonal
884 * non-negative and smaller than the diagonal element on the same row.
885 * We first normalize x_0 by making the same property hold in the affine
886 * T matrix.
887 * The rows i of G with a 1 on the diagonal do not impose any modulo
888 * constraint and simply express x_i = x'_i.
889 * For each of the remaining rows i, we introduce a div and a corresponding
890 * equality. In particular
891 *
892 * g_ii e_j = x_i - g_i(x')
893 *
894 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
895 * corresponding div (if g_kk != 1).
896 *
897 * If there are any equalities not involving any div, then we
898 * first apply a variable compression on the variables x:
899 *
900 * x = C x'' x'' = C_2 x
901 *
902 * and perform the above parameter compression on A C instead of on A.
903 * The resulting compression is then of the form
904 *
905 * x'' = T(x') = x_0 + G x'
906 *
907 * and in constructing the new divs and the corresponding equalities,
908 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
909 * by the corresponding row from C_2.
910 */
913 {
922 isl_int v;
956 }
957 }
966 }
972 }
982 }
989 }
994 /* We have to be careful because dropping equalities may reorder them */
1004 }
1005 }
1011 else
1012 needed++;
1013 }
1019 }
1029 else
1037 else
1040 }
1044 }
1051 done:
1055 error:
1061 }
1065 {
1075 }
1077 /* Check whether it is ok to define a div based on an inequality.
1078 * To avoid the introduction of circular definitions of divs, we
1079 * do not allow such a definition if the resulting expression would refer to
1080 * any other undefined divs or if any known div is defined in
1081 * terms of the unknown div.
1082 */
1085 {
1089 /* Not defined in terms of unknown divs */
1097 }
1099 /* No other div defined in terms of this one => avoid loops */
1107 }
1110 }
1112 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1113 * be a better expression than the current one?
1114 *
1115 * If we do not have any expression yet, then any expression would be better.
1116 * Otherwise we check if the last variable involved in the inequality
1117 * (disregarding the div that it would define) is in an earlier position
1118 * than the last variable involved in the current div expression.
1119 */
1122 {
1139 }
1141 /* Given two constraints "k" and "l" that are opposite to each other,
1142 * except for the constant term, check if we can use them
1143 * to obtain an expression for one of the hitherto unknown divs or
1144 * a "better" expression for a div for which we already have an expression.
1145 * "sum" is the sum of the constant terms of the constraints.
1146 * If this sum is strictly smaller than the coefficient of one
1147 * of the divs, then this pair can be used define the div.
1148 * To avoid the introduction of circular definitions of divs, we
1149 * do not use the pair if the resulting expression would refer to
1150 * any other undefined divs or if any known div is defined in
1151 * terms of the unknown div.
1152 */
1156 {
1161 isl_bool set_div;
1176 else
1181 }
1183 }
1187 {
1191 isl_int sum;
1206 }
1214 }
1229 }
1231 /* We need to break out of the loop after these
1232 * changes since the contents of the hash
1233 * will no longer be valid.
1234 * Plus, we probably we want to regauss first.
1235 */
1243 }
1248 }
1250 /* Detect all pairs of inequalities that form an equality.
1251 *
1252 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1253 * Call it repeatedly while it is making progress.
1254 */
1257 {
1269 }
1271 /* Eliminate knowns divs from constraints where they appear with
1272 * a (positive or negative) unit coefficient.
1273 *
1274 * That is, replace
1275 *
1276 * floor(e/m) + f >= 0
1277 *
1278 * by
1279 *
1280 * e + m f >= 0
1281 *
1282 * and
1283 *
1284 * -floor(e/m) + f >= 0
1285 *
1286 * by
1287 *
1288 * -e + m f + m - 1 >= 0
1289 *
1290 * The first conversion is valid because floor(e/m) >= -f is equivalent
1291 * to e/m >= -f because -f is an integral expression.
1292 * The second conversion follows from the fact that
1293 *
1294 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1295 *
1296 *
1297 * Note that one of the div constraints may have been eliminated
1298 * due to being redundant with respect to the constraint that is
1299 * being modified by this function. The modified constraint may
1300 * no longer imply this div constraint, so we add it back to make
1301 * sure we do not lose any information.
1302 *
1303 * We skip integral divs, i.e., those with denominator 1, as we would
1304 * risk eliminating the div from the div constraints. We do not need
1305 * to handle those divs here anyway since the div constraints will turn
1306 * out to form an equality and this equality can then be used to eliminate
1307 * the div from all constraints.
1308 */
1311 {
1343 else
1353 }
1359 }
1360 }
1363 }
1366 {
1371 isl_bool empty;
1387 /* requires equalities in normal form */
1393 }
1395 }
1398 {
1400 }
1405 {
1437 }
1441 {
1443 }
1446 /* If the only constraints a div d=floor(f/m)
1447 * appears in are its two defining constraints
1448 *
1449 * f - m d >=0
1450 * -(f - (m - 1)) + m d >= 0
1451 *
1452 * then it can safely be removed.
1453 */
1455 {
1464 isl_bool red;
1471 }
1478 }
1481 }
1483 /*
1484 * Remove divs that don't occur in any of the constraints or other divs.
1485 * These can arise when dropping constraints from a basic map or
1486 * when the divs of a basic map have been temporarily aligned
1487 * with the divs of another basic map.
1488 */
1491 {
1500 isl_bool redundant;
1510 }
1512 }
1514 /* Mark "bmap" as final, without checking for obviously redundant
1515 * integer divisions. This function should be used when "bmap"
1516 * is known not to involve any such integer divisions.
1517 */
1520 {
1525 }
1527 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1528 */
1530 {
1534 }
1537 {
1539 }
1541 /* Remove definition of any div that is defined in terms of the given variable.
1542 * The div itself is not removed. Functions such as
1543 * eliminate_divs_ineq depend on the other divs remaining in place.
1544 */
1547 {
1561 }
1563 }
1565 /* Eliminate the specified variables from the constraints using
1566 * Fourier-Motzkin. The variables themselves are not removed.
1567 */
1570 {
1604 }
1611 n_lower++;
1613 n_upper++;
1614 }
1638 }
1641 }
1653 }
1654 }
1658 error:
1661 }
1665 {
1668 }
1670 /* Eliminate the specified n dimensions starting at first from the
1671 * constraints, without removing the dimensions from the space.
1672 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1673 * Otherwise, they are projected out and the original space is restored.
1674 */
1678 {
1693 }
1700 }
1705 {
1707 }
1709 /* Remove all constraints from "bmap" that reference any unknown local
1710 * variables (directly or indirectly).
1711 *
1712 * Dropping all constraints on a local variable will make it redundant,
1713 * so it will get removed implicitly by
1714 * isl_basic_map_drop_constraints_involving_dims. Some other local
1715 * variables may also end up becoming redundant if they only appear
1716 * in constraints together with the unknown local variable.
1717 * Therefore, start over after calling
1718 * isl_basic_map_drop_constraints_involving_dims.
1719 */
1722 {
1723 isl_bool known;
1748 }
1751 }
1753 /* Remove all constraints from "map" that reference any unknown local
1754 * variables (directly or indirectly).
1755 *
1756 * Since constraints may get dropped from the basic maps,
1757 * they may no longer be disjoint from each other.
1758 */
1761 {
1763 isl_bool known;
1781 }
1787 }
1789 /* Don't assume equalities are in order, because align_divs
1790 * may have changed the order of the divs.
1791 */
1793 {
1806 }
1807 }
1808 }
1812 {
1814 }
1818 {
1832 }
1834 }
1836 }
1840 {
1843 }
1847 {
1857 }
1878 error:
1882 }
1884 /* For each inequality in "ineq" that is a shifted (more relaxed)
1885 * copy of an inequality in "context", mark the corresponding entry
1886 * in "row" with -1.
1887 * If an inequality only has a non-negative constant term, then
1888 * mark it as well.
1889 */
1892 {
1909 isl_bool redundant;
1915 }
1922 }
1925 error:
1928 }
1932 {
1945 isl_bool redundant;
1957 }
1960 error:
1963 }
1965 /* Remove constraints from "bmap" that are identical to constraints
1966 * in "context" or that are more relaxed (greater constant term).
1967 *
1968 * We perform the test for shifted copies on the pure constraints
1969 * in remove_shifted_constraints.
1970 */
1973 {
1982 }
1995 error:
1999 }
2001 /* Does the (linear part of a) constraint "c" involve any of the "len"
2002 * "relevant" dimensions?
2003 */
2005 {
2013 }
2016 }
2018 /* Drop constraints from "bmap" that do not involve any of
2019 * the dimensions marked "relevant".
2020 */
2023 {
2038 }
2045 }
2048 }
2050 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2051 *
2052 * In particular, for any variable involved in the constraint,
2053 * find the actual group id from before and replace the group
2054 * of the corresponding variable by the minimal group of all
2055 * the variables involved in the constraint considered so far
2056 * (if this minimum is smaller) or replace the minimum by this group
2057 * (if the minimum is larger).
2058 *
2059 * At the end, all the variables in "c" will (indirectly) point
2060 * to the minimal of the groups that they referred to originally.
2061 */
2063 {
2080 }
2081 }
2083 /* Allocate an array of groups of variables, one for each variable
2084 * in "context", initialized to zero.
2085 */
2087 {
2094 }
2096 /* Drop constraints from "bmap" that only involve variables that are
2097 * not related to any of the variables marked with a "-1" in "group".
2098 *
2099 * We construct groups of variables that collect variables that
2100 * (indirectly) appear in some common constraint of "bmap".
2101 * Each group is identified by the first variable in the group,
2102 * except for the special group of variables that was already identified
2103 * in the input as -1 (or are related to those variables).
2104 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2105 * otherwise the group of i is the group of group[i].
2106 *
2107 * We first initialize groups for the remaining variables.
2108 * Then we iterate over the constraints of "bmap" and update the
2109 * group of the variables in the constraint by the smallest group.
2110 * Finally, we resolve indirect references to groups by running over
2111 * the variables.
2112 *
2113 * After computing the groups, we drop constraints that do not involve
2114 * any variables in the -1 group.
2115 */
2118 {
2135 }
2153 }
2155 /* Drop constraints from "context" that are irrelevant for computing
2156 * the gist of "bset".
2157 *
2158 * In particular, drop constraints in variables that are not related
2159 * to any of the variables involved in the constraints of "bset"
2160 * in the sense that there is no sequence of constraints that connects them.
2161 *
2162 * We first mark all variables that appear in "bset" as belonging
2163 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2164 */
2167 {
2188 }
2194 }
2197 }
2199 /* Drop constraints from "context" that are irrelevant for computing
2200 * the gist of the inequalities "ineq".
2201 * Inequalities in "ineq" for which the corresponding element of row
2202 * is set to -1 have already been marked for removal and should be ignored.
2203 *
2204 * In particular, drop constraints in variables that are not related
2205 * to any of the variables involved in "ineq"
2206 * in the sense that there is no sequence of constraints that connects them.
2207 *
2208 * We first mark all variables that appear in "bset" as belonging
2209 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2210 */
2213 {
2234 }
2237 }
2240 }
2242 /* Do all "n" entries of "row" contain a negative value?
2243 */
2245 {
2253 }
2255 /* Update the inequalities in "bset" based on the information in "row"
2256 * and "tab".
2257 *
2258 * In particular, the array "row" contains either -1, meaning that
2259 * the corresponding inequality of "bset" is redundant, or the index
2260 * of an inequality in "tab".
2261 *
2262 * If the row entry is -1, then drop the inequality.
2263 * Otherwise, if the constraint is marked redundant in the tableau,
2264 * then drop the inequality. Similarly, if it is marked as an equality
2265 * in the tableau, then turn the inequality into an equality and
2266 * perform Gaussian elimination.
2267 */
2270 {
2287 }
2297 }
2298 }
2304 }
2306 /* Update the inequalities in "bset" based on the information in "row"
2307 * and "tab" and free all arguments (other than "bset").
2308 */
2313 {
2322 }
2324 /* Remove all information from bset that is redundant in the context
2325 * of context.
2326 * "ineq" contains the (possibly transformed) inequalities of "bset",
2327 * in the same order.
2328 * The (explicit) equalities of "bset" are assumed to have been taken
2329 * into account by the transformation such that only the inequalities
2330 * are relevant.
2331 * "context" is assumed not to be empty.
2332 *
2333 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2334 * A value of -1 means that the inequality is obviously redundant and may
2335 * not even appear in "tab".
2336 *
2337 * We first mark the inequalities of "bset"
2338 * that are obviously redundant with respect to some inequality in "context".
2339 * Then we remove those constraints from "context" that have become
2340 * irrelevant for computing the gist of "bset".
2341 * Note that this removal of constraints cannot be replaced by
2342 * a factorization because factors in "bset" may still be connected
2343 * to each other through constraints in "context".
2344 *
2345 * If there are any inequalities left, we construct a tableau for
2346 * the context and then add the inequalities of "bset".
2347 * Before adding these inequalities, we freeze all constraints such that
2348 * they won't be considered redundant in terms of the constraints of "bset".
2349 * Then we detect all redundant constraints (among the
2350 * constraints that weren't frozen), first by checking for redundancy in the
2351 * the tableau and then by checking if replacing a constraint by its negation
2352 * would lead to an empty set. This last step is fairly expensive
2353 * and could be optimized by more reuse of the tableau.
2354 * Finally, we update bset according to the results.
2355 */
2358 {
2373 }
2409 }
2432 }
2437 }
2441 error:
2449 }
2451 /* Extract the inequalities of "bset" as an isl_mat.
2452 */
2454 {
2468 }
2470 /* Remove all information from "bset" that is redundant in the context
2471 * of "context", for the case where both "bset" and "context" are
2472 * full-dimensional.
2473 */
2476 {
2481 }
2483 /* Remove all information from "bset" that is redundant in the context
2484 * of "context", for the case where the combined equalities of
2485 * "bset" and "context" allow for a compression that can be obtained
2486 * by preapplication of "T".
2487 *
2488 * "bset" itself is not transformed by "T". Instead, the inequalities
2489 * are extracted from "bset" and those are transformed by "T".
2490 * uset_gist_full then determines which of the transformed inequalities
2491 * are redundant with respect to the transformed "context" and removes
2492 * the corresponding inequalities from "bset".
2493 *
2494 * After preapplying "T" to the inequalities, any common factor is
2495 * removed from the coefficients. If this results in a tightening
2496 * of the constant term, then the same tightening is applied to
2497 * the corresponding untransformed inequality in "bset".
2498 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2499 *
2500 * g f'(x) + r >= 0
2501 *
2502 * with 0 <= r < g, then it is equivalent to
2503 *
2504 * f'(x) >= 0
2505 *
2506 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2507 * subspace compressed by T since the latter would be transformed to
2508 *
2509 * g f'(x) >= 0
2510 */
2514 {
2518 isl_int rem;
2530 }
2553 }
2557 error:
2562 }
2564 /* Project "bset" onto the variables that are involved in "template".
2565 */
2568 {
2577 isl_bool involved;
2586 }
2589 }
2591 /* Remove all information from bset that is redundant in the context
2592 * of context. In particular, equalities that are linear combinations
2593 * of those in context are removed. Then the inequalities that are
2594 * redundant in the context of the equalities and inequalities of
2595 * context are removed.
2596 *
2597 * First of all, we drop those constraints from "context"
2598 * that are irrelevant for computing the gist of "bset".
2599 * Alternatively, we could factorize the intersection of "context" and "bset".
2600 *
2601 * We first compute the intersection of the integer affine hulls
2602 * of "bset" and "context",
2603 * compute the gist inside this intersection and then reduce
2604 * the constraints with respect to the equalities of the context
2605 * that only involve variables already involved in the input.
2606 *
2607 * If two constraints are mutually redundant, then uset_gist_full
2608 * will remove the second of those constraints. We therefore first
2609 * sort the constraints so that constraints not involving existentially
2610 * quantified variables are given precedence over those that do.
2611 * We have to perform this sorting before the variable compression,
2612 * because that may effect the order of the variables.
2613 */
2616 {
2641 }
2646 }
2656 }
2667 }
2670 error:
2674 }
2676 /* Return the number of equality constraints in "bmap" that involve
2677 * local variables. This function assumes that Gaussian elimination
2678 * has been applied to the equality constraints.
2679 */
2681 {
2701 }
2703 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2704 * The constraints are assumed not to involve any local variables.
2705 */
2708 {
2726 }
2731 error:
2736 }
2738 /* Construct and return a variable compression based on the equality
2739 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2740 * "n1" is the number of (initial) equality constraints in "bmap1"
2741 * that do involve local variables.
2742 * "n2" is the number of (initial) equality constraints in "bmap2"
2743 * that do involve local variables.
2744 * "total" is the total number of other variables.
2745 * This function assumes that Gaussian elimination
2746 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2747 * such that the equality constraints not involving local variables
2748 * are those that start at "n1" or "n2".
2749 *
2750 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2751 * then simply compute the compression based on the equality constraints
2752 * in the other basic map.
2753 * Otherwise, combine the equality constraints from both into a new
2754 * basic map such that Gaussian elimination can be applied to this combination
2755 * and then construct a variable compression from the resulting
2756 * equality constraints.
2757 */
2761 {
2771 }
2776 }
2791 }
2793 /* Extract the stride constraints from "bmap", compressed
2794 * with respect to both the stride constraints in "context" and
2795 * the remaining equality constraints in both "bmap" and "context".
2796 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2797 * "context_n_eq" is the number of (initial) stride constraints in "context".
2798 *
2799 * Let x be all variables in "bmap" (and "context") other than the local
2800 * variables. First compute a variable compression
2801 *
2802 * x = V x'
2803 *
2804 * based on the non-stride equality constraints in "bmap" and "context".
2805 * Consider the stride constraints of "context",
2806 *
2807 * A(x) + B(y) = 0
2808 *
2809 * with y the local variables and plug in the variable compression,
2810 * resulting in
2811 *
2812 * A(V x') + B(y) = 0
2813 *
2814 * Use these constraints to compute a parameter compression on x'
2815 *
2816 * x' = T x''
2817 *
2818 * Now consider the stride constraints of "bmap"
2819 *
2820 * C(x) + D(y) = 0
2821 *
2822 * and plug in x = V*T x''.
2823 * That is, return A = [C*V*T D].
2824 */
2828 {
2857 }
2859 /* Remove the prime factors from *g that have an exponent that
2860 * is strictly smaller than the exponent in "c".
2861 * All exponents in *g are known to be smaller than or equal
2862 * to those in "c".
2863 *
2864 * That is, if *g is equal to
2865 *
2866 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2867 *
2868 * and "c" is equal to
2869 *
2870 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2871 *
2872 * then update *g to
2873 *
2874 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2875 * p_n^{e_n * (e_n = f_n)}
2876 *
2877 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2878 * neither does the gcd of *g and c / *g.
2879 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2880 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2881 * Dividing *g by this gcd therefore strictly reduces the exponent
2882 * of the prime factors that need to be removed, while leaving the
2883 * other prime factors untouched.
2884 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2885 * removes all undesired factors, without removing any others.
2886 */
2888 {
2889 isl_int t;
2898 }
2900 }
2902 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2903 * of the same stride constraints in a compressed space that exploits
2904 * all equalities in the context and the other equalities in "bmap".
2905 *
2906 * If the stride constraints of "bmap" are of the form
2907 *
2908 * C(x) + D(y) = 0
2909 *
2910 * then A is of the form
2911 *
2912 * B(x') + D(y) = 0
2913 *
2914 * If any of these constraints involves only a single local variable y,
2915 * then the constraint appears as
2916 *
2917 * f(x) + m y_i = 0
2918 *
2919 * in "bmap" and as
2920 *
2921 * h(x') + m y_i = 0
2922 *
2923 * in "A".
2924 *
2925 * Let g be the gcd of m and the coefficients of h.
2926 * Then, in particular, g is a divisor of the coefficients of h and
2927 *
2928 * f(x) = h(x')
2929 *
2930 * is known to be a multiple of g.
2931 * If some prime factor in m appears with the same exponent in g,
2932 * then it can be removed from m because f(x) is already known
2933 * to be a multiple of g and therefore in particular of this power
2934 * of the prime factors.
2935 * Prime factors that appear with a smaller exponent in g cannot
2936 * be removed from m.
2937 * Let g' be the divisor of g containing all prime factors that
2938 * appear with the same exponent in m and g, then
2939 *
2940 * f(x) + m y_i = 0
2941 *
2942 * can be replaced by
2943 *
2944 * f(x) + m/g' y_i' = 0
2945 *
2946 * Note that (if g' != 1) this changes the explicit representation
2947 * of y_i to that of y_i', so the integer division at position i
2948 * is marked unknown and later recomputed by a call to
2949 * isl_basic_map_gauss.
2950 */
2953 {
2957 isl_int gcd;
2991 }
2998 error:
3002 }
3004 /* Simplify the stride constraints in "bmap" based on
3005 * the remaining equality constraints in "bmap" and all equality
3006 * constraints in "context".
3007 * Only do this if both "bmap" and "context" have stride constraints.
3008 *
3009 * First extract a copy of the stride constraints in "bmap" in a compressed
3010 * space exploiting all the other equality constraints and then
3011 * use this compressed copy to simplify the original stride constraints.
3012 */
3015 {
3037 }
3039 /* Return a basic map that has the same intersection with "context" as "bmap"
3040 * and that is as "simple" as possible.
3041 *
3042 * The core computation is performed on the pure constraints.
3043 * When we add back the meaning of the integer divisions, we need
3044 * to (re)introduce the div constraints. If we happen to have
3045 * discovered that some of these integer divisions are equal to
3046 * some affine combination of other variables, then these div
3047 * constraints may end up getting simplified in terms of the equalities,
3048 * resulting in extra inequalities on the other variables that
3049 * may have been removed already or that may not even have been
3050 * part of the input. We try and remove those constraints of
3051 * this form that are most obviously redundant with respect to
3052 * the context. We also remove those div constraints that are
3053 * redundant with respect to the other constraints in the result.
3054 *
3055 * The stride constraints among the equality constraints in "bmap" are
3056 * also simplified with respecting to the other equality constraints
3057 * in "bmap" and with respect to all equality constraints in "context".
3058 */
3061 {
3072 }
3078 }
3082 }
3104 }
3123 error:
3127 }
3129 /*
3130 * Assumes context has no implicit divs.
3131 */
3134 {
3145 }
3165 }
3166 }
3170 error:
3174 }
3176 /* Drop all inequalities from "bmap" that also appear in "context".
3177 * "context" is assumed to have only known local variables and
3178 * the initial local variables of "bmap" are assumed to be the same
3179 * as those of "context".
3180 * The constraints of both "bmap" and "context" are assumed
3181 * to have been sorted using isl_basic_map_sort_constraints.
3182 *
3183 * Run through the inequality constraints of "bmap" and "context"
3184 * in sorted order.
3185 * If a constraint of "bmap" involves variables not in "context",
3186 * then it cannot appear in "context".
3187 * If a matching constraint is found, it is removed from "bmap".
3188 */
3191 {
3210 }
3216 }
3220 }
3225 }
3228 }
3231 }
3233 /* Drop all equalities from "bmap" that also appear in "context".
3234 * "context" is assumed to have only known local variables and
3235 * the initial local variables of "bmap" are assumed to be the same
3236 * as those of "context".
3237 *
3238 * Run through the equality constraints of "bmap" and "context"
3239 * in sorted order.
3240 * If a constraint of "bmap" involves variables not in "context",
3241 * then it cannot appear in "context".
3242 * If a matching constraint is found, it is removed from "bmap".
3243 */
3246 {
3270 }
3274 }
3279 }
3282 }
3285 }
3287 /* Remove the constraints in "context" from "bmap".
3288 * "context" is assumed to have explicit representations
3289 * for all local variables.
3290 *
3291 * First align the divs of "bmap" to those of "context" and
3292 * sort the constraints. Then drop all constraints from "bmap"
3293 * that appear in "context".
3294 */
3297 {
3312 }
3331 error:
3335 }
3337 /* Replace "map" by the disjunct at position "pos" and free "context".
3338 */
3341 {
3348 }
3350 /* Remove the constraints in "context" from "map".
3351 * If any of the disjuncts in the result turns out to be the universe,
3352 * then return this universe.
3353 * "context" is assumed to have explicit representations
3354 * for all local variables.
3355 */
3358 {
3368 }
3387 }
3394 error:
3398 }
3400 /* Remove the constraints in "context" from "set".
3401 * If any of the disjuncts in the result turns out to be the universe,
3402 * then return this universe.
3403 * "context" is assumed to have explicit representations
3404 * for all local variables.
3405 */
3408 {
3411 }
3413 /* Remove the constraints in "context" from "map".
3414 * If any of the disjuncts in the result turns out to be the universe,
3415 * then return this universe.
3416 * "context" is assumed to consist of a single disjunct and
3417 * to have explicit representations for all local variables.
3418 */
3421 {
3426 }
3428 /* Replace "map" by a universe map in the same space and free "drop".
3429 */
3432 {
3439 }
3441 /* Return a map that has the same intersection with "context" as "map"
3442 * and that is as "simple" as possible.
3443 *
3444 * If "map" is already the universe, then we cannot make it any simpler.
3445 * Similarly, if "context" is the universe, then we cannot exploit it
3446 * to simplify "map"
3447 * If "map" and "context" are identical to each other, then we can
3448 * return the corresponding universe.
3449 *
3450 * If either "map" or "context" consists of multiple disjuncts,
3451 * then check if "context" happens to be a subset of "map",
3452 * in which case all constraints can be removed.
3453 * In case of multiple disjuncts, the standard procedure
3454 * may not be able to detect that all constraints can be removed.
3455 *
3456 * If none of these cases apply, we have to work a bit harder.
3457 * During this computation, we make use of a single disjunct context,
3458 * so if the original context consists of more than one disjunct
3459 * then we need to approximate the context by a single disjunct set.
3460 * Simply taking the simple hull may drop constraints that are
3461 * only implicitly available in each disjunct. We therefore also
3462 * look for constraints among those defining "map" that are valid
3463 * for the context. These can then be used to simplify away
3464 * the corresponding constraints in "map".
3465 */
3468 {
3472 isl_bool subset;
3483 }
3499 }
3515 list);
3516 }
3518 error:
3522 }
3526 {
3528 }
3532 {
3535 }
3539 {
3542 }
3546 {
3551 }
3555 {
3557 }
3559 /* Compute the gist of "bmap" with respect to the constraints "context"
3560 * on the domain.
3561 */
3564 {
3570 }
3574 {
3578 }
3582 {
3586 }
3590 {
3594 }
3598 {
3600 }
3602 /* Quick check to see if two basic maps are disjoint.
3603 * In particular, we reduce the equalities and inequalities of
3604 * one basic map in the context of the equalities of the other
3605 * basic map and check if we get a contradiction.
3606 */
3609 {
3641 }
3649 }
3658 }
3662 disjoint:
3666 error:
3670 }
3674 {
3677 }
3679 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3680 */
3684 {
3695 }
3696 }
3699 }
3701 /* Are "map1" and "map2" obviously disjoint, based on information
3702 * that can be derived without looking at the individual basic maps?
3703 *
3704 * In particular, if one of them is empty or if they live in different spaces
3705 * (ignoring parameters), then they are clearly disjoint.
3706 */
3709 {
3710 isl_bool disjoint;
3711 isl_bool match;
3735 }
3737 /* Are "map1" and "map2" obviously disjoint?
3738 *
3739 * If one of them is empty or if they live in different spaces (ignoring
3740 * parameters), then they are clearly disjoint.
3741 * This is checked by isl_map_plain_is_disjoint_global.
3742 *
3743 * If they have different parameters, then we skip any further tests.
3744 *
3745 * If they are obviously equal, but not obviously empty, then we will
3746 * not be able to detect if they are disjoint.
3747 *
3748 * Otherwise we check if each basic map in "map1" is obviously disjoint
3749 * from each basic map in "map2".
3750 */
3753 {
3754 isl_bool disjoint;
3755 isl_bool intersect;
3756 isl_bool match;
3771 }
3773 /* Are "map1" and "map2" disjoint?
3774 * The parameters are assumed to have been aligned.
3775 *
3776 * In particular, check whether all pairs of basic maps are disjoint.
3777 */
3780 {
3782 }
3784 /* Are "map1" and "map2" disjoint?
3785 *
3786 * They are disjoint if they are "obviously disjoint" or if one of them
3787 * is empty. Otherwise, they are not disjoint if one of them is universal.
3788 * If the two inputs are (obviously) equal and not empty, then they are
3789 * not disjoint.
3790 * If none of these cases apply, then check if all pairs of basic maps
3791 * are disjoint after aligning the parameters.
3792 */
3794 {
3795 isl_bool disjoint;
3796 isl_bool intersect;
3824 }
3826 /* Are "bmap1" and "bmap2" disjoint?
3827 *
3828 * They are disjoint if they are "obviously disjoint" or if one of them
3829 * is empty. Otherwise, they are not disjoint if one of them is universal.
3830 * If none of these cases apply, we compute the intersection and see if
3831 * the result is empty.
3832 */
3835 {
3836 isl_bool disjoint;
3837 isl_bool intersect;
3866 }
3868 /* Are "bset1" and "bset2" disjoint?
3869 */
3872 {
3874 }
3878 {
3880 }
3882 /* Are "set1" and "set2" disjoint?
3883 */
3885 {
3887 }
3889 /* Is "v" equal to 0, 1 or -1?
3890 */
3892 {
3894 }
3896 /* Are the "n" coefficients starting at "first" of inequality constraints
3897 * "i" and "j" of "bmap" opposite to each other?
3898 */
3901 {
3903 }
3905 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
3906 * apart from the constant term?
3907 */
3909 {
3914 }
3916 /* Check if we can combine a given div with lower bound l and upper
3917 * bound u with some other div and if so return that other div.
3918 * Otherwise, return a position beyond the integer divisions.
3919 * Return -1 on error.
3920 *
3921 * We first check that
3922 * - the bounds are opposites of each other (except for the constant
3923 * term)
3924 * - the bounds do not reference any other div
3925 * - no div is defined in terms of this div
3926 *
3927 * Let m be the size of the range allowed on the div by the bounds.
3928 * That is, the bounds are of the form
3929 *
3930 * e <= a <= e + m - 1
3931 *
3932 * with e some expression in the other variables.
3933 * We look for another div b such that no third div is defined in terms
3934 * of this second div b and such that in any constraint that contains
3935 * a (except for the given lower and upper bound), also contains b
3936 * with a coefficient that is m times that of b.
3937 * That is, all constraints (except for the lower and upper bound)
3938 * are of the form
3939 *
3940 * e + f (a + m b) >= 0
3941 *
3942 * Furthermore, in the constraints that only contain b, the coefficient
3943 * of b should be equal to 1 or -1.
3944 * If so, we return b so that "a + m b" can be replaced by
3945 * a single div "c = a + m b".
3946 */
3949 {
3954 isl_bool opp;
3976 }
3986 }
3999 }
4010 }
4023 }
4028 }
4032 }
4034 /* Internal data structure used during the construction and/or evaluation of
4035 * an inequality that ensures that a pair of bounds always allows
4036 * for an integer value.
4037 *
4038 * "tab" is the tableau in which the inequality is evaluated. It may
4039 * be NULL until it is actually needed.
4040 * "v" contains the inequality coefficients.
4041 * "g", "fl" and "fu" are temporary scalars used during the construction and
4042 * evaluation.
4043 */
4047 isl_int g;
4048 isl_int fl;
4049 isl_int fu;
4050 };
4052 /* Free all the memory allocated by the fields of "data".
4053 */
4055 {
4061 }
4063 /* Is the inequality stored in data->v satisfied by "bmap"?
4064 * That is, does it only attain non-negative values?
4065 * data->tab is a tableau corresponding to "bmap".
4066 */
4069 {
4080 }
4082 /* Given a lower and an upper bound on div i, do they always allow
4083 * for an integer value of the given div?
4084 * Determine this property by constructing an inequality
4085 * such that the property is guaranteed when the inequality is nonnegative.
4086 * The lower bound is inequality l, while the upper bound is inequality u.
4087 * The constructed inequality is stored in data->v.
4088 *
4089 * Let the upper bound be
4090 *
4091 * -n_u a + e_u >= 0
4092 *
4093 * and the lower bound
4094 *
4095 * n_l a + e_l >= 0
4096 *
4097 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4098 * We have
4099 *
4100 * - f_u e_l <= f_u f_l g a <= f_l e_u
4101 *
4102 * Since all variables are integer valued, this is equivalent to
4103 *
4104 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4105 *
4106 * If this interval is at least f_u f_l g, then it contains at least
4107 * one integer value for a.
4108 * That is, the test constraint is
4109 *
4110 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4111 *
4112 * or
4113 *
4114 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4115 *
4116 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4117 * then the constraint can be scaled down by a factor g',
4118 * with the constant term replaced by
4119 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4120 * Note that the result of applying Fourier-Motzkin to this pair
4121 * of constraints is
4122 *
4123 * f_l e_u + f_u e_l >= 0
4124 *
4125 * If the constant term of the scaled down version of this constraint,
4126 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4127 * term of the scaled down test constraint, then the test constraint
4128 * is known to hold and no explicit evaluation is required.
4129 * This is essentially the Omega test.
4130 *
4131 * If the test constraint consists of only a constant term, then
4132 * it is sufficient to look at the sign of this constant term.
4133 */
4136 {
4161 }
4171 }
4173 /* Remove more kinds of divs that are not strictly needed.
4174 * In particular, if all pairs of lower and upper bounds on a div
4175 * are such that they allow at least one integer value of the div,
4176 * then we can eliminate the div using Fourier-Motzkin without
4177 * introducing any spurious solutions.
4178 *
4179 * If at least one of the two constraints has a unit coefficient for the div,
4180 * then the presence of such a value is guaranteed so there is no need to check.
4181 * In particular, the value attained by the bound with unit coefficient
4182 * can serve as this intermediate value.
4183 */
4186 {
4209 isl_bool has_int;
4217 }
4238 }
4241 }
4245 }
4249 }
4252 }
4263 error:
4268 }
4270 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4271 * and the upper bound u, div1 always occurs together with div2 in the form
4272 * (div1 + m div2), where m is the constant range on the variable div1
4273 * allowed by l and u, replace the pair div1 and div2 by a single
4274 * div that is equal to div1 + m div2.
4275 *
4276 * The new div will appear in the location that contains div2.
4277 * We need to modify all constraints that contain
4278 * div2 = (div - div1) / m
4279 * The coefficient of div2 is known to be equal to 1 or -1.
4280 * (If a constraint does not contain div2, it will also not contain div1.)
4281 * If the constraint also contains div1, then we know they appear
4282 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4283 * i.e., the coefficient of div is f.
4284 *
4285 * Otherwise, we first need to introduce div1 into the constraint.
4286 * Let l be
4287 *
4288 * div1 + f >=0
4289 *
4290 * and u
4291 *
4292 * -div1 + f' >= 0
4293 *
4294 * A lower bound on div2
4295 *
4296 * div2 + t >= 0
4297 *
4298 * can be replaced by
4299 *
4300 * m div2 + div1 + m t + f >= 0
4301 *
4302 * An upper bound
4303 *
4304 * -div2 + t >= 0
4305 *
4306 * can be replaced by
4307 *
4308 * -(m div2 + div1) + m t + f' >= 0
4309 *
4310 * These constraint are those that we would obtain from eliminating
4311 * div1 using Fourier-Motzkin.
4312 *
4313 * After all constraints have been modified, we drop the lower and upper
4314 * bound and then drop div1.
4315 * Since the new div is only placed in the same location that used
4316 * to store div2, but otherwise has a different meaning, any possible
4317 * explicit representation of the original div2 is removed.
4318 */
4321 {
4323 isl_int m;
4348 else
4351 }
4355 }
4364 }
4368 }
4370 /* First check if we can coalesce any pair of divs and
4371 * then continue with dropping more redundant divs.
4372 *
4373 * We loop over all pairs of lower and upper bounds on a div
4374 * with coefficient 1 and -1, respectively, check if there
4375 * is any other div "c" with which we can coalesce the div
4376 * and if so, perform the coalescing.
4377 */
4380 {
4409 }
4410 }
4411 }
4416 }
4419 error:
4423 }
4425 /* Are the "n" coefficients starting at "first" of inequality constraints
4426 * "i" and "j" of "bmap" equal to each other?
4427 */
4430 {
4432 }
4434 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4435 * apart from the constant term and the coefficient at position "pos"?
4436 */
4439 {
4445 }
4447 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4448 * apart from the constant term and the coefficient at position "pos"?
4449 */
4452 {
4458 }
4460 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4461 * been modified, simplying it if "simplify" is set.
4462 * Free the temporary data structure "pairs" that was associated
4463 * to the old version of "bmap".
4464 */
4467 {
4472 }
4474 /* Is "div" the single unknown existentially quantified variable
4475 * in inequality constraint "ineq" of "bmap"?
4476 * "div" is known to have a non-zero coefficient in "ineq".
4477 */
4480 {
4483 isl_bool known;
4493 isl_bool known;
4502 }
4505 }
4507 /* Does integer division "div" have coefficient 1 in inequality constraint
4508 * "ineq" of "map"?
4509 */
4511 {
4519 }
4521 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4522 * then try and drop redundant divs again,
4523 * freeing the temporary data structure "pairs" that was associated
4524 * to the old version of "bmap".
4525 */
4528 {
4532 }
4534 /* Drop the integer division at position "div", along with the two
4535 * inequality constraints "ineq1" and "ineq2" in which it appears
4536 * from "bmap" and then try and drop redundant divs again,
4537 * freeing the temporary data structure "pairs" that was associated
4538 * to the old version of "bmap".
4539 */
4543 {
4550 }
4553 }
4555 /* Given two inequality constraints
4556 *
4557 * f(x) + n d + c >= 0, (ineq)
4558 *
4559 * with d the variable at position "pos", and
4560 *
4561 * f(x) + c0 >= 0, (lower)
4562 *
4563 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4564 * determined by the first constraint.
4565 * That is, store
4566 *
4567 * ceil((c0 - c)/n)
4568 *
4569 * in *l.
4570 */
4573 {
4577 }
4579 /* Given two inequality constraints
4580 *
4581 * f(x) + n d + c >= 0, (ineq)
4582 *
4583 * with d the variable at position "pos", and
4584 *
4585 * -f(x) - c0 >= 0, (upper)
4586 *
4587 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4588 * determined by the first constraint.
4589 * That is, store
4590 *
4591 * ceil((-c1 - c)/n)
4592 *
4593 * in *u.
4594 */
4597 {
4601 }
4603 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4604 * does the corresponding lower bound have a fixed value in "bmap"?
4605 *
4606 * In particular, "ineq" is of the form
4607 *
4608 * f(x) + n d + c >= 0
4609 *
4610 * with n > 0, c the constant term and
4611 * d the existentially quantified variable "div".
4612 * That is, the lower bound is
4613 *
4614 * ceil((-f(x) - c)/n)
4615 *
4616 * Look for a pair of constraints
4617 *
4618 * f(x) + c0 >= 0
4619 * -f(x) + c1 >= 0
4620 *
4621 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4622 * That is, check that
4623 *
4624 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4625 *
4626 * If so, return the index of inequality f(x) + c0 >= 0.
4627 * Otherwise, return bmap->n_ineq.
4628 * Return -1 on error.
4629 */
4631 {
4654 }
4662 }
4679 }
4681 /* Given a lower bound constraint "ineq" on the existentially quantified
4682 * variable "div", such that the corresponding lower bound has
4683 * a fixed value in "bmap", assign this fixed value to the variable and
4684 * then try and drop redundant divs again,
4685 * freeing the temporary data structure "pairs" that was associated
4686 * to the old version of "bmap".
4687 * "lower" determines the constant value for the lower bound.
4688 *
4689 * In particular, "ineq" is of the form
4690 *
4691 * f(x) + n d + c >= 0,
4692 *
4693 * while "lower" is of the form
4694 *
4695 * f(x) + c0 >= 0
4696 *
4697 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4698 * is ceil((c0 - c)/n).
4699 */
4702 {
4703 isl_int c;
4716 }
4718 /* Remove divs that are not strictly needed based on the inequality
4719 * constraints.
4720 * In particular, if a div only occurs positively (or negatively)
4721 * in constraints, then it can simply be dropped.
4722 * Also, if a div occurs in only two constraints and if moreover
4723 * those two constraints are opposite to each other, except for the constant
4724 * term and if the sum of the constant terms is such that for any value
4725 * of the other values, there is always at least one integer value of the
4726 * div, i.e., if one plus this sum is greater than or equal to
4727 * the (absolute value) of the coefficient of the div in the constraints,
4728 * then we can also simply drop the div.
4729 *
4730 * If an existentially quantified variable does not have an explicit
4731 * representation, appears in only a single lower bound that does not
4732 * involve any other such existentially quantified variables and appears
4733 * in this lower bound with coefficient 1,
4734 * then fix the variable to the value of the lower bound. That is,
4735 * turn the inequality into an equality.
4736 * If for any value of the other variables, there is any value
4737 * for the existentially quantified variable satisfying the constraints,
4738 * then this lower bound also satisfies the constraints.
4739 * It is therefore safe to pick this lower bound.
4740 *
4741 * The same reasoning holds even if the coefficient is not one.
4742 * However, fixing the variable to the value of the lower bound may
4743 * in general introduce an extra integer division, in which case
4744 * it may be better to pick another value.
4745 * If this integer division has a known constant value, then plugging
4746 * in this constant value removes the existentially quantified variable
4747 * completely. In particular, if the lower bound is of the form
4748 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4749 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4750 * then the existentially quantified variable can be assigned this
4751 * shared value.
4752 *
4753 * We skip divs that appear in equalities or in the definition of other divs.
4754 * Divs that appear in the definition of other divs usually occur in at least
4755 * 4 constraints, but the constraints may have been simplified.
4756 *
4757 * If any divs are left after these simple checks then we move on
4758 * to more complicated cases in drop_more_redundant_divs.
4759 */
4762 {
4806 }
4810 }
4811 }
4819 }
4822 else
4842 pairs);
4848 pairs);
4850 }
4867 else
4874 }
4877 }
4884 error:
4888 }
4890 /* Consider the coefficients at "c" as a row vector and replace
4891 * them with their product with "T". "T" is assumed to be a square matrix.
4892 */
4894 {
4916 }
4918 /* Plug in T for the variables in "bmap" starting at "pos".
4919 * T is a linear unimodular matrix, i.e., without constant term.
4920 */
4923 {
4950 }
4954 error:
4958 }
4960 /* Remove divs that are not strictly needed.
4961 *
4962 * First look for an equality constraint involving two or more
4963 * existentially quantified variables without an explicit
4964 * representation. Replace the combination that appears
4965 * in the equality constraint by a single existentially quantified
4966 * variable such that the equality can be used to derive
4967 * an explicit representation for the variable.
4968 * If there are no more such equality constraints, then continue
4969 * with isl_basic_map_drop_redundant_divs_ineq.
4970 *
4971 * In particular, if the equality constraint is of the form
4972 *
4973 * f(x) + \sum_i c_i a_i = 0
4974 *
4975 * with a_i existentially quantified variable without explicit
4976 * representation, then apply a transformation on the existentially
4977 * quantified variables to turn the constraint into
4978 *
4979 * f(x) + g a_1' = 0
4980 *
4981 * with g the gcd of the c_i.
4982 * In order to easily identify which existentially quantified variables
4983 * have a complete explicit representation, i.e., without being defined
4984 * in terms of other existentially quantified variables without
4985 * an explicit representation, the existentially quantified variables
4986 * are first sorted.
4987 *
4988 * The variable transformation is computed by extending the row
4989 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4990 *
4991 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4992 * [a_2'] [ a_2 ]
4993 * ... = U ....
4994 * [a_n'] [ a_n ]
4995 *
4996 * with [c_1/g ... c_n/g] representing the first row of U.
4997 * The inverse of U is then plugged into the original constraints.
4998 * The call to isl_basic_map_simplify makes sure the explicit
4999 * representation for a_1' is extracted from the equality constraint.
5000 */
5003 {
5038 }
5057 }
5059 /* Does "bmap" satisfy any equality that involves more than 2 variables
5060 * and/or has coefficients different from -1 and 1?
5061 */
5063 {
5092 }
5095 }
5097 /* Remove any common factor g from the constraint coefficients in "v".
5098 * The constant term is stored in the first position and is replaced
5099 * by floor(c/g). If any common factor is removed and if this results
5100 * in a tightening of the constraint, then set *tightened.
5101 */
5104 {
5124 }
5126 /* If "bmap" is an integer set that satisfies any equality involving
5127 * more than 2 variables and/or has coefficients different from -1 and 1,
5128 * then use variable compression to reduce the coefficients by removing
5129 * any (hidden) common factor.
5130 * In particular, apply the variable compression to each constraint,
5131 * factor out any common factor in the non-constant coefficients and
5132 * then apply the inverse of the compression.
5133 * At the end, we mark the basic map as having reduced constants.
5134 * If this flag is still set on the next invocation of this function,
5135 * then we skip the computation.
5136 *
5137 * Removing a common factor may result in a tightening of some of
5138 * the constraints. If this happens, then we may end up with two
5139 * opposite inequalities that can be replaced by an equality.
5140 * We therefore call isl_basic_map_detect_inequality_pairs,
5141 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5142 * and isl_basic_map_gauss if such a pair was found.
5143 *
5144 * Note that this function may leave the result in an inconsistent state.
5145 * In particular, the constraints may not be gaussed.
5146 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5147 * for some of the test cases to pass successfully.
5148 * Any potential modification of the representation is therefore only
5149 * performed on a single copy of the basic map.
5150 */
5153 {
5155 isl_bool multi;
5191 }
5206 }
5221 }
5222 }
5225 error:
5230 }
5232 /* Shift the integer division at position "div" of "bmap"
5233 * by "shift" times the variable at position "pos".
5234 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5235 * corresponds to the constant term.
5236 *
5237 * That is, if the integer division has the form
5238 *
5239 * floor(f(x)/d)
5240 *
5241 * then replace it by
5242 *
5243 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5244 */
5247 {
5266 }
5272 }
5280 }
5283 }