| blob | 06806d9e9841968f4bb00d90c536b9bde2cc01ab |
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 */
238 {
254 }
256 /* Remove any common factor in numerator and denominator of a div expression,
257 * not taking into account the constant term.
258 * That is, look for any div of the form
259 *
260 * floor((a + m f(x))/(m d))
261 *
262 * and replace it by
263 *
264 * floor((floor(a/m) + f(x))/d)
265 *
266 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
267 * and can therefore not influence the result of the floor.
268 */
271 {
283 }
285 /* Assumes divs have been ordered if keep_divs is set.
286 */
289 {
307 }
317 }
326 /* We need to be careful about circular definitions,
327 * so for now we just remove the definition of div k
328 * if the equality contains any divs.
329 * If keep_divs is set, then the divs have been ordered
330 * and we can keep the definition as long as the result
331 * is still ordered.
332 */
340 }
341 }
343 /* Assumes divs have been ordered if keep_divs is set.
344 */
347 {
355 }
357 /* Check if elimination of div "div" using equality "eq" would not
358 * result in a div depending on a later div.
359 */
362 {
377 }
380 }
382 /* Eliminate divs based on equalities
383 */
386 {
401 isl_bool ok;
417 }
418 }
422 }
424 /* Eliminate divs based on inequalities
425 */
428 {
458 }
460 }
462 /* Does the equality constraint at position "eq" in "bmap" involve
463 * any local variables in the range [first, first + n)
464 * that are not marked as having an explicit representation?
465 */
468 {
477 isl_bool unknown;
486 }
489 }
491 /* The last local variable involved in the equality constraint
492 * at position "eq" in "bmap" is the local variable at position "div".
493 * It can therefore be used to extract an explicit representation
494 * for that variable.
495 * Do so unless the local variable already has an explicit representation or
496 * the explicit representation would involve any other local variables
497 * that in turn do not have an explicit representation.
498 * An equality constraint involving local variables without an explicit
499 * representation can be used in isl_basic_map_drop_redundant_divs
500 * to separate out an independent local variable. Introducing
501 * an explicit representation here would block this transformation,
502 * while the partial explicit representation in itself is not very useful.
503 * Set *progress if anything is changed.
504 *
505 * The equality constraint is of the form
506 *
507 * f(x) + n e >= 0
508 *
509 * with n a positive number. The explicit representation derived from
510 * this constraint is
511 *
512 * floor((-f(x))/n)
513 */
516 {
518 isl_bool involves;
542 }
546 {
569 }
578 progress);
585 }
592 }
595 }
599 {
601 progress));
602 }
606 {
612 }
614 }
616 /* Hash table of inequalities in a basic map.
617 * "index" is an array of addresses of inequalities in the basic map, some
618 * of which are NULL. The inequalities are hashed on the coefficients
619 * except the constant term.
620 * "size" is the number of elements in the array and is always a power of two
621 * "bits" is the number of bits need to represent an index into the array.
622 * "total" is the total dimension of the basic map.
623 */
629 };
631 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
632 */
635 {
652 }
654 /* Free the memory allocated by create_constraint_index.
655 */
657 {
659 }
661 /* Return the position in ci->index that contains the address of
662 * an inequality that is equal to *ineq up to the constant term,
663 * provided this address is not identical to "ineq".
664 * If there is no such inequality, then return the position where
665 * such an inequality should be inserted.
666 */
668 {
676 }
678 /* Return the position in ci->index that contains the address of
679 * an inequality that is equal to the k'th inequality of "bmap"
680 * up to the constant term, provided it does not point to the very
681 * same inequality.
682 * If there is no such inequality, then return the position where
683 * such an inequality should be inserted.
684 */
687 {
689 }
693 {
695 }
697 /* Fill in the "ci" data structure with the inequalities of "bset".
698 */
701 {
710 }
713 }
715 /* Is the inequality ineq (obviously) redundant with respect
716 * to the constraints in "ci"?
717 *
718 * Look for an inequality in "ci" with the same coefficients and then
719 * check if the contant term of "ineq" is greater than or equal
720 * to the constant term of that inequality. If so, "ineq" is clearly
721 * redundant.
722 *
723 * Note that hash_index_ineq ignores a stored constraint if it has
724 * the same address as the passed inequality. It is ok to pass
725 * the address of a local variable here since it will never be
726 * the same as the address of a constraint in "ci".
727 */
730 {
737 }
739 /* If we can eliminate more than one div, then we need to make
740 * sure we do it from last div to first div, in order not to
741 * change the position of the other divs that still need to
742 * be removed.
743 */
746 {
800 }
802 }
814 }
817 out:
821 }
824 {
836 }
838 }
840 /* Normalize divs that appear in equalities.
841 *
842 * In particular, we assume that bmap contains some equalities
843 * of the form
844 *
845 * a x = m * e_i
846 *
847 * and we want to replace the set of e_i by a minimal set and
848 * such that the new e_i have a canonical representation in terms
849 * of the vector x.
850 * If any of the equalities involves more than one divs, then
851 * we currently simply bail out.
852 *
853 * Let us first additionally assume that all equalities involve
854 * a div. The equalities then express modulo constraints on the
855 * remaining variables and we can use "parameter compression"
856 * to find a minimal set of constraints. The result is a transformation
857 *
858 * x = T(x') = x_0 + G x'
859 *
860 * with G a lower-triangular matrix with all elements below the diagonal
861 * non-negative and smaller than the diagonal element on the same row.
862 * We first normalize x_0 by making the same property hold in the affine
863 * T matrix.
864 * The rows i of G with a 1 on the diagonal do not impose any modulo
865 * constraint and simply express x_i = x'_i.
866 * For each of the remaining rows i, we introduce a div and a corresponding
867 * equality. In particular
868 *
869 * g_ii e_j = x_i - g_i(x')
870 *
871 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
872 * corresponding div (if g_kk != 1).
873 *
874 * If there are any equalities not involving any div, then we
875 * first apply a variable compression on the variables x:
876 *
877 * x = C x'' x'' = C_2 x
878 *
879 * and perform the above parameter compression on A C instead of on A.
880 * The resulting compression is then of the form
881 *
882 * x'' = T(x') = x_0 + G x'
883 *
884 * and in constructing the new divs and the corresponding equalities,
885 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
886 * by the corresponding row from C_2.
887 */
890 {
899 isl_int v;
931 }
932 }
941 }
947 }
957 }
964 }
969 /* We have to be careful because dropping equalities may reorder them */
979 }
980 }
986 else
987 needed++;
988 }
994 }
1004 else
1012 else
1015 }
1019 }
1026 done:
1030 error:
1036 }
1040 {
1050 }
1052 /* Check whether it is ok to define a div based on an inequality.
1053 * To avoid the introduction of circular definitions of divs, we
1054 * do not allow such a definition if the resulting expression would refer to
1055 * any other undefined divs or if any known div is defined in
1056 * terms of the unknown div.
1057 */
1060 {
1064 /* Not defined in terms of unknown divs */
1072 }
1074 /* No other div defined in terms of this one => avoid loops */
1082 }
1085 }
1087 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1088 * be a better expression than the current one?
1089 *
1090 * If we do not have any expression yet, then any expression would be better.
1091 * Otherwise we check if the last variable involved in the inequality
1092 * (disregarding the div that it would define) is in an earlier position
1093 * than the last variable involved in the current div expression.
1094 */
1097 {
1114 }
1116 /* Given two constraints "k" and "l" that are opposite to each other,
1117 * except for the constant term, check if we can use them
1118 * to obtain an expression for one of the hitherto unknown divs or
1119 * a "better" expression for a div for which we already have an expression.
1120 * "sum" is the sum of the constant terms of the constraints.
1121 * If this sum is strictly smaller than the coefficient of one
1122 * of the divs, then this pair can be used define the div.
1123 * To avoid the introduction of circular definitions of divs, we
1124 * do not use the pair if the resulting expression would refer to
1125 * any other undefined divs or if any known div is defined in
1126 * terms of the unknown div.
1127 */
1131 {
1136 isl_bool set_div;
1151 else
1156 }
1158 }
1162 {
1166 isl_int sum;
1181 }
1189 }
1204 }
1206 /* We need to break out of the loop after these
1207 * changes since the contents of the hash
1208 * will no longer be valid.
1209 * Plus, we probably we want to regauss first.
1210 */
1218 }
1223 }
1225 /* Detect all pairs of inequalities that form an equality.
1226 *
1227 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1228 * Call it repeatedly while it is making progress.
1229 */
1232 {
1244 }
1246 /* Eliminate knowns divs from constraints where they appear with
1247 * a (positive or negative) unit coefficient.
1248 *
1249 * That is, replace
1250 *
1251 * floor(e/m) + f >= 0
1252 *
1253 * by
1254 *
1255 * e + m f >= 0
1256 *
1257 * and
1258 *
1259 * -floor(e/m) + f >= 0
1260 *
1261 * by
1262 *
1263 * -e + m f + m - 1 >= 0
1264 *
1265 * The first conversion is valid because floor(e/m) >= -f is equivalent
1266 * to e/m >= -f because -f is an integral expression.
1267 * The second conversion follows from the fact that
1268 *
1269 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1270 *
1271 *
1272 * Note that one of the div constraints may have been eliminated
1273 * due to being redundant with respect to the constraint that is
1274 * being modified by this function. The modified constraint may
1275 * no longer imply this div constraint, so we add it back to make
1276 * sure we do not lose any information.
1277 *
1278 * We skip integral divs, i.e., those with denominator 1, as we would
1279 * risk eliminating the div from the div constraints. We do not need
1280 * to handle those divs here anyway since the div constraints will turn
1281 * out to form an equality and this equality can then be used to eliminate
1282 * the div from all constraints.
1283 */
1286 {
1318 else
1328 }
1333 }
1334 }
1337 }
1340 {
1345 isl_bool empty;
1361 /* requires equalities in normal form */
1367 }
1369 }
1372 {
1374 }
1379 {
1411 }
1415 {
1417 }
1420 /* If the only constraints a div d=floor(f/m)
1421 * appears in are its two defining constraints
1422 *
1423 * f - m d >=0
1424 * -(f - (m - 1)) + m d >= 0
1425 *
1426 * then it can safely be removed.
1427 */
1429 {
1438 isl_bool red;
1445 }
1452 }
1455 }
1457 /*
1458 * Remove divs that don't occur in any of the constraints or other divs.
1459 * These can arise when dropping constraints from a basic map or
1460 * when the divs of a basic map have been temporarily aligned
1461 * with the divs of another basic map.
1462 */
1465 {
1472 isl_bool redundant;
1480 }
1482 }
1484 /* Mark "bmap" as final, without checking for obviously redundant
1485 * integer divisions. This function should be used when "bmap"
1486 * is known not to involve any such integer divisions.
1487 */
1490 {
1495 }
1497 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1498 */
1500 {
1504 }
1507 {
1509 }
1511 /* Remove definition of any div that is defined in terms of the given variable.
1512 * The div itself is not removed. Functions such as
1513 * eliminate_divs_ineq depend on the other divs remaining in place.
1514 */
1517 {
1531 }
1533 }
1535 /* Eliminate the specified variables from the constraints using
1536 * Fourier-Motzkin. The variables themselves are not removed.
1537 */
1540 {
1572 }
1579 n_lower++;
1581 n_upper++;
1582 }
1606 }
1609 }
1621 }
1622 }
1627 error:
1630 }
1634 {
1637 }
1639 /* Eliminate the specified n dimensions starting at first from the
1640 * constraints, without removing the dimensions from the space.
1641 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1642 * Otherwise, they are projected out and the original space is restored.
1643 */
1647 {
1663 }
1670 error:
1673 }
1678 {
1680 }
1682 /* Remove all constraints from "bmap" that reference any unknown local
1683 * variables (directly or indirectly).
1684 *
1685 * Dropping all constraints on a local variable will make it redundant,
1686 * so it will get removed implicitly by
1687 * isl_basic_map_drop_constraints_involving_dims. Some other local
1688 * variables may also end up becoming redundant if they only appear
1689 * in constraints together with the unknown local variable.
1690 * Therefore, start over after calling
1691 * isl_basic_map_drop_constraints_involving_dims.
1692 */
1695 {
1696 isl_bool known;
1721 }
1724 }
1726 /* Remove all constraints from "map" that reference any unknown local
1727 * variables (directly or indirectly).
1728 *
1729 * Since constraints may get dropped from the basic maps,
1730 * they may no longer be disjoint from each other.
1731 */
1734 {
1736 isl_bool known;
1754 }
1760 }
1762 /* Don't assume equalities are in order, because align_divs
1763 * may have changed the order of the divs.
1764 */
1766 {
1779 }
1780 }
1781 }
1785 {
1787 }
1791 {
1805 }
1807 }
1809 }
1813 {
1816 }
1820 {
1830 }
1851 error:
1855 }
1857 /* For each inequality in "ineq" that is a shifted (more relaxed)
1858 * copy of an inequality in "context", mark the corresponding entry
1859 * in "row" with -1.
1860 * If an inequality only has a non-negative constant term, then
1861 * mark it as well.
1862 */
1865 {
1882 isl_bool redundant;
1888 }
1895 }
1898 error:
1901 }
1905 {
1918 isl_bool redundant;
1930 }
1933 error:
1936 }
1938 /* Remove constraints from "bmap" that are identical to constraints
1939 * in "context" or that are more relaxed (greater constant term).
1940 *
1941 * We perform the test for shifted copies on the pure constraints
1942 * in remove_shifted_constraints.
1943 */
1946 {
1955 }
1968 error:
1972 }
1974 /* Does the (linear part of a) constraint "c" involve any of the "len"
1975 * "relevant" dimensions?
1976 */
1978 {
1986 }
1989 }
1991 /* Drop constraints from "bmap" that do not involve any of
1992 * the dimensions marked "relevant".
1993 */
1996 {
2011 }
2018 }
2021 }
2023 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2024 *
2025 * In particular, for any variable involved in the constraint,
2026 * find the actual group id from before and replace the group
2027 * of the corresponding variable by the minimal group of all
2028 * the variables involved in the constraint considered so far
2029 * (if this minimum is smaller) or replace the minimum by this group
2030 * (if the minimum is larger).
2031 *
2032 * At the end, all the variables in "c" will (indirectly) point
2033 * to the minimal of the groups that they referred to originally.
2034 */
2036 {
2053 }
2054 }
2056 /* Allocate an array of groups of variables, one for each variable
2057 * in "context", initialized to zero.
2058 */
2060 {
2067 }
2069 /* Drop constraints from "bmap" that only involve variables that are
2070 * not related to any of the variables marked with a "-1" in "group".
2071 *
2072 * We construct groups of variables that collect variables that
2073 * (indirectly) appear in some common constraint of "bmap".
2074 * Each group is identified by the first variable in the group,
2075 * except for the special group of variables that was already identified
2076 * in the input as -1 (or are related to those variables).
2077 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2078 * otherwise the group of i is the group of group[i].
2079 *
2080 * We first initialize groups for the remaining variables.
2081 * Then we iterate over the constraints of "bmap" and update the
2082 * group of the variables in the constraint by the smallest group.
2083 * Finally, we resolve indirect references to groups by running over
2084 * the variables.
2085 *
2086 * After computing the groups, we drop constraints that do not involve
2087 * any variables in the -1 group.
2088 */
2091 {
2108 }
2126 }
2128 /* Drop constraints from "context" that are irrelevant for computing
2129 * the gist of "bset".
2130 *
2131 * In particular, drop constraints in variables that are not related
2132 * to any of the variables involved in the constraints of "bset"
2133 * in the sense that there is no sequence of constraints that connects them.
2134 *
2135 * We first mark all variables that appear in "bset" as belonging
2136 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2137 */
2140 {
2161 }
2167 }
2170 }
2172 /* Drop constraints from "context" that are irrelevant for computing
2173 * the gist of the inequalities "ineq".
2174 * Inequalities in "ineq" for which the corresponding element of row
2175 * is set to -1 have already been marked for removal and should be ignored.
2176 *
2177 * In particular, drop constraints in variables that are not related
2178 * to any of the variables involved in "ineq"
2179 * in the sense that there is no sequence of constraints that connects them.
2180 *
2181 * We first mark all variables that appear in "bset" as belonging
2182 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2183 */
2186 {
2207 }
2210 }
2213 }
2215 /* Do all "n" entries of "row" contain a negative value?
2216 */
2218 {
2226 }
2228 /* Update the inequalities in "bset" based on the information in "row"
2229 * and "tab".
2230 *
2231 * In particular, the array "row" contains either -1, meaning that
2232 * the corresponding inequality of "bset" is redundant, or the index
2233 * of an inequality in "tab".
2234 *
2235 * If the row entry is -1, then drop the inequality.
2236 * Otherwise, if the constraint is marked redundant in the tableau,
2237 * then drop the inequality. Similarly, if it is marked as an equality
2238 * in the tableau, then turn the inequality into an equality and
2239 * perform Gaussian elimination.
2240 */
2243 {
2260 }
2270 }
2271 }
2277 }
2279 /* Update the inequalities in "bset" based on the information in "row"
2280 * and "tab" and free all arguments (other than "bset").
2281 */
2286 {
2295 }
2297 /* Remove all information from bset that is redundant in the context
2298 * of context.
2299 * "ineq" contains the (possibly transformed) inequalities of "bset",
2300 * in the same order.
2301 * The (explicit) equalities of "bset" are assumed to have been taken
2302 * into account by the transformation such that only the inequalities
2303 * are relevant.
2304 * "context" is assumed not to be empty.
2305 *
2306 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2307 * A value of -1 means that the inequality is obviously redundant and may
2308 * not even appear in "tab".
2309 *
2310 * We first mark the inequalities of "bset"
2311 * that are obviously redundant with respect to some inequality in "context".
2312 * Then we remove those constraints from "context" that have become
2313 * irrelevant for computing the gist of "bset".
2314 * Note that this removal of constraints cannot be replaced by
2315 * a factorization because factors in "bset" may still be connected
2316 * to each other through constraints in "context".
2317 *
2318 * If there are any inequalities left, we construct a tableau for
2319 * the context and then add the inequalities of "bset".
2320 * Before adding these inequalities, we freeze all constraints such that
2321 * they won't be considered redundant in terms of the constraints of "bset".
2322 * Then we detect all redundant constraints (among the
2323 * constraints that weren't frozen), first by checking for redundancy in the
2324 * the tableau and then by checking if replacing a constraint by its negation
2325 * would lead to an empty set. This last step is fairly expensive
2326 * and could be optimized by more reuse of the tableau.
2327 * Finally, we update bset according to the results.
2328 */
2331 {
2346 }
2382 }
2406 }
2411 }
2415 error:
2423 }
2425 /* Extract the inequalities of "bset" as an isl_mat.
2426 */
2428 {
2442 }
2444 /* Remove all information from "bset" that is redundant in the context
2445 * of "context", for the case where both "bset" and "context" are
2446 * full-dimensional.
2447 */
2450 {
2455 }
2457 /* Remove all information from "bset" that is redundant in the context
2458 * of "context", for the case where the combined equalities of
2459 * "bset" and "context" allow for a compression that can be obtained
2460 * by preapplication of "T".
2461 *
2462 * "bset" itself is not transformed by "T". Instead, the inequalities
2463 * are extracted from "bset" and those are transformed by "T".
2464 * uset_gist_full then determines which of the transformed inequalities
2465 * are redundant with respect to the transformed "context" and removes
2466 * the corresponding inequalities from "bset".
2467 *
2468 * After preapplying "T" to the inequalities, any common factor is
2469 * removed from the coefficients. If this results in a tightening
2470 * of the constant term, then the same tightening is applied to
2471 * the corresponding untransformed inequality in "bset".
2472 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2473 *
2474 * g f'(x) + r >= 0
2475 *
2476 * with 0 <= r < g, then it is equivalent to
2477 *
2478 * f'(x) >= 0
2479 *
2480 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2481 * subspace compressed by T since the latter would be transformed to
2482 *
2483 * g f'(x) >= 0
2484 */
2488 {
2492 isl_int rem;
2504 }
2527 }
2531 error:
2536 }
2538 /* Project "bset" onto the variables that are involved in "template".
2539 */
2542 {
2551 isl_bool involved;
2560 }
2563 }
2565 /* Remove all information from bset that is redundant in the context
2566 * of context. In particular, equalities that are linear combinations
2567 * of those in context are removed. Then the inequalities that are
2568 * redundant in the context of the equalities and inequalities of
2569 * context are removed.
2570 *
2571 * First of all, we drop those constraints from "context"
2572 * that are irrelevant for computing the gist of "bset".
2573 * Alternatively, we could factorize the intersection of "context" and "bset".
2574 *
2575 * We first compute the intersection of the integer affine hulls
2576 * of "bset" and "context",
2577 * compute the gist inside this intersection and then reduce
2578 * the constraints with respect to the equalities of the context
2579 * that only involve variables already involved in the input.
2580 *
2581 * If two constraints are mutually redundant, then uset_gist_full
2582 * will remove the second of those constraints. We therefore first
2583 * sort the constraints so that constraints not involving existentially
2584 * quantified variables are given precedence over those that do.
2585 * We have to perform this sorting before the variable compression,
2586 * because that may effect the order of the variables.
2587 */
2590 {
2615 }
2620 }
2630 }
2641 }
2644 error:
2648 }
2650 /* Return the number of equality constraints in "bmap" that involve
2651 * local variables. This function assumes that Gaussian elimination
2652 * has been applied to the equality constraints.
2653 */
2655 {
2675 }
2677 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2678 * The constraints are assumed not to involve any local variables.
2679 */
2682 {
2700 }
2705 error:
2710 }
2712 /* Construct and return a variable compression based on the equality
2713 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2714 * "n1" is the number of (initial) equality constraints in "bmap1"
2715 * that do involve local variables.
2716 * "n2" is the number of (initial) equality constraints in "bmap2"
2717 * that do involve local variables.
2718 * "total" is the total number of other variables.
2719 * This function assumes that Gaussian elimination
2720 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2721 * such that the equality constraints not involving local variables
2722 * are those that start at "n1" or "n2".
2723 *
2724 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2725 * then simply compute the compression based on the equality constraints
2726 * in the other basic map.
2727 * Otherwise, combine the equality constraints from both into a new
2728 * basic map such that Gaussian elimination can be applied to this combination
2729 * and then construct a variable compression from the resulting
2730 * equality constraints.
2731 */
2735 {
2745 }
2750 }
2765 }
2767 /* Extract the stride constraints from "bmap", compressed
2768 * with respect to both the stride constraints in "context" and
2769 * the remaining equality constraints in both "bmap" and "context".
2770 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2771 * "context_n_eq" is the number of (initial) stride constraints in "context".
2772 *
2773 * Let x be all variables in "bmap" (and "context") other than the local
2774 * variables. First compute a variable compression
2775 *
2776 * x = V x'
2777 *
2778 * based on the non-stride equality constraints in "bmap" and "context".
2779 * Consider the stride constraints of "context",
2780 *
2781 * A(x) + B(y) = 0
2782 *
2783 * with y the local variables and plug in the variable compression,
2784 * resulting in
2785 *
2786 * A(V x') + B(y) = 0
2787 *
2788 * Use these constraints to compute a parameter compression on x'
2789 *
2790 * x' = T x''
2791 *
2792 * Now consider the stride constraints of "bmap"
2793 *
2794 * C(x) + D(y) = 0
2795 *
2796 * and plug in x = V*T x''.
2797 * That is, return A = [C*V*T D].
2798 */
2802 {
2831 }
2833 /* Remove the prime factors from *g that have an exponent that
2834 * is strictly smaller than the exponent in "c".
2835 * All exponents in *g are known to be smaller than or equal
2836 * to those in "c".
2837 *
2838 * That is, if *g is equal to
2839 *
2840 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2841 *
2842 * and "c" is equal to
2843 *
2844 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2845 *
2846 * then update *g to
2847 *
2848 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2849 * p_n^{e_n * (e_n = f_n)}
2850 *
2851 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2852 * neither does the gcd of *g and c / *g.
2853 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2854 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2855 * Dividing *g by this gcd therefore strictly reduces the exponent
2856 * of the prime factors that need to be removed, while leaving the
2857 * other prime factors untouched.
2858 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2859 * removes all undesired factors, without removing any others.
2860 */
2862 {
2863 isl_int t;
2872 }
2874 }
2876 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2877 * of the same stride constraints in a compressed space that exploits
2878 * all equalities in the context and the other equalities in "bmap".
2879 *
2880 * If the stride constraints of "bmap" are of the form
2881 *
2882 * C(x) + D(y) = 0
2883 *
2884 * then A is of the form
2885 *
2886 * B(x') + D(y) = 0
2887 *
2888 * If any of these constraints involves only a single local variable y,
2889 * then the constraint appears as
2890 *
2891 * f(x) + m y_i = 0
2892 *
2893 * in "bmap" and as
2894 *
2895 * h(x') + m y_i = 0
2896 *
2897 * in "A".
2898 *
2899 * Let g be the gcd of m and the coefficients of h.
2900 * Then, in particular, g is a divisor of the coefficients of h and
2901 *
2902 * f(x) = h(x')
2903 *
2904 * is known to be a multiple of g.
2905 * If some prime factor in m appears with the same exponent in g,
2906 * then it can be removed from m because f(x) is already known
2907 * to be a multiple of g and therefore in particular of this power
2908 * of the prime factors.
2909 * Prime factors that appear with a smaller exponent in g cannot
2910 * be removed from m.
2911 * Let g' be the divisor of g containing all prime factors that
2912 * appear with the same exponent in m and g, then
2913 *
2914 * f(x) + m y_i = 0
2915 *
2916 * can be replaced by
2917 *
2918 * f(x) + m/g' y_i' = 0
2919 *
2920 * Note that (if g' != 1) this changes the explicit representation
2921 * of y_i to that of y_i', so the integer division at position i
2922 * is marked unknown and later recomputed by a call to
2923 * isl_basic_map_gauss.
2924 */
2927 {
2931 isl_int gcd;
2965 }
2972 error:
2976 }
2978 /* Simplify the stride constraints in "bmap" based on
2979 * the remaining equality constraints in "bmap" and all equality
2980 * constraints in "context".
2981 * Only do this if both "bmap" and "context" have stride constraints.
2982 *
2983 * First extract a copy of the stride constraints in "bmap" in a compressed
2984 * space exploiting all the other equality constraints and then
2985 * use this compressed copy to simplify the original stride constraints.
2986 */
2989 {
3011 }
3013 /* Return a basic map that has the same intersection with "context" as "bmap"
3014 * and that is as "simple" as possible.
3015 *
3016 * The core computation is performed on the pure constraints.
3017 * When we add back the meaning of the integer divisions, we need
3018 * to (re)introduce the div constraints. If we happen to have
3019 * discovered that some of these integer divisions are equal to
3020 * some affine combination of other variables, then these div
3021 * constraints may end up getting simplified in terms of the equalities,
3022 * resulting in extra inequalities on the other variables that
3023 * may have been removed already or that may not even have been
3024 * part of the input. We try and remove those constraints of
3025 * this form that are most obviously redundant with respect to
3026 * the context. We also remove those div constraints that are
3027 * redundant with respect to the other constraints in the result.
3028 *
3029 * The stride constraints among the equality constraints in "bmap" are
3030 * also simplified with respecting to the other equality constraints
3031 * in "bmap" and with respect to all equality constraints in "context".
3032 */
3035 {
3046 }
3052 }
3056 }
3078 }
3097 error:
3101 }
3103 /*
3104 * Assumes context has no implicit divs.
3105 */
3108 {
3119 }
3139 }
3140 }
3144 error:
3148 }
3150 /* Drop all inequalities from "bmap" that also appear in "context".
3151 * "context" is assumed to have only known local variables and
3152 * the initial local variables of "bmap" are assumed to be the same
3153 * as those of "context".
3154 * The constraints of both "bmap" and "context" are assumed
3155 * to have been sorted using isl_basic_map_sort_constraints.
3156 *
3157 * Run through the inequality constraints of "bmap" and "context"
3158 * in sorted order.
3159 * If a constraint of "bmap" involves variables not in "context",
3160 * then it cannot appear in "context".
3161 * If a matching constraint is found, it is removed from "bmap".
3162 */
3165 {
3184 }
3190 }
3194 }
3199 }
3202 }
3205 }
3207 /* Drop all equalities from "bmap" that also appear in "context".
3208 * "context" is assumed to have only known local variables and
3209 * the initial local variables of "bmap" are assumed to be the same
3210 * as those of "context".
3211 *
3212 * Run through the equality constraints of "bmap" and "context"
3213 * in sorted order.
3214 * If a constraint of "bmap" involves variables not in "context",
3215 * then it cannot appear in "context".
3216 * If a matching constraint is found, it is removed from "bmap".
3217 */
3220 {
3244 }
3248 }
3253 }
3256 }
3259 }
3261 /* Remove the constraints in "context" from "bmap".
3262 * "context" is assumed to have explicit representations
3263 * for all local variables.
3264 *
3265 * First align the divs of "bmap" to those of "context" and
3266 * sort the constraints. Then drop all constraints from "bmap"
3267 * that appear in "context".
3268 */
3271 {
3286 }
3305 error:
3309 }
3311 /* Replace "map" by the disjunct at position "pos" and free "context".
3312 */
3315 {
3322 }
3324 /* Remove the constraints in "context" from "map".
3325 * If any of the disjuncts in the result turns out to be the universe,
3326 * then return this universe.
3327 * "context" is assumed to have explicit representations
3328 * for all local variables.
3329 */
3332 {
3342 }
3361 }
3368 error:
3372 }
3374 /* Remove the constraints in "context" from "set".
3375 * If any of the disjuncts in the result turns out to be the universe,
3376 * then return this universe.
3377 * "context" is assumed to have explicit representations
3378 * for all local variables.
3379 */
3382 {
3385 }
3387 /* Remove the constraints in "context" from "map".
3388 * If any of the disjuncts in the result turns out to be the universe,
3389 * then return this universe.
3390 * "context" is assumed to consist of a single disjunct and
3391 * to have explicit representations for all local variables.
3392 */
3395 {
3400 }
3402 /* Replace "map" by a universe map in the same space and free "drop".
3403 */
3406 {
3413 }
3415 /* Return a map that has the same intersection with "context" as "map"
3416 * and that is as "simple" as possible.
3417 *
3418 * If "map" is already the universe, then we cannot make it any simpler.
3419 * Similarly, if "context" is the universe, then we cannot exploit it
3420 * to simplify "map"
3421 * If "map" and "context" are identical to each other, then we can
3422 * return the corresponding universe.
3423 *
3424 * If either "map" or "context" consists of multiple disjuncts,
3425 * then check if "context" happens to be a subset of "map",
3426 * in which case all constraints can be removed.
3427 * In case of multiple disjuncts, the standard procedure
3428 * may not be able to detect that all constraints can be removed.
3429 *
3430 * If none of these cases apply, we have to work a bit harder.
3431 * During this computation, we make use of a single disjunct context,
3432 * so if the original context consists of more than one disjunct
3433 * then we need to approximate the context by a single disjunct set.
3434 * Simply taking the simple hull may drop constraints that are
3435 * only implicitly available in each disjunct. We therefore also
3436 * look for constraints among those defining "map" that are valid
3437 * for the context. These can then be used to simplify away
3438 * the corresponding constraints in "map".
3439 */
3442 {
3446 isl_bool subset;
3457 }
3473 }
3489 list);
3490 }
3492 error:
3496 }
3500 {
3502 }
3506 {
3509 }
3513 {
3516 }
3520 {
3525 }
3529 {
3531 }
3533 /* Compute the gist of "bmap" with respect to the constraints "context"
3534 * on the domain.
3535 */
3538 {
3544 }
3548 {
3552 }
3556 {
3560 }
3564 {
3568 }
3572 {
3574 }
3576 /* Quick check to see if two basic maps are disjoint.
3577 * In particular, we reduce the equalities and inequalities of
3578 * one basic map in the context of the equalities of the other
3579 * basic map and check if we get a contradiction.
3580 */
3583 {
3615 }
3623 }
3632 }
3636 disjoint:
3640 error:
3644 }
3648 {
3651 }
3653 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3654 */
3658 {
3669 }
3670 }
3673 }
3675 /* Are "map1" and "map2" obviously disjoint, based on information
3676 * that can be derived without looking at the individual basic maps?
3677 *
3678 * In particular, if one of them is empty or if they live in different spaces
3679 * (ignoring parameters), then they are clearly disjoint.
3680 */
3683 {
3684 isl_bool disjoint;
3685 isl_bool match;
3709 }
3711 /* Are "map1" and "map2" obviously disjoint?
3712 *
3713 * If one of them is empty or if they live in different spaces (ignoring
3714 * parameters), then they are clearly disjoint.
3715 * This is checked by isl_map_plain_is_disjoint_global.
3716 *
3717 * If they have different parameters, then we skip any further tests.
3718 *
3719 * If they are obviously equal, but not obviously empty, then we will
3720 * not be able to detect if they are disjoint.
3721 *
3722 * Otherwise we check if each basic map in "map1" is obviously disjoint
3723 * from each basic map in "map2".
3724 */
3727 {
3728 isl_bool disjoint;
3729 isl_bool intersect;
3730 isl_bool match;
3745 }
3747 /* Are "map1" and "map2" disjoint?
3748 * The parameters are assumed to have been aligned.
3749 *
3750 * In particular, check whether all pairs of basic maps are disjoint.
3751 */
3754 {
3756 }
3758 /* Are "map1" and "map2" disjoint?
3759 *
3760 * They are disjoint if they are "obviously disjoint" or if one of them
3761 * is empty. Otherwise, they are not disjoint if one of them is universal.
3762 * If the two inputs are (obviously) equal and not empty, then they are
3763 * not disjoint.
3764 * If none of these cases apply, then check if all pairs of basic maps
3765 * are disjoint after aligning the parameters.
3766 */
3768 {
3769 isl_bool disjoint;
3770 isl_bool intersect;
3798 }
3800 /* Are "bmap1" and "bmap2" disjoint?
3801 *
3802 * They are disjoint if they are "obviously disjoint" or if one of them
3803 * is empty. Otherwise, they are not disjoint if one of them is universal.
3804 * If none of these cases apply, we compute the intersection and see if
3805 * the result is empty.
3806 */
3809 {
3810 isl_bool disjoint;
3811 isl_bool intersect;
3840 }
3842 /* Are "bset1" and "bset2" disjoint?
3843 */
3846 {
3848 }
3852 {
3854 }
3856 /* Are "set1" and "set2" disjoint?
3857 */
3859 {
3861 }
3863 /* Is "v" equal to 0, 1 or -1?
3864 */
3866 {
3868 }
3870 /* Check if we can combine a given div with lower bound l and upper
3871 * bound u with some other div and if so return that other div.
3872 * Otherwise return -1.
3873 *
3874 * We first check that
3875 * - the bounds are opposites of each other (except for the constant
3876 * term)
3877 * - the bounds do not reference any other div
3878 * - no div is defined in terms of this div
3879 *
3880 * Let m be the size of the range allowed on the div by the bounds.
3881 * That is, the bounds are of the form
3882 *
3883 * e <= a <= e + m - 1
3884 *
3885 * with e some expression in the other variables.
3886 * We look for another div b such that no third div is defined in terms
3887 * of this second div b and such that in any constraint that contains
3888 * a (except for the given lower and upper bound), also contains b
3889 * with a coefficient that is m times that of b.
3890 * That is, all constraints (except for the lower and upper bound)
3891 * are of the form
3892 *
3893 * e + f (a + m b) >= 0
3894 *
3895 * Furthermore, in the constraints that only contain b, the coefficient
3896 * of b should be equal to 1 or -1.
3897 * If so, we return b so that "a + m b" can be replaced by
3898 * a single div "c = a + m b".
3899 */
3902 {
3924 }
3934 }
3946 }
3957 }
3970 }
3975 }
3979 }
3981 /* Internal data structure used during the construction and/or evaluation of
3982 * an inequality that ensures that a pair of bounds always allows
3983 * for an integer value.
3984 *
3985 * "tab" is the tableau in which the inequality is evaluated. It may
3986 * be NULL until it is actually needed.
3987 * "v" contains the inequality coefficients.
3988 * "g", "fl" and "fu" are temporary scalars used during the construction and
3989 * evaluation.
3990 */
3994 isl_int g;
3995 isl_int fl;
3996 isl_int fu;
3997 };
3999 /* Free all the memory allocated by the fields of "data".
4000 */
4002 {
4008 }
4010 /* Is the inequality stored in data->v satisfied by "bmap"?
4011 * That is, does it only attain non-negative values?
4012 * data->tab is a tableau corresponding to "bmap".
4013 */
4016 {
4027 }
4029 /* Given a lower and an upper bound on div i, do they always allow
4030 * for an integer value of the given div?
4031 * Determine this property by constructing an inequality
4032 * such that the property is guaranteed when the inequality is nonnegative.
4033 * The lower bound is inequality l, while the upper bound is inequality u.
4034 * The constructed inequality is stored in data->v.
4035 *
4036 * Let the upper bound be
4037 *
4038 * -n_u a + e_u >= 0
4039 *
4040 * and the lower bound
4041 *
4042 * n_l a + e_l >= 0
4043 *
4044 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4045 * We have
4046 *
4047 * - f_u e_l <= f_u f_l g a <= f_l e_u
4048 *
4049 * Since all variables are integer valued, this is equivalent to
4050 *
4051 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4052 *
4053 * If this interval is at least f_u f_l g, then it contains at least
4054 * one integer value for a.
4055 * That is, the test constraint is
4056 *
4057 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4058 *
4059 * or
4060 *
4061 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4062 *
4063 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4064 * then the constraint can be scaled down by a factor g',
4065 * with the constant term replaced by
4066 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4067 * Note that the result of applying Fourier-Motzkin to this pair
4068 * of constraints is
4069 *
4070 * f_l e_u + f_u e_l >= 0
4071 *
4072 * If the constant term of the scaled down version of this constraint,
4073 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4074 * term of the scaled down test constraint, then the test constraint
4075 * is known to hold and no explicit evaluation is required.
4076 * This is essentially the Omega test.
4077 *
4078 * If the test constraint consists of only a constant term, then
4079 * it is sufficient to look at the sign of this constant term.
4080 */
4083 {
4108 }
4118 }
4120 /* Remove more kinds of divs that are not strictly needed.
4121 * In particular, if all pairs of lower and upper bounds on a div
4122 * are such that they allow at least one integer value of the div,
4123 * then we can eliminate the div using Fourier-Motzkin without
4124 * introducing any spurious solutions.
4125 *
4126 * If at least one of the two constraints has a unit coefficient for the div,
4127 * then the presence of such a value is guaranteed so there is no need to check.
4128 * In particular, the value attained by the bound with unit coefficient
4129 * can serve as this intermediate value.
4130 */
4133 {
4156 isl_bool has_int;
4164 }
4185 }
4188 }
4192 }
4196 }
4199 }
4210 error:
4215 }
4217 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4218 * and the upper bound u, div1 always occurs together with div2 in the form
4219 * (div1 + m div2), where m is the constant range on the variable div1
4220 * allowed by l and u, replace the pair div1 and div2 by a single
4221 * div that is equal to div1 + m div2.
4222 *
4223 * The new div will appear in the location that contains div2.
4224 * We need to modify all constraints that contain
4225 * div2 = (div - div1) / m
4226 * The coefficient of div2 is known to be equal to 1 or -1.
4227 * (If a constraint does not contain div2, it will also not contain div1.)
4228 * If the constraint also contains div1, then we know they appear
4229 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4230 * i.e., the coefficient of div is f.
4231 *
4232 * Otherwise, we first need to introduce div1 into the constraint.
4233 * Let l be
4234 *
4235 * div1 + f >=0
4236 *
4237 * and u
4238 *
4239 * -div1 + f' >= 0
4240 *
4241 * A lower bound on div2
4242 *
4243 * div2 + t >= 0
4244 *
4245 * can be replaced by
4246 *
4247 * m div2 + div1 + m t + f >= 0
4248 *
4249 * An upper bound
4250 *
4251 * -div2 + t >= 0
4252 *
4253 * can be replaced by
4254 *
4255 * -(m div2 + div1) + m t + f' >= 0
4256 *
4257 * These constraint are those that we would obtain from eliminating
4258 * div1 using Fourier-Motzkin.
4259 *
4260 * After all constraints have been modified, we drop the lower and upper
4261 * bound and then drop div1.
4262 * Since the new div is only placed in the same location that used
4263 * to store div2, but otherwise has a different meaning, any possible
4264 * explicit representation of the original div2 is removed.
4265 */
4268 {
4270 isl_int m;
4292 else
4295 }
4299 }
4308 }
4312 }
4314 /* First check if we can coalesce any pair of divs and
4315 * then continue with dropping more redundant divs.
4316 *
4317 * We loop over all pairs of lower and upper bounds on a div
4318 * with coefficient 1 and -1, respectively, check if there
4319 * is any other div "c" with which we can coalesce the div
4320 * and if so, perform the coalescing.
4321 */
4324 {
4347 }
4348 }
4349 }
4354 }
4357 }
4359 /* Are the "n" coefficients starting at "first" of inequality constraints
4360 * "i" and "j" of "bmap" equal to each other?
4361 */
4364 {
4366 }
4368 /* Are the "n" coefficients starting at "first" of inequality constraints
4369 * "i" and "j" of "bmap" opposite to each other?
4370 */
4373 {
4375 }
4377 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4378 * apart from the constant term?
4379 */
4381 {
4386 }
4388 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4389 * apart from the constant term and the coefficient at position "pos"?
4390 */
4393 {
4399 }
4401 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4402 * apart from the constant term and the coefficient at position "pos"?
4403 */
4406 {
4412 }
4414 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4415 * been modified, simplying it if "simplify" is set.
4416 * Free the temporary data structure "pairs" that was associated
4417 * to the old version of "bmap".
4418 */
4421 {
4426 }
4428 /* Is "div" the single unknown existentially quantified variable
4429 * in inequality constraint "ineq" of "bmap"?
4430 * "div" is known to have a non-zero coefficient in "ineq".
4431 */
4434 {
4437 isl_bool known;
4447 isl_bool known;
4456 }
4459 }
4461 /* Does integer division "div" have coefficient 1 in inequality constraint
4462 * "ineq" of "map"?
4463 */
4465 {
4473 }
4475 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4476 * then try and drop redundant divs again,
4477 * freeing the temporary data structure "pairs" that was associated
4478 * to the old version of "bmap".
4479 */
4482 {
4486 }
4488 /* Drop the integer division at position "div", along with the two
4489 * inequality constraints "ineq1" and "ineq2" in which it appears
4490 * from "bmap" and then try and drop redundant divs again,
4491 * freeing the temporary data structure "pairs" that was associated
4492 * to the old version of "bmap".
4493 */
4497 {
4504 }
4507 }
4509 /* Given two inequality constraints
4510 *
4511 * f(x) + n d + c >= 0, (ineq)
4512 *
4513 * with d the variable at position "pos", and
4514 *
4515 * f(x) + c0 >= 0, (lower)
4516 *
4517 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4518 * determined by the first constraint.
4519 * That is, store
4520 *
4521 * ceil((c0 - c)/n)
4522 *
4523 * in *l.
4524 */
4527 {
4531 }
4533 /* Given two inequality constraints
4534 *
4535 * f(x) + n d + c >= 0, (ineq)
4536 *
4537 * with d the variable at position "pos", and
4538 *
4539 * -f(x) - c0 >= 0, (upper)
4540 *
4541 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4542 * determined by the first constraint.
4543 * That is, store
4544 *
4545 * ceil((-c1 - c)/n)
4546 *
4547 * in *u.
4548 */
4551 {
4555 }
4557 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4558 * does the corresponding lower bound have a fixed value in "bmap"?
4559 *
4560 * In particular, "ineq" is of the form
4561 *
4562 * f(x) + n d + c >= 0
4563 *
4564 * with n > 0, c the constant term and
4565 * d the existentially quantified variable "div".
4566 * That is, the lower bound is
4567 *
4568 * ceil((-f(x) - c)/n)
4569 *
4570 * Look for a pair of constraints
4571 *
4572 * f(x) + c0 >= 0
4573 * -f(x) + c1 >= 0
4574 *
4575 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4576 * That is, check that
4577 *
4578 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4579 *
4580 * If so, return the index of inequality f(x) + c0 >= 0.
4581 * Otherwise, return -1.
4582 */
4584 {
4601 }
4605 }
4606 }
4623 }
4625 /* Given a lower bound constraint "ineq" on the existentially quantified
4626 * variable "div", such that the corresponding lower bound has
4627 * a fixed value in "bmap", assign this fixed value to the variable 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 * "lower" determines the constant value for the lower bound.
4632 *
4633 * In particular, "ineq" is of the form
4634 *
4635 * f(x) + n d + c >= 0,
4636 *
4637 * while "lower" is of the form
4638 *
4639 * f(x) + c0 >= 0
4640 *
4641 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4642 * is ceil((c0 - c)/n).
4643 */
4646 {
4647 isl_int c;
4660 }
4662 /* Remove divs that are not strictly needed based on the inequality
4663 * constraints.
4664 * In particular, if a div only occurs positively (or negatively)
4665 * in constraints, then it can simply be dropped.
4666 * Also, if a div occurs in only two constraints and if moreover
4667 * those two constraints are opposite to each other, except for the constant
4668 * term and if the sum of the constant terms is such that for any value
4669 * of the other values, there is always at least one integer value of the
4670 * div, i.e., if one plus this sum is greater than or equal to
4671 * the (absolute value) of the coefficient of the div in the constraints,
4672 * then we can also simply drop the div.
4673 *
4674 * If an existentially quantified variable does not have an explicit
4675 * representation, appears in only a single lower bound that does not
4676 * involve any other such existentially quantified variables and appears
4677 * in this lower bound with coefficient 1,
4678 * then fix the variable to the value of the lower bound. That is,
4679 * turn the inequality into an equality.
4680 * If for any value of the other variables, there is any value
4681 * for the existentially quantified variable satisfying the constraints,
4682 * then this lower bound also satisfies the constraints.
4683 * It is therefore safe to pick this lower bound.
4684 *
4685 * The same reasoning holds even if the coefficient is not one.
4686 * However, fixing the variable to the value of the lower bound may
4687 * in general introduce an extra integer division, in which case
4688 * it may be better to pick another value.
4689 * If this integer division has a known constant value, then plugging
4690 * in this constant value removes the existentially quantified variable
4691 * completely. In particular, if the lower bound is of the form
4692 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4693 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4694 * then the existentially quantified variable can be assigned this
4695 * shared value.
4696 *
4697 * We skip divs that appear in equalities or in the definition of other divs.
4698 * Divs that appear in the definition of other divs usually occur in at least
4699 * 4 constraints, but the constraints may have been simplified.
4700 *
4701 * If any divs are left after these simple checks then we move on
4702 * to more complicated cases in drop_more_redundant_divs.
4703 */
4706 {
4746 }
4750 }
4751 }
4759 }
4762 else
4782 pairs);
4786 pairs);
4788 }
4805 else
4812 }
4815 }
4822 error:
4826 }
4828 /* Consider the coefficients at "c" as a row vector and replace
4829 * them with their product with "T". "T" is assumed to be a square matrix.
4830 */
4832 {
4854 }
4856 /* Plug in T for the variables in "bmap" starting at "pos".
4857 * T is a linear unimodular matrix, i.e., without constant term.
4858 */
4861 {
4890 }
4894 error:
4898 }
4900 /* Remove divs that are not strictly needed.
4901 *
4902 * First look for an equality constraint involving two or more
4903 * existentially quantified variables without an explicit
4904 * representation. Replace the combination that appears
4905 * in the equality constraint by a single existentially quantified
4906 * variable such that the equality can be used to derive
4907 * an explicit representation for the variable.
4908 * If there are no more such equality constraints, then continue
4909 * with isl_basic_map_drop_redundant_divs_ineq.
4910 *
4911 * In particular, if the equality constraint is of the form
4912 *
4913 * f(x) + \sum_i c_i a_i = 0
4914 *
4915 * with a_i existentially quantified variable without explicit
4916 * representation, then apply a transformation on the existentially
4917 * quantified variables to turn the constraint into
4918 *
4919 * f(x) + g a_1' = 0
4920 *
4921 * with g the gcd of the c_i.
4922 * In order to easily identify which existentially quantified variables
4923 * have a complete explicit representation, i.e., without being defined
4924 * in terms of other existentially quantified variables without
4925 * an explicit representation, the existentially quantified variables
4926 * are first sorted.
4927 *
4928 * The variable transformation is computed by extending the row
4929 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4930 *
4931 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4932 * [a_2'] [ a_2 ]
4933 * ... = U ....
4934 * [a_n'] [ a_n ]
4935 *
4936 * with [c_1/g ... c_n/g] representing the first row of U.
4937 * The inverse of U is then plugged into the original constraints.
4938 * The call to isl_basic_map_simplify makes sure the explicit
4939 * representation for a_1' is extracted from the equality constraint.
4940 */
4943 {
4978 }
4997 }
4999 /* Does "bmap" satisfy any equality that involves more than 2 variables
5000 * and/or has coefficients different from -1 and 1?
5001 */
5003 {
5032 }
5035 }
5037 /* Remove any common factor g from the constraint coefficients in "v".
5038 * The constant term is stored in the first position and is replaced
5039 * by floor(c/g). If any common factor is removed and if this results
5040 * in a tightening of the constraint, then set *tightened.
5041 */
5044 {
5064 }
5066 /* If "bmap" is an integer set that satisfies any equality involving
5067 * more than 2 variables and/or has coefficients different from -1 and 1,
5068 * then use variable compression to reduce the coefficients by removing
5069 * any (hidden) common factor.
5070 * In particular, apply the variable compression to each constraint,
5071 * factor out any common factor in the non-constant coefficients and
5072 * then apply the inverse of the compression.
5073 * At the end, we mark the basic map as having reduced constants.
5074 * If this flag is still set on the next invocation of this function,
5075 * then we skip the computation.
5076 *
5077 * Removing a common factor may result in a tightening of some of
5078 * the constraints. If this happens, then we may end up with two
5079 * opposite inequalities that can be replaced by an equality.
5080 * We therefore call isl_basic_map_detect_inequality_pairs,
5081 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5082 * and isl_basic_map_gauss if such a pair was found.
5083 *
5084 * Note that this function may leave the result in an inconsistent state.
5085 * In particular, the constraints may not be gaussed.
5086 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5087 * for some of the test cases to pass successfully.
5088 * Any potential modification of the representation is therefore only
5089 * performed on a single copy of the basic map.
5090 */
5093 {
5127 }
5142 }
5157 }
5158 }
5161 error:
5166 }
5168 /* Shift the integer division at position "div" of "bmap"
5169 * by "shift" times the variable at position "pos".
5170 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5171 * corresponds to the constant term.
5172 *
5173 * That is, if the integer division has the form
5174 *
5175 * floor(f(x)/d)
5176 *
5177 * then replace it by
5178 *
5179 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5180 */
5183 {
5202 }
5208 }
5216 }
5219 }