| blob | 82892f2b89724ac98516d07649b9d7a45d41a761 |
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 }
47 /* Scale down the inequality constraint "ineq" of length "len"
48 * by a factor of "f".
49 * All the coefficients, except the constant term,
50 * are assumed to be multiples of "f".
51 *
52 * If the factor is 1, then no scaling needs to be performed.
53 */
56 {
67 }
71 {
73 isl_int gcd;
86 }
90 }
98 }
100 }
108 }
112 }
118 }
122 error:
126 }
130 {
133 }
135 /* Reduce the coefficient of the variable at position "pos"
136 * in integer division "div", such that it lies in the half-open
137 * interval (1/2,1/2], extracting any excess value from this integer division.
138 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
139 * corresponds to the constant term.
140 *
141 * That is, the integer division is of the form
142 *
143 * floor((... + (c * d + r) * x_pos + ...)/d)
144 *
145 * with -d < 2 * r <= d.
146 * Replace it by
147 *
148 * floor((... + r * x_pos + ...)/d) + c * x_pos
149 *
150 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
151 * Otherwise, c = floor((c * d + r)/d) + 1.
152 *
153 * This is the same normalization that is performed by isl_aff_floor.
154 */
157 {
158 isl_int shift;
173 }
175 /* Does the coefficient of the variable at position "pos"
176 * in integer division "div" need to be reduced?
177 * That is, does it lie outside the half-open interval (1/2,1/2]?
178 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
179 * 2 * c != d.
180 */
183 {
184 isl_bool r;
196 }
198 /* Reduce the coefficients (including the constant term) of
199 * integer division "div", if needed.
200 * In particular, make sure all coefficients lie in
201 * the half-open interval (1/2,1/2].
202 */
205 {
207 isl_size total;
213 isl_bool reduce;
223 }
226 }
228 /* Reduce the coefficients (including the constant term) of
229 * the known integer divisions, if needed
230 * In particular, make sure all coefficients lie in
231 * the half-open interval (1/2,1/2].
232 */
235 {
249 }
252 }
254 /* Remove any common factor in numerator and denominator of the div expression,
255 * not taking into account the constant term.
256 * That is, if the div is of the form
257 *
258 * floor((a + m f(x))/(m d))
259 *
260 * then replace it by
261 *
262 * floor((floor(a/m) + f(x))/d)
263 *
264 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
265 * and can therefore not influence the result of the floor.
266 */
269 {
289 }
291 /* Remove any common factor in numerator and denominator of a div expression,
292 * not taking into account the constant term.
293 * That is, look for any div of the form
294 *
295 * floor((a + m f(x))/(m d))
296 *
297 * and replace it by
298 *
299 * floor((floor(a/m) + f(x))/d)
300 *
301 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
302 * and can therefore not influence the result of the floor.
303 */
306 {
318 }
320 /* Assumes divs have been ordered if keep_divs is set.
321 */
325 {
326 isl_size total;
327 isl_size v_div;
347 }
360 }
369 /* We need to be careful about circular definitions,
370 * so for now we just remove the definition of div k
371 * if the equality contains any divs.
372 * If keep_divs is set, then the divs have been ordered
373 * and we can keep the definition as long as the result
374 * is still ordered.
375 */
384 }
387 }
389 /* Assumes divs have been ordered if keep_divs is set.
390 */
393 {
394 isl_size v_div;
406 }
408 /* Check if elimination of div "div" using equality "eq" would not
409 * result in a div depending on a later div.
410 */
413 {
416 isl_size v_div;
433 }
436 }
438 /* Eliminate divs based on equalities
439 */
442 {
457 isl_bool ok;
473 }
474 }
478 }
480 /* Eliminate divs based on inequalities
481 */
484 {
514 }
516 }
518 /* Does the equality constraint at position "eq" in "bmap" involve
519 * any local variables in the range [first, first + n)
520 * that are not marked as having an explicit representation?
521 */
524 {
533 isl_bool unknown;
542 }
545 }
547 /* The last local variable involved in the equality constraint
548 * at position "eq" in "bmap" is the local variable at position "div".
549 * It can therefore be used to extract an explicit representation
550 * for that variable.
551 * Do so unless the local variable already has an explicit representation or
552 * the explicit representation would involve any other local variables
553 * that in turn do not have an explicit representation.
554 * An equality constraint involving local variables without an explicit
555 * representation can be used in isl_basic_map_drop_redundant_divs
556 * to separate out an independent local variable. Introducing
557 * an explicit representation here would block this transformation,
558 * while the partial explicit representation in itself is not very useful.
559 * Set *progress if anything is changed.
560 *
561 * The equality constraint is of the form
562 *
563 * f(x) + n e >= 0
564 *
565 * with n a positive number. The explicit representation derived from
566 * this constraint is
567 *
568 * floor((-f(x))/n)
569 */
572 {
573 isl_size total;
575 isl_bool involves;
600 }
602 /* Perform fangcheng (Gaussian elimination) on the equality
603 * constraints of "bmap".
604 * That is, put them into row-echelon form, starting from the last column
605 * backward and use them to eliminate the corresponding coefficients
606 * from all constraints.
607 *
608 * If "progress" is not NULL, then it gets set if the elimination
609 * results in any changes.
610 * The elimination process may result in some equality constraints
611 * getting interchanged or removed.
612 * If "swap" or "drop" are not NULL, then they get called when
613 * two equality constraints get interchanged or
614 * when a number of final equality constraints get removed.
615 * As a special case, if the input turns out to be empty,
616 * then drop gets called with the number of removed equality
617 * constraints set to the total number of equality constraints.
618 * If "swap" or "drop" are not NULL, then the local variables (if any)
619 * are assumed to be in a valid order.
620 */
625 {
630 isl_size total;
650 }
657 }
669 }
678 }
684 }
688 {
690 }
694 {
696 progress));
697 }
701 {
707 }
709 }
711 /* Hash table of inequalities in a basic map.
712 * "index" is an array of addresses of inequalities in the basic map, some
713 * of which are NULL. The inequalities are hashed on the coefficients
714 * except the constant term.
715 * "size" is the number of elements in the array and is always a power of two
716 * "bits" is the number of bits need to represent an index into the array.
717 * "total" is the total dimension of the basic map.
718 */
723 isl_size total;
724 };
726 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
727 */
730 {
749 }
751 /* Free the memory allocated by create_constraint_index.
752 */
754 {
756 }
758 /* Return the position in ci->index that contains the address of
759 * an inequality that is equal to *ineq up to the constant term,
760 * provided this address is not identical to "ineq".
761 * If there is no such inequality, then return the position where
762 * such an inequality should be inserted.
763 */
765 {
773 }
775 /* Return the position in ci->index that contains the address of
776 * an inequality that is equal to the k'th inequality of "bmap"
777 * up to the constant term, provided it does not point to the very
778 * same inequality.
779 * If there is no such inequality, then return the position where
780 * such an inequality should be inserted.
781 */
784 {
786 }
790 {
792 }
794 /* Fill in the "ci" data structure with the inequalities of "bset".
795 */
798 {
807 }
810 }
812 /* Is the inequality ineq (obviously) redundant with respect
813 * to the constraints in "ci"?
814 *
815 * Look for an inequality in "ci" with the same coefficients and then
816 * check if the contant term of "ineq" is greater than or equal
817 * to the constant term of that inequality. If so, "ineq" is clearly
818 * redundant.
819 *
820 * Note that hash_index_ineq ignores a stored constraint if it has
821 * the same address as the passed inequality. It is ok to pass
822 * the address of a local variable here since it will never be
823 * the same as the address of a constraint in "ci".
824 */
827 {
834 }
836 /* If we can eliminate more than one div, then we need to make
837 * sure we do it from last div to first div, in order not to
838 * change the position of the other divs that still need to
839 * be removed.
840 */
843 {
850 isl_size v_div;
899 }
901 }
913 }
916 out:
920 }
923 {
925 isl_size v_div;
937 }
939 }
941 /* Normalize divs that appear in equalities.
942 *
943 * In particular, we assume that bmap contains some equalities
944 * of the form
945 *
946 * a x = m * e_i
947 *
948 * and we want to replace the set of e_i by a minimal set and
949 * such that the new e_i have a canonical representation in terms
950 * of the vector x.
951 * If any of the equalities involves more than one divs, then
952 * we currently simply bail out.
953 *
954 * Let us first additionally assume that all equalities involve
955 * a div. The equalities then express modulo constraints on the
956 * remaining variables and we can use "parameter compression"
957 * to find a minimal set of constraints. The result is a transformation
958 *
959 * x = T(x') = x_0 + G x'
960 *
961 * with G a lower-triangular matrix with all elements below the diagonal
962 * non-negative and smaller than the diagonal element on the same row.
963 * We first normalize x_0 by making the same property hold in the affine
964 * T matrix.
965 * The rows i of G with a 1 on the diagonal do not impose any modulo
966 * constraint and simply express x_i = x'_i.
967 * For each of the remaining rows i, we introduce a div and a corresponding
968 * equality. In particular
969 *
970 * g_ii e_j = x_i - g_i(x')
971 *
972 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
973 * corresponding div (if g_kk != 1).
974 *
975 * If there are any equalities not involving any div, then we
976 * first apply a variable compression on the variables x:
977 *
978 * x = C x'' x'' = C_2 x
979 *
980 * and perform the above parameter compression on A C instead of on A.
981 * The resulting compression is then of the form
982 *
983 * x'' = T(x') = x_0 + G x'
984 *
985 * and in constructing the new divs and the corresponding equalities,
986 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
987 * by the corresponding row from C_2.
988 */
991 {
993 isl_size v_div;
1000 isl_int v;
1034 }
1035 }
1044 }
1050 }
1060 }
1067 }
1072 /* We have to be careful because dropping equalities may reorder them */
1083 }
1084 }
1090 else
1091 needed++;
1092 }
1097 }
1107 else
1115 else
1118 }
1122 }
1129 done:
1133 error:
1140 }
1144 {
1154 }
1156 /* Check whether it is ok to define a div based on an inequality.
1157 * To avoid the introduction of circular definitions of divs, we
1158 * do not allow such a definition if the resulting expression would refer to
1159 * any other undefined divs or if any known div is defined in
1160 * terms of the unknown div.
1161 */
1164 {
1168 /* Not defined in terms of unknown divs */
1176 }
1178 /* No other div defined in terms of this one => avoid loops */
1186 }
1189 }
1191 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1192 * be a better expression than the current one?
1193 *
1194 * If we do not have any expression yet, then any expression would be better.
1195 * Otherwise we check if the last variable involved in the inequality
1196 * (disregarding the div that it would define) is in an earlier position
1197 * than the last variable involved in the current div expression.
1198 */
1201 {
1218 }
1220 /* Given two constraints "k" and "l" that are opposite to each other,
1221 * except for the constant term, check if we can use them
1222 * to obtain an expression for one of the hitherto unknown divs or
1223 * a "better" expression for a div for which we already have an expression.
1224 * "sum" is the sum of the constant terms of the constraints.
1225 * If this sum is strictly smaller than the coefficient of one
1226 * of the divs, then this pair can be used to define the div.
1227 * To avoid the introduction of circular definitions of divs, we
1228 * do not use the pair if the resulting expression would refer to
1229 * any other undefined divs or if any known div is defined in
1230 * terms of the unknown div.
1231 */
1235 {
1240 isl_bool set_div;
1255 else
1260 }
1262 }
1266 {
1270 isl_int sum;
1285 }
1293 }
1308 }
1310 /* We need to break out of the loop after these
1311 * changes since the contents of the hash
1312 * will no longer be valid.
1313 * Plus, we probably we want to regauss first.
1314 */
1322 }
1327 }
1329 /* Detect all pairs of inequalities that form an equality.
1330 *
1331 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1332 * Call it repeatedly while it is making progress.
1333 */
1336 {
1348 }
1350 /* Given a known integer division "div" that is not integral
1351 * (with denominator 1), eliminate it from the constraints in "bmap"
1352 * where it appears with a (positive or negative) unit coefficient.
1353 * If "progress" is not NULL, then it gets set if the elimination
1354 * results in any changes.
1355 *
1356 * That is, replace
1357 *
1358 * floor(e/m) + f >= 0
1359 *
1360 * by
1361 *
1362 * e + m f >= 0
1363 *
1364 * and
1365 *
1366 * -floor(e/m) + f >= 0
1367 *
1368 * by
1369 *
1370 * -e + m f + m - 1 >= 0
1371 *
1372 * The first conversion is valid because floor(e/m) >= -f is equivalent
1373 * to e/m >= -f because -f is an integral expression.
1374 * The second conversion follows from the fact that
1375 *
1376 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1377 *
1378 *
1379 * Note that one of the div constraints may have been eliminated
1380 * due to being redundant with respect to the constraint that is
1381 * being modified by this function. The modified constraint may
1382 * no longer imply this div constraint, so we add it back to make
1383 * sure we do not lose any information.
1384 */
1387 {
1415 else
1424 }
1430 }
1433 }
1435 /* Eliminate selected known divs from constraints where they appear with
1436 * a (positive or negative) unit coefficient.
1437 * In particular, only handle those for which "select" returns isl_bool_true.
1438 * If "progress" is not NULL, then it gets set if the elimination
1439 * results in any changes.
1440 *
1441 * We skip integral divs, i.e., those with denominator 1, as we would
1442 * risk eliminating the div from the div constraints. We do not need
1443 * to handle those divs here anyway since the div constraints will turn
1444 * out to form an equality and this equality can then be used to eliminate
1445 * the div from all constraints.
1446 */
1451 {
1458 isl_bool selected;
1472 }
1475 }
1477 /* eliminate_selected_unit_divs callback that selects every
1478 * integer division.
1479 */
1481 {
1483 }
1485 /* Eliminate known divs from constraints where they appear with
1486 * a (positive or negative) unit coefficient.
1487 * If "progress" is not NULL, then it gets set if the elimination
1488 * results in any changes.
1489 */
1492 {
1494 }
1496 /* eliminate_selected_unit_divs callback that selects
1497 * integer divisions that only appear with
1498 * a (positive or negative) unit coefficient
1499 * (outside their div constraints).
1500 */
1502 {
1512 isl_bool skip;
1525 }
1528 }
1530 /* Eliminate known divs from constraints where they appear with
1531 * a (positive or negative) unit coefficient,
1532 * but only if they do not appear in any other constraints
1533 * (other than the div constraints).
1534 */
1537 {
1539 }
1542 {
1547 isl_bool empty;
1563 /* requires equalities in normal form */
1567 }
1569 }
1573 {
1575 }
1580 {
1612 }
1614 /* If the only constraints a div d=floor(f/m)
1615 * appears in are its two defining constraints
1616 *
1617 * f - m d >=0
1618 * -(f - (m - 1)) + m d >= 0
1619 *
1620 * then it can safely be removed.
1621 */
1623 {
1636 isl_bool red;
1643 }
1650 }
1653 }
1655 /*
1656 * Remove divs that don't occur in any of the constraints or other divs.
1657 * These can arise when dropping constraints from a basic map or
1658 * when the divs of a basic map have been temporarily aligned
1659 * with the divs of another basic map.
1660 */
1663 {
1665 isl_size v_div;
1672 isl_bool redundant;
1682 }
1684 }
1686 /* Mark "bmap" as final, without checking for obviously redundant
1687 * integer divisions. This function should be used when "bmap"
1688 * is known not to involve any such integer divisions.
1689 */
1692 {
1697 }
1699 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1700 */
1702 {
1706 }
1710 {
1712 }
1714 /* Remove definition of any div that is defined in terms of the given variable.
1715 * The div itself is not removed. Functions such as
1716 * eliminate_divs_ineq depend on the other divs remaining in place.
1717 */
1720 {
1734 }
1736 }
1738 /* Eliminate the specified variables from the constraints using
1739 * Fourier-Motzkin. The variables themselves are not removed.
1740 */
1743 {
1746 isl_size total;
1777 }
1784 n_lower++;
1786 n_upper++;
1787 }
1811 }
1814 }
1826 }
1827 }
1831 error:
1834 }
1838 {
1841 }
1843 /* Eliminate the specified n dimensions starting at first from the
1844 * constraints, without removing the dimensions from the space.
1845 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1846 * Otherwise, they are projected out and the original space is restored.
1847 */
1851 {
1866 }
1873 }
1878 {
1880 }
1882 /* Remove all constraints from "bmap" that reference any unknown local
1883 * variables (directly or indirectly).
1884 *
1885 * Dropping all constraints on a local variable will make it redundant,
1886 * so it will get removed implicitly by
1887 * isl_basic_map_drop_constraints_involving_dims. Some other local
1888 * variables may also end up becoming redundant if they only appear
1889 * in constraints together with the unknown local variable.
1890 * Therefore, start over after calling
1891 * isl_basic_map_drop_constraints_involving_dims.
1892 */
1895 {
1896 isl_bool known;
1897 isl_size n_div;
1924 }
1927 }
1929 /* Remove all constraints from "bset" that reference any unknown local
1930 * variables (directly or indirectly).
1931 */
1934 {
1940 }
1942 /* Remove all constraints from "map" that reference any unknown local
1943 * variables (directly or indirectly).
1944 *
1945 * Since constraints may get dropped from the basic maps,
1946 * they may no longer be disjoint from each other.
1947 */
1950 {
1952 isl_bool known;
1970 }
1976 }
1978 /* Don't assume equalities are in order, because align_divs
1979 * may have changed the order of the divs.
1980 */
1983 {
1994 }
1995 }
1996 }
2000 {
2002 }
2006 {
2018 }
2020 }
2022 }
2026 {
2029 }
2033 {
2036 isl_size dim;
2044 }
2066 error:
2070 }
2072 /* For each inequality in "ineq" that is a shifted (more relaxed)
2073 * copy of an inequality in "context", mark the corresponding entry
2074 * in "row" with -1.
2075 * If an inequality only has a non-negative constant term, then
2076 * mark it as well.
2077 */
2080 {
2100 isl_bool redundant;
2106 }
2113 }
2116 error:
2119 }
2123 {
2136 isl_bool redundant;
2148 }
2151 error:
2154 }
2156 /* Remove constraints from "bmap" that are identical to constraints
2157 * in "context" or that are more relaxed (greater constant term).
2158 *
2159 * We perform the test for shifted copies on the pure constraints
2160 * in remove_shifted_constraints.
2161 */
2164 {
2173 }
2187 error:
2191 }
2193 /* Does the (linear part of a) constraint "c" involve any of the "len"
2194 * "relevant" dimensions?
2195 */
2197 {
2205 }
2208 }
2210 /* Drop constraints from "bmap" that do not involve any of
2211 * the dimensions marked "relevant".
2212 */
2215 {
2217 isl_size dim;
2233 }
2240 }
2243 }
2245 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2246 *
2247 * In particular, for any variable involved in the constraint,
2248 * find the actual group id from before and replace the group
2249 * of the corresponding variable by the minimal group of all
2250 * the variables involved in the constraint considered so far
2251 * (if this minimum is smaller) or replace the minimum by this group
2252 * (if the minimum is larger).
2253 *
2254 * At the end, all the variables in "c" will (indirectly) point
2255 * to the minimal of the groups that they referred to originally.
2256 */
2258 {
2275 }
2276 }
2278 /* Allocate an array of groups of variables, one for each variable
2279 * in "context", initialized to zero.
2280 */
2282 {
2284 isl_size dim;
2291 }
2293 /* Drop constraints from "bmap" that only involve variables that are
2294 * not related to any of the variables marked with a "-1" in "group".
2295 *
2296 * We construct groups of variables that collect variables that
2297 * (indirectly) appear in some common constraint of "bmap".
2298 * Each group is identified by the first variable in the group,
2299 * except for the special group of variables that was already identified
2300 * in the input as -1 (or are related to those variables).
2301 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2302 * otherwise the group of i is the group of group[i].
2303 *
2304 * We first initialize groups for the remaining variables.
2305 * Then we iterate over the constraints of "bmap" and update the
2306 * group of the variables in the constraint by the smallest group.
2307 * Finally, we resolve indirect references to groups by running over
2308 * the variables.
2309 *
2310 * After computing the groups, we drop constraints that do not involve
2311 * any variables in the -1 group.
2312 */
2315 {
2316 isl_size dim;
2331 }
2349 }
2351 /* Drop constraints from "context" that are irrelevant for computing
2352 * the gist of "bset".
2353 *
2354 * In particular, drop constraints in variables that are not related
2355 * to any of the variables involved in the constraints of "bset"
2356 * in the sense that there is no sequence of constraints that connects them.
2357 *
2358 * We first mark all variables that appear in "bset" as belonging
2359 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2360 */
2363 {
2365 isl_size dim;
2384 }
2390 }
2393 }
2395 /* Drop constraints from "context" that are irrelevant for computing
2396 * the gist of the inequalities "ineq".
2397 * Inequalities in "ineq" for which the corresponding element of row
2398 * is set to -1 have already been marked for removal and should be ignored.
2399 *
2400 * In particular, drop constraints in variables that are not related
2401 * to any of the variables involved in "ineq"
2402 * in the sense that there is no sequence of constraints that connects them.
2403 *
2404 * We first mark all variables that appear in "bset" as belonging
2405 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2406 */
2409 {
2411 isl_size dim;
2413 isl_size n;
2431 }
2434 }
2437 }
2439 /* Do all "n" entries of "row" contain a negative value?
2440 */
2442 {
2450 }
2452 /* Update the inequalities in "bset" based on the information in "row"
2453 * and "tab".
2454 *
2455 * In particular, the array "row" contains either -1, meaning that
2456 * the corresponding inequality of "bset" is redundant, or the index
2457 * of an inequality in "tab".
2458 *
2459 * If the row entry is -1, then drop the inequality.
2460 * Otherwise, if the constraint is marked redundant in the tableau,
2461 * then drop the inequality. Similarly, if it is marked as an equality
2462 * in the tableau, then turn the inequality into an equality and
2463 * perform Gaussian elimination.
2464 */
2467 {
2484 }
2494 }
2495 }
2501 }
2503 /* Update the inequalities in "bset" based on the information in "row"
2504 * and "tab" and free all arguments (other than "bset").
2505 */
2510 {
2519 }
2521 /* Remove all information from bset that is redundant in the context
2522 * of context.
2523 * "ineq" contains the (possibly transformed) inequalities of "bset",
2524 * in the same order.
2525 * The (explicit) equalities of "bset" are assumed to have been taken
2526 * into account by the transformation such that only the inequalities
2527 * are relevant.
2528 * "context" is assumed not to be empty.
2529 *
2530 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2531 * A value of -1 means that the inequality is obviously redundant and may
2532 * not even appear in "tab".
2533 *
2534 * We first mark the inequalities of "bset"
2535 * that are obviously redundant with respect to some inequality in "context".
2536 * Then we remove those constraints from "context" that have become
2537 * irrelevant for computing the gist of "bset".
2538 * Note that this removal of constraints cannot be replaced by
2539 * a factorization because factors in "bset" may still be connected
2540 * to each other through constraints in "context".
2541 *
2542 * If there are any inequalities left, we construct a tableau for
2543 * the context and then add the inequalities of "bset".
2544 * Before adding these inequalities, we freeze all constraints such that
2545 * they won't be considered redundant in terms of the constraints of "bset".
2546 * Then we detect all redundant constraints (among the
2547 * constraints that weren't frozen), first by checking for redundancy in the
2548 * the tableau and then by checking if replacing a constraint by its negation
2549 * would lead to an empty set. This last step is fairly expensive
2550 * and could be optimized by more reuse of the tableau.
2551 * Finally, we update bset according to the results.
2552 */
2555 {
2570 }
2606 }
2629 }
2634 }
2638 error:
2646 }
2648 /* Extract the inequalities of "bset" as an isl_mat.
2649 */
2651 {
2652 isl_size total;
2665 }
2667 /* Remove all information from "bset" that is redundant in the context
2668 * of "context", for the case where both "bset" and "context" are
2669 * full-dimensional.
2670 */
2673 {
2678 }
2680 /* Replace "bset" by an empty basic set in the same space.
2681 */
2684 {
2690 }
2692 /* Remove all information from "bset" that is redundant in the context
2693 * of "context", for the case where the combined equalities of
2694 * "bset" and "context" allow for a compression that can be obtained
2695 * by preapplication of "T".
2696 * If the compression of "context" is empty, meaning that "bset" and
2697 * "context" do not intersect, then return the empty set.
2698 *
2699 * "bset" itself is not transformed by "T". Instead, the inequalities
2700 * are extracted from "bset" and those are transformed by "T".
2701 * uset_gist_full then determines which of the transformed inequalities
2702 * are redundant with respect to the transformed "context" and removes
2703 * the corresponding inequalities from "bset".
2704 *
2705 * After preapplying "T" to the inequalities, any common factor is
2706 * removed from the coefficients. If this results in a tightening
2707 * of the constant term, then the same tightening is applied to
2708 * the corresponding untransformed inequality in "bset".
2709 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2710 *
2711 * g f'(x) + r >= 0
2712 *
2713 * with 0 <= r < g, then it is equivalent to
2714 *
2715 * f'(x) >= 0
2716 *
2717 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2718 * subspace compressed by T since the latter would be transformed to
2719 *
2720 * g f'(x) >= 0
2721 */
2725 {
2730 isl_int rem;
2742 }
2767 }
2771 error:
2776 }
2778 /* Project "bset" onto the variables that are involved in "template".
2779 */
2782 {
2784 isl_size n;
2791 isl_bool involved;
2800 }
2803 }
2805 /* Remove all information from bset that is redundant in the context
2806 * of context. In particular, equalities that are linear combinations
2807 * of those in context are removed. Then the inequalities that are
2808 * redundant in the context of the equalities and inequalities of
2809 * context are removed.
2810 *
2811 * First of all, we drop those constraints from "context"
2812 * that are irrelevant for computing the gist of "bset".
2813 * Alternatively, we could factorize the intersection of "context" and "bset".
2814 *
2815 * We first compute the intersection of the integer affine hulls
2816 * of "bset" and "context",
2817 * compute the gist inside this intersection and then reduce
2818 * the constraints with respect to the equalities of the context
2819 * that only involve variables already involved in the input.
2820 * If the intersection of the affine hulls turns out to be empty,
2821 * then return the empty set.
2822 *
2823 * If two constraints are mutually redundant, then uset_gist_full
2824 * will remove the second of those constraints. We therefore first
2825 * sort the constraints so that constraints not involving existentially
2826 * quantified variables are given precedence over those that do.
2827 * We have to perform this sorting before the variable compression,
2828 * because that may effect the order of the variables.
2829 */
2832 {
2837 isl_size total;
2858 }
2863 }
2872 }
2883 }
2886 error:
2890 }
2892 /* Return the number of equality constraints in "bmap" that involve
2893 * local variables. This function assumes that Gaussian elimination
2894 * has been applied to the equality constraints.
2895 */
2897 {
2919 }
2921 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2922 * The constraints are assumed not to involve any local variables.
2923 */
2926 {
2928 isl_size total;
2946 }
2951 error:
2956 }
2958 /* Construct and return a variable compression based on the equality
2959 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2960 * "n1" is the number of (initial) equality constraints in "bmap1"
2961 * that do involve local variables.
2962 * "n2" is the number of (initial) equality constraints in "bmap2"
2963 * that do involve local variables.
2964 * "total" is the total number of other variables.
2965 * This function assumes that Gaussian elimination
2966 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2967 * such that the equality constraints not involving local variables
2968 * are those that start at "n1" or "n2".
2969 *
2970 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2971 * then simply compute the compression based on the equality constraints
2972 * in the other basic map.
2973 * Otherwise, combine the equality constraints from both into a new
2974 * basic map such that Gaussian elimination can be applied to this combination
2975 * and then construct a variable compression from the resulting
2976 * equality constraints.
2977 */
2981 {
2991 }
2996 }
3011 }
3013 /* Extract the stride constraints from "bmap", compressed
3014 * with respect to both the stride constraints in "context" and
3015 * the remaining equality constraints in both "bmap" and "context".
3016 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
3017 * "context_n_eq" is the number of (initial) stride constraints in "context".
3018 *
3019 * Let x be all variables in "bmap" (and "context") other than the local
3020 * variables. First compute a variable compression
3021 *
3022 * x = V x'
3023 *
3024 * based on the non-stride equality constraints in "bmap" and "context".
3025 * Consider the stride constraints of "context",
3026 *
3027 * A(x) + B(y) = 0
3028 *
3029 * with y the local variables and plug in the variable compression,
3030 * resulting in
3031 *
3032 * A(V x') + B(y) = 0
3033 *
3034 * Use these constraints to compute a parameter compression on x'
3035 *
3036 * x' = T x''
3037 *
3038 * Now consider the stride constraints of "bmap"
3039 *
3040 * C(x) + D(y) = 0
3041 *
3042 * and plug in x = V*T x''.
3043 * That is, return A = [C*V*T D].
3044 */
3048 {
3074 else
3082 }
3084 /* Remove the prime factors from *g that have an exponent that
3085 * is strictly smaller than the exponent in "c".
3086 * All exponents in *g are known to be smaller than or equal
3087 * to those in "c".
3088 *
3089 * That is, if *g is equal to
3090 *
3091 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
3092 *
3093 * and "c" is equal to
3094 *
3095 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
3096 *
3097 * then update *g to
3098 *
3099 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
3100 * p_n^{e_n * (e_n = f_n)}
3101 *
3102 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
3103 * neither does the gcd of *g and c / *g.
3104 * If e_i < f_i, then the gcd of *g and c / *g has a positive
3105 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
3106 * Dividing *g by this gcd therefore strictly reduces the exponent
3107 * of the prime factors that need to be removed, while leaving the
3108 * other prime factors untouched.
3109 * Repeating this process until gcd(*g, c / *g) = 1 therefore
3110 * removes all undesired factors, without removing any others.
3111 */
3113 {
3114 isl_int t;
3123 }
3125 }
3127 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3128 * of the same stride constraints in a compressed space that exploits
3129 * all equalities in the context and the other equalities in "bmap".
3130 *
3131 * If the stride constraints of "bmap" are of the form
3132 *
3133 * C(x) + D(y) = 0
3134 *
3135 * then A is of the form
3136 *
3137 * B(x') + D(y) = 0
3138 *
3139 * If any of these constraints involves only a single local variable y,
3140 * then the constraint appears as
3141 *
3142 * f(x) + m y_i = 0
3143 *
3144 * in "bmap" and as
3145 *
3146 * h(x') + m y_i = 0
3147 *
3148 * in "A".
3149 *
3150 * Let g be the gcd of m and the coefficients of h.
3151 * Then, in particular, g is a divisor of the coefficients of h and
3152 *
3153 * f(x) = h(x')
3154 *
3155 * is known to be a multiple of g.
3156 * If some prime factor in m appears with the same exponent in g,
3157 * then it can be removed from m because f(x) is already known
3158 * to be a multiple of g and therefore in particular of this power
3159 * of the prime factors.
3160 * Prime factors that appear with a smaller exponent in g cannot
3161 * be removed from m.
3162 * Let g' be the divisor of g containing all prime factors that
3163 * appear with the same exponent in m and g, then
3164 *
3165 * f(x) + m y_i = 0
3166 *
3167 * can be replaced by
3168 *
3169 * f(x) + m/g' y_i' = 0
3170 *
3171 * Note that (if g' != 1) this changes the explicit representation
3172 * of y_i to that of y_i', so the integer division at position i
3173 * is marked unknown and later recomputed by a call to
3174 * isl_basic_map_gauss.
3175 */
3178 {
3182 isl_int gcd;
3215 }
3222 error:
3226 }
3228 /* Simplify the stride constraints in "bmap" based on
3229 * the remaining equality constraints in "bmap" and all equality
3230 * constraints in "context".
3231 * Only do this if both "bmap" and "context" have stride constraints.
3232 *
3233 * First extract a copy of the stride constraints in "bmap" in a compressed
3234 * space exploiting all the other equality constraints and then
3235 * use this compressed copy to simplify the original stride constraints.
3236 */
3239 {
3261 }
3263 /* Return a basic map that has the same intersection with "context" as "bmap"
3264 * and that is as "simple" as possible.
3265 *
3266 * The core computation is performed on the pure constraints.
3267 * When we add back the meaning of the integer divisions, we need
3268 * to (re)introduce the div constraints. If we happen to have
3269 * discovered that some of these integer divisions are equal to
3270 * some affine combination of other variables, then these div
3271 * constraints may end up getting simplified in terms of the equalities,
3272 * resulting in extra inequalities on the other variables that
3273 * may have been removed already or that may not even have been
3274 * part of the input. We try and remove those constraints of
3275 * this form that are most obviously redundant with respect to
3276 * the context. We also remove those div constraints that are
3277 * redundant with respect to the other constraints in the result.
3278 *
3279 * The stride constraints among the equality constraints in "bmap" are
3280 * also simplified with respecting to the other equality constraints
3281 * in "bmap" and with respect to all equality constraints in "context".
3282 */
3285 {
3297 }
3303 }
3307 }
3331 }
3348 error:
3352 }
3354 /*
3355 * Assumes context has no implicit divs.
3356 */
3359 {
3370 }
3389 }
3390 }
3394 error:
3398 }
3400 /* Drop all inequalities from "bmap" that also appear in "context".
3401 * "context" is assumed to have only known local variables and
3402 * the initial local variables of "bmap" are assumed to be the same
3403 * as those of "context".
3404 * The constraints of both "bmap" and "context" are assumed
3405 * to have been sorted using isl_basic_map_sort_constraints.
3406 *
3407 * Run through the inequality constraints of "bmap" and "context"
3408 * in sorted order.
3409 * If a constraint of "bmap" involves variables not in "context",
3410 * then it cannot appear in "context".
3411 * If a matching constraint is found, it is removed from "bmap".
3412 */
3415 {
3436 }
3442 }
3446 }
3451 }
3454 }
3457 }
3459 /* Drop all equalities from "bmap" that also appear in "context".
3460 * "context" is assumed to have only known local variables and
3461 * the initial local variables of "bmap" are assumed to be the same
3462 * as those of "context".
3463 *
3464 * Run through the equality constraints of "bmap" and "context"
3465 * in sorted order.
3466 * If a constraint of "bmap" involves variables not in "context",
3467 * then it cannot appear in "context".
3468 * If a matching constraint is found, it is removed from "bmap".
3469 */
3472 {
3498 }
3502 }
3507 }
3510 }
3513 }
3515 /* Remove the constraints in "context" from "bmap".
3516 * "context" is assumed to have explicit representations
3517 * for all local variables.
3518 *
3519 * First align the divs of "bmap" to those of "context" and
3520 * sort the constraints. Then drop all constraints from "bmap"
3521 * that appear in "context".
3522 */
3525 {
3540 }
3560 error:
3564 }
3566 /* Replace "map" by the disjunct at position "pos" and free "context".
3567 */
3570 {
3577 }
3579 /* Remove the constraints in "context" from "map".
3580 * If any of the disjuncts in the result turns out to be the universe,
3581 * then return this universe.
3582 * "context" is assumed to have explicit representations
3583 * for all local variables.
3584 */
3587 {
3597 }
3616 }
3623 error:
3627 }
3629 /* Remove the constraints in "context" from "set".
3630 * If any of the disjuncts in the result turns out to be the universe,
3631 * then return this universe.
3632 * "context" is assumed to have explicit representations
3633 * for all local variables.
3634 */
3637 {
3640 }
3642 /* Remove the constraints in "context" from "map".
3643 * If any of the disjuncts in the result turns out to be the universe,
3644 * then return this universe.
3645 * "context" is assumed to consist of a single disjunct and
3646 * to have explicit representations for all local variables.
3647 */
3650 {
3655 }
3657 /* Replace "map" by a universe map in the same space and free "drop".
3658 */
3661 {
3668 }
3670 /* Return a map that has the same intersection with "context" as "map"
3671 * and that is as "simple" as possible.
3672 *
3673 * If "map" is already the universe, then we cannot make it any simpler.
3674 * Similarly, if "context" is the universe, then we cannot exploit it
3675 * to simplify "map"
3676 * If "map" and "context" are identical to each other, then we can
3677 * return the corresponding universe.
3678 *
3679 * If either "map" or "context" consists of multiple disjuncts,
3680 * then check if "context" happens to be a subset of "map",
3681 * in which case all constraints can be removed.
3682 * In case of multiple disjuncts, the standard procedure
3683 * may not be able to detect that all constraints can be removed.
3684 *
3685 * If none of these cases apply, we have to work a bit harder.
3686 * During this computation, we make use of a single disjunct context,
3687 * so if the original context consists of more than one disjunct
3688 * then we need to approximate the context by a single disjunct set.
3689 * Simply taking the simple hull may drop constraints that are
3690 * only implicitly available in each disjunct. We therefore also
3691 * look for constraints among those defining "map" that are valid
3692 * for the context. These can then be used to simplify away
3693 * the corresponding constraints in "map".
3694 */
3697 {
3701 isl_bool subset;
3712 }
3731 }
3747 list);
3748 }
3750 error:
3754 }
3758 {
3761 }
3765 {
3768 }
3772 {
3777 }
3781 {
3783 }
3785 /* Compute the gist of "bmap" with respect to the constraints "context"
3786 * on the domain.
3787 */
3790 {
3796 }
3800 {
3804 }
3808 {
3812 }
3816 {
3820 }
3824 {
3826 }
3828 /* Quick check to see if two basic maps are disjoint.
3829 * In particular, we reduce the equalities and inequalities of
3830 * one basic map in the context of the equalities of the other
3831 * basic map and check if we get a contradiction.
3832 */
3835 {
3838 isl_size total;
3867 }
3875 }
3884 }
3888 disjoint:
3892 error:
3896 }
3900 {
3903 }
3905 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3906 */
3910 {
3921 }
3922 }
3925 }
3927 /* Are "map1" and "map2" obviously disjoint, based on information
3928 * that can be derived without looking at the individual basic maps?
3929 *
3930 * In particular, if one of them is empty or if they live in different spaces
3931 * (ignoring parameters), then they are clearly disjoint.
3932 */
3935 {
3936 isl_bool disjoint;
3937 isl_bool match;
3959 }
3961 /* Are "map1" and "map2" obviously disjoint?
3962 *
3963 * If one of them is empty or if they live in different spaces (ignoring
3964 * parameters), then they are clearly disjoint.
3965 * This is checked by isl_map_plain_is_disjoint_global.
3966 *
3967 * If they have different parameters, then we skip any further tests.
3968 *
3969 * If they are obviously equal, but not obviously empty, then we will
3970 * not be able to detect if they are disjoint.
3971 *
3972 * Otherwise we check if each basic map in "map1" is obviously disjoint
3973 * from each basic map in "map2".
3974 */
3977 {
3978 isl_bool disjoint;
3979 isl_bool intersect;
3980 isl_bool match;
3995 }
3997 /* Are "map1" and "map2" disjoint?
3998 * The parameters are assumed to have been aligned.
3999 *
4000 * In particular, check whether all pairs of basic maps are disjoint.
4001 */
4004 {
4006 }
4008 /* Are "map1" and "map2" disjoint?
4009 *
4010 * They are disjoint if they are "obviously disjoint" or if one of them
4011 * is empty. Otherwise, they are not disjoint if one of them is universal.
4012 * If the two inputs are (obviously) equal and not empty, then they are
4013 * not disjoint.
4014 * If none of these cases apply, then check if all pairs of basic maps
4015 * are disjoint after aligning the parameters.
4016 */
4018 {
4019 isl_bool disjoint;
4020 isl_bool intersect;
4048 }
4050 /* Are "bmap1" and "bmap2" disjoint?
4051 *
4052 * They are disjoint if they are "obviously disjoint" or if one of them
4053 * is empty. Otherwise, they are not disjoint if one of them is universal.
4054 * If none of these cases apply, we compute the intersection and see if
4055 * the result is empty.
4056 */
4059 {
4060 isl_bool disjoint;
4061 isl_bool intersect;
4090 }
4092 /* Are "bset1" and "bset2" disjoint?
4093 */
4096 {
4098 }
4102 {
4104 }
4106 /* Are "set1" and "set2" disjoint?
4107 */
4109 {
4111 }
4113 /* Is "v" equal to 0, 1 or -1?
4114 */
4116 {
4118 }
4120 /* Are the "n" coefficients starting at "first" of inequality constraints
4121 * "i" and "j" of "bmap" opposite to each other?
4122 */
4125 {
4127 }
4129 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4130 * apart from the constant term?
4131 */
4133 {
4134 isl_size total;
4140 }
4142 /* Check if we can combine a given div with lower bound l and upper
4143 * bound u with some other div and if so return that other div.
4144 * Otherwise, return a position beyond the integer divisions.
4145 * Return -1 on error.
4146 *
4147 * We first check that
4148 * - the bounds are opposites of each other (except for the constant
4149 * term)
4150 * - the bounds do not reference any other div
4151 * - no div is defined in terms of this div
4152 *
4153 * Let m be the size of the range allowed on the div by the bounds.
4154 * That is, the bounds are of the form
4155 *
4156 * e <= a <= e + m - 1
4157 *
4158 * with e some expression in the other variables.
4159 * We look for another div b such that no third div is defined in terms
4160 * of this second div b and such that in any constraint that contains
4161 * a (except for the given lower and upper bound), also contains b
4162 * with a coefficient that is m times that of b.
4163 * That is, all constraints (except for the lower and upper bound)
4164 * are of the form
4165 *
4166 * e + f (a + m b) >= 0
4167 *
4168 * Furthermore, in the constraints that only contain b, the coefficient
4169 * of b should be equal to 1 or -1.
4170 * If so, we return b so that "a + m b" can be replaced by
4171 * a single div "c = a + m b".
4172 */
4175 {
4178 isl_size v_div;
4180 isl_bool opp;
4202 }
4212 }
4225 }
4236 }
4249 }
4254 }
4258 }
4260 /* Internal data structure used during the construction and/or evaluation of
4261 * an inequality that ensures that a pair of bounds always allows
4262 * for an integer value.
4263 *
4264 * "tab" is the tableau in which the inequality is evaluated. It may
4265 * be NULL until it is actually needed.
4266 * "v" contains the inequality coefficients.
4267 * "g", "fl" and "fu" are temporary scalars used during the construction and
4268 * evaluation.
4269 */
4273 isl_int g;
4274 isl_int fl;
4275 isl_int fu;
4276 };
4278 /* Free all the memory allocated by the fields of "data".
4279 */
4281 {
4287 }
4289 /* Is the inequality stored in data->v satisfied by "bmap"?
4290 * That is, does it only attain non-negative values?
4291 * data->tab is a tableau corresponding to "bmap".
4292 */
4295 {
4306 }
4308 /* Given a lower and an upper bound on div i, do they always allow
4309 * for an integer value of the given div?
4310 * Determine this property by constructing an inequality
4311 * such that the property is guaranteed when the inequality is nonnegative.
4312 * The lower bound is inequality l, while the upper bound is inequality u.
4313 * The constructed inequality is stored in data->v.
4314 *
4315 * Let the upper bound be
4316 *
4317 * -n_u a + e_u >= 0
4318 *
4319 * and the lower bound
4320 *
4321 * n_l a + e_l >= 0
4322 *
4323 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4324 * We have
4325 *
4326 * - f_u e_l <= f_u f_l g a <= f_l e_u
4327 *
4328 * Since all variables are integer valued, this is equivalent to
4329 *
4330 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4331 *
4332 * If this interval is at least f_u f_l g, then it contains at least
4333 * one integer value for a.
4334 * That is, the test constraint is
4335 *
4336 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4337 *
4338 * or
4339 *
4340 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4341 *
4342 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4343 * then the constraint can be scaled down by a factor g',
4344 * with the constant term replaced by
4345 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4346 * Note that the result of applying Fourier-Motzkin to this pair
4347 * of constraints is
4348 *
4349 * f_l e_u + f_u e_l >= 0
4350 *
4351 * If the constant term of the scaled down version of this constraint,
4352 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4353 * term of the scaled down test constraint, then the test constraint
4354 * is known to hold and no explicit evaluation is required.
4355 * This is essentially the Omega test.
4356 *
4357 * If the test constraint consists of only a constant term, then
4358 * it is sufficient to look at the sign of this constant term.
4359 */
4362 {
4364 isl_size n_div;
4391 }
4401 }
4403 /* Remove more kinds of divs that are not strictly needed.
4404 * In particular, if all pairs of lower and upper bounds on a div
4405 * are such that they allow at least one integer value of the div,
4406 * then we can eliminate the div using Fourier-Motzkin without
4407 * introducing any spurious solutions.
4408 *
4409 * If at least one of the two constraints has a unit coefficient for the div,
4410 * then the presence of such a value is guaranteed so there is no need to check.
4411 * In particular, the value attained by the bound with unit coefficient
4412 * can serve as this intermediate value.
4413 */
4416 {
4420 isl_size n_div;
4440 isl_bool has_int;
4448 }
4469 }
4472 }
4476 }
4480 }
4483 }
4494 error:
4499 }
4501 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4502 * and the upper bound u, div1 always occurs together with div2 in the form
4503 * (div1 + m div2), where m is the constant range on the variable div1
4504 * allowed by l and u, replace the pair div1 and div2 by a single
4505 * div that is equal to div1 + m div2.
4506 *
4507 * The new div will appear in the location that contains div2.
4508 * We need to modify all constraints that contain
4509 * div2 = (div - div1) / m
4510 * The coefficient of div2 is known to be equal to 1 or -1.
4511 * (If a constraint does not contain div2, it will also not contain div1.)
4512 * If the constraint also contains div1, then we know they appear
4513 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4514 * i.e., the coefficient of div is f.
4515 *
4516 * Otherwise, we first need to introduce div1 into the constraint.
4517 * Let l be
4518 *
4519 * div1 + f >=0
4520 *
4521 * and u
4522 *
4523 * -div1 + f' >= 0
4524 *
4525 * A lower bound on div2
4526 *
4527 * div2 + t >= 0
4528 *
4529 * can be replaced by
4530 *
4531 * m div2 + div1 + m t + f >= 0
4532 *
4533 * An upper bound
4534 *
4535 * -div2 + t >= 0
4536 *
4537 * can be replaced by
4538 *
4539 * -(m div2 + div1) + m t + f' >= 0
4540 *
4541 * These constraint are those that we would obtain from eliminating
4542 * div1 using Fourier-Motzkin.
4543 *
4544 * After all constraints have been modified, we drop the lower and upper
4545 * bound and then drop div1.
4546 * Since the new div is only placed in the same location that used
4547 * to store div2, but otherwise has a different meaning, any possible
4548 * explicit representation of the original div2 is removed.
4549 */
4552 {
4554 isl_int m;
4555 isl_size v_div;
4579 else
4582 }
4586 }
4595 }
4599 }
4601 /* First check if we can coalesce any pair of divs and
4602 * then continue with dropping more redundant divs.
4603 *
4604 * We loop over all pairs of lower and upper bounds on a div
4605 * with coefficient 1 and -1, respectively, check if there
4606 * is any other div "c" with which we can coalesce the div
4607 * and if so, perform the coalescing.
4608 */
4611 {
4613 isl_size v_div;
4614 isl_size n_div;
4640 }
4641 }
4642 }
4647 }
4650 error:
4654 }
4656 /* Are the "n" coefficients starting at "first" of inequality constraints
4657 * "i" and "j" of "bmap" equal to each other?
4658 */
4661 {
4663 }
4665 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4666 * apart from the constant term and the coefficient at position "pos"?
4667 */
4670 {
4671 isl_size total;
4678 }
4680 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4681 * apart from the constant term and the coefficient at position "pos"?
4682 */
4685 {
4686 isl_size total;
4693 }
4695 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4696 * been modified, simplying it if "simplify" is set.
4697 * Free the temporary data structure "pairs" that was associated
4698 * to the old version of "bmap".
4699 */
4702 {
4707 }
4709 /* Is "div" the single unknown existentially quantified variable
4710 * in inequality constraint "ineq" of "bmap"?
4711 * "div" is known to have a non-zero coefficient in "ineq".
4712 */
4715 {
4717 isl_size n_div;
4719 isl_bool known;
4731 isl_bool known;
4740 }
4743 }
4745 /* Does integer division "div" have coefficient 1 in inequality constraint
4746 * "ineq" of "map"?
4747 */
4749 {
4757 }
4759 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4760 * then try and drop redundant divs again,
4761 * freeing the temporary data structure "pairs" that was associated
4762 * to the old version of "bmap".
4763 */
4766 {
4770 }
4772 /* Drop the integer division at position "div", along with the two
4773 * inequality constraints "ineq1" and "ineq2" in which it appears
4774 * from "bmap" and then try and drop redundant divs again,
4775 * freeing the temporary data structure "pairs" that was associated
4776 * to the old version of "bmap".
4777 */
4781 {
4788 }
4791 }
4793 /* Given two inequality constraints
4794 *
4795 * f(x) + n d + c >= 0, (ineq)
4796 *
4797 * with d the variable at position "pos", and
4798 *
4799 * f(x) + c0 >= 0, (lower)
4800 *
4801 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4802 * determined by the first constraint.
4803 * That is, store
4804 *
4805 * ceil((c0 - c)/n)
4806 *
4807 * in *l.
4808 */
4811 {
4815 }
4817 /* Given two inequality constraints
4818 *
4819 * f(x) + n d + c >= 0, (ineq)
4820 *
4821 * with d the variable at position "pos", and
4822 *
4823 * -f(x) - c0 >= 0, (upper)
4824 *
4825 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4826 * determined by the first constraint.
4827 * That is, store
4828 *
4829 * ceil((-c1 - c)/n)
4830 *
4831 * in *u.
4832 */
4835 {
4839 }
4841 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4842 * does the corresponding lower bound have a fixed value in "bmap"?
4843 *
4844 * In particular, "ineq" is of the form
4845 *
4846 * f(x) + n d + c >= 0
4847 *
4848 * with n > 0, c the constant term and
4849 * d the existentially quantified variable "div".
4850 * That is, the lower bound is
4851 *
4852 * ceil((-f(x) - c)/n)
4853 *
4854 * Look for a pair of constraints
4855 *
4856 * f(x) + c0 >= 0
4857 * -f(x) + c1 >= 0
4858 *
4859 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4860 * That is, check that
4861 *
4862 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4863 *
4864 * If so, return the index of inequality f(x) + c0 >= 0.
4865 * Otherwise, return bmap->n_ineq.
4866 * Return -1 on error.
4867 */
4869 {
4892 }
4900 }
4917 }
4919 /* Given a lower bound constraint "ineq" on the existentially quantified
4920 * variable "div", such that the corresponding lower bound has
4921 * a fixed value in "bmap", assign this fixed value to the variable and
4922 * then try and drop redundant divs again,
4923 * freeing the temporary data structure "pairs" that was associated
4924 * to the old version of "bmap".
4925 * "lower" determines the constant value for the lower bound.
4926 *
4927 * In particular, "ineq" is of the form
4928 *
4929 * f(x) + n d + c >= 0,
4930 *
4931 * while "lower" is of the form
4932 *
4933 * f(x) + c0 >= 0
4934 *
4935 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4936 * is ceil((c0 - c)/n).
4937 */
4940 {
4941 isl_int c;
4954 }
4956 /* Do any of the integer divisions of "bmap" involve integer division "div"?
4957 *
4958 * The integer division "div" could only ever appear in any later
4959 * integer division (with an explicit representation).
4960 */
4962 {
4972 isl_bool unknown;
4981 }
4984 }
4986 /* Remove divs that are not strictly needed based on the inequality
4987 * constraints.
4988 * In particular, if a div only occurs positively (or negatively)
4989 * in constraints, then it can simply be dropped.
4990 * Also, if a div occurs in only two constraints and if moreover
4991 * those two constraints are opposite to each other, except for the constant
4992 * term and if the sum of the constant terms is such that for any value
4993 * of the other values, there is always at least one integer value of the
4994 * div, i.e., if one plus this sum is greater than or equal to
4995 * the (absolute value) of the coefficient of the div in the constraints,
4996 * then we can also simply drop the div.
4997 *
4998 * If an existentially quantified variable does not have an explicit
4999 * representation, appears in only a single lower bound that does not
5000 * involve any other such existentially quantified variables and appears
5001 * in this lower bound with coefficient 1,
5002 * then fix the variable to the value of the lower bound. That is,
5003 * turn the inequality into an equality.
5004 * If for any value of the other variables, there is any value
5005 * for the existentially quantified variable satisfying the constraints,
5006 * then this lower bound also satisfies the constraints.
5007 * It is therefore safe to pick this lower bound.
5008 *
5009 * The same reasoning holds even if the coefficient is not one.
5010 * However, fixing the variable to the value of the lower bound may
5011 * in general introduce an extra integer division, in which case
5012 * it may be better to pick another value.
5013 * If this integer division has a known constant value, then plugging
5014 * in this constant value removes the existentially quantified variable
5015 * completely. In particular, if the lower bound is of the form
5016 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
5017 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
5018 * then the existentially quantified variable can be assigned this
5019 * shared value.
5020 *
5021 * We skip divs that appear in equalities or in the definition of other divs.
5022 * Divs that appear in the definition of other divs usually occur in at least
5023 * 4 constraints, but the constraints may have been simplified.
5024 *
5025 * If any divs are left after these simple checks then we move on
5026 * to more complicated cases in drop_more_redundant_divs.
5027 */
5030 {
5032 isl_size off;
5035 isl_size n_ineq;
5076 }
5080 }
5081 }
5089 }
5092 else
5112 pairs);
5118 pairs);
5120 }
5137 else
5144 }
5147 }
5154 error:
5158 }
5160 /* Consider the coefficients at "c" as a row vector and replace
5161 * them with their product with "T". "T" is assumed to be a square matrix.
5162 */
5164 {
5165 isl_size n;
5186 }
5188 /* Plug in T for the variables in "bmap" starting at "pos".
5189 * T is a linear unimodular matrix, i.e., without constant term.
5190 */
5193 {
5221 }
5225 error:
5229 }
5231 /* Remove divs that are not strictly needed.
5232 *
5233 * First look for an equality constraint involving two or more
5234 * existentially quantified variables without an explicit
5235 * representation. Replace the combination that appears
5236 * in the equality constraint by a single existentially quantified
5237 * variable such that the equality can be used to derive
5238 * an explicit representation for the variable.
5239 * If there are no more such equality constraints, then continue
5240 * with isl_basic_map_drop_redundant_divs_ineq.
5241 *
5242 * In particular, if the equality constraint is of the form
5243 *
5244 * f(x) + \sum_i c_i a_i = 0
5245 *
5246 * with a_i existentially quantified variable without explicit
5247 * representation, then apply a transformation on the existentially
5248 * quantified variables to turn the constraint into
5249 *
5250 * f(x) + g a_1' = 0
5251 *
5252 * with g the gcd of the c_i.
5253 * In order to easily identify which existentially quantified variables
5254 * have a complete explicit representation, i.e., without being defined
5255 * in terms of other existentially quantified variables without
5256 * an explicit representation, the existentially quantified variables
5257 * are first sorted.
5258 *
5259 * The variable transformation is computed by extending the row
5260 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5261 *
5262 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5263 * [a_2'] [ a_2 ]
5264 * ... = U ....
5265 * [a_n'] [ a_n ]
5266 *
5267 * with [c_1/g ... c_n/g] representing the first row of U.
5268 * The inverse of U is then plugged into the original constraints.
5269 * The call to isl_basic_map_simplify makes sure the explicit
5270 * representation for a_1' is extracted from the equality constraint.
5271 */
5274 {
5278 isl_size n_div;
5312 }
5331 }
5333 /* Does "bmap" satisfy any equality that involves more than 2 variables
5334 * and/or has coefficients different from -1 and 1?
5335 */
5337 {
5339 isl_size total;
5368 }
5371 }
5373 /* Remove any common factor g from the constraint coefficients in "v".
5374 * The constant term is stored in the first position and is replaced
5375 * by floor(c/g). If any common factor is removed and if this results
5376 * in a tightening of the constraint, then set *tightened.
5377 */
5380 {
5400 }
5402 /* Internal representation used by isl_basic_map_reduce_coefficients.
5403 *
5404 * "total" is the total dimensionality of the original basic map.
5405 * "v" is a temporary vector of size 1 + total that can be used
5406 * to store constraint coefficients.
5407 * "T" is the variable compression.
5408 * "T2" is the inverse transformation.
5409 * "tightened" is set if any constant term got tightened
5410 * while reducing the coefficients.
5411 */
5413 isl_size total;
5418 };
5420 /* Free all memory allocated in "data".
5421 */
5424 {
5428 }
5430 /* Initialize "data" for "bmap", freeing all allocated memory
5431 * if anything goes wrong.
5432 *
5433 * In particular, construct a variable compression
5434 * from the equality constraints of "bmap" and
5435 * allocate a temporary vector.
5436 */
5440 {
5464 error:
5467 }
5469 /* Reduce the coefficients of "bmap" by applying the variable compression
5470 * in "data".
5471 * In particular, apply the variable compression to each constraint,
5472 * factor out any common factor in the non-constant coefficients and
5473 * then apply the inverse of the compression.
5474 *
5475 * Only apply the reduction on a single copy of the basic map
5476 * since the reduction may leave the result in an inconsistent state.
5477 * In particular, the constraints may not be gaussed.
5478 */
5482 {
5484 isl_size total;
5506 }
5511 }
5513 /* If "bmap" is an integer set that satisfies any equality involving
5514 * more than 2 variables and/or has coefficients different from -1 and 1,
5515 * then use variable compression to reduce the coefficients by removing
5516 * any (hidden) common factor.
5517 * In particular, apply the variable compression to each constraint,
5518 * factor out any common factor in the non-constant coefficients and
5519 * then apply the inverse of the compression.
5520 * At the end, we mark the basic map as having reduced constants.
5521 * If this flag is still set on the next invocation of this function,
5522 * then we skip the computation.
5523 *
5524 * Removing a common factor may result in a tightening of some of
5525 * the constraints. If this happens, then we may end up with two
5526 * opposite inequalities that can be replaced by an equality.
5527 * We therefore call isl_basic_map_detect_inequality_pairs,
5528 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5529 * and isl_basic_map_gauss if such a pair was found.
5530 * This call to isl_basic_map_gauss may undo much of the effect
5531 * of the reduction on which isl_map_coalesce depends.
5532 * In particular, constraints in terms of (compressed) local variables
5533 * get reformulated in terms of the set variables again.
5534 * The reduction is therefore applied again afterwards.
5535 * This has to be done before the call to eliminate_divs_eq, however,
5536 * since that may remove some local variables, while
5537 * the data used during the reduction is formulated in terms
5538 * of the original variables.
5539 *
5540 * Tightening may also result in some other constraints becoming
5541 * (rationally) redundant with respect to the tightened constraint
5542 * (in combination with other constraints). The basic map may
5543 * therefore no longer be assumed to have no redundant constraints.
5544 *
5545 * Note that this function may leave the result in an inconsistent state.
5546 * In particular, the constraints may not be gaussed.
5547 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5548 * for some of the test cases to pass successfully.
5549 * Any potential modification of the representation is therefore only
5550 * performed on a single copy of the basic map.
5551 */
5554 {
5556 isl_bool multi;
5578 }
5593 }
5594 }
5599 error:
5602 }
5604 /* Shift the integer division at position "div" of "bmap"
5605 * by "shift" times the variable at position "pos".
5606 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5607 * corresponds to the constant term.
5608 *
5609 * That is, if the integer division has the form
5610 *
5611 * floor(f(x)/d)
5612 *
5613 * then replace it by
5614 *
5615 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5616 */
5619 {
5638 }
5644 }
5652 }
5655 }