/* * Copyright 2008-2009 Katholieke Universiteit Leuven * Copyright 2010 INRIA Saclay * Copyright 2012-2013 Ecole Normale Superieure * Copyright 2014 INRIA Rocquencourt * Copyright 2016 INRIA Paris * Copyright 2020 Cerebras Systems * * Use of this software is governed by the MIT license * * Written by Sven Verdoolaege, K.U.Leuven, Departement * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite, * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt, * B.P. 105 - 78153 Le Chesnay, France * and Centre de Recherche Inria de Paris, 2 rue Simone Iff - Voie DQ12, * CS 42112, 75589 Paris Cedex 12, France * and Cerebras Systems, 175 S San Antonio Rd, Los Altos, CA, USA */ #include <isl_ctx_private.h> #include "isl_map_private.h" #include <isl_seq.h> #include <isl/options.h> #include "isl_tab.h" #include <isl_mat_private.h> #include <isl_local_space_private.h> #include <isl_val_private.h> #include <isl_vec_private.h> #include <isl_aff_private.h> #include <isl_equalities.h> #include <isl_constraint_private.h> #include <set_to_map.c> #include <set_from_map.c> #define STATUS_ERROR … #define STATUS_REDUNDANT … #define STATUS_VALID … #define STATUS_SEPARATE … #define STATUS_CUT … #define STATUS_ADJ_EQ … #define STATUS_ADJ_INEQ … static int status_in(isl_int *ineq, struct isl_tab *tab) { … } /* Compute the position of the equalities of basic map "bmap_i" * with respect to the basic map represented by "tab_j". * The resulting array has twice as many entries as the number * of equalities corresponding to the two inequalities to which * each equality corresponds. */ static int *eq_status_in(__isl_keep isl_basic_map *bmap_i, struct isl_tab *tab_j) { … } /* Compute the position of the inequalities of basic map "bmap_i" * (also represented by "tab_i", if not NULL) with respect to the basic map * represented by "tab_j". */ static int *ineq_status_in(__isl_keep isl_basic_map *bmap_i, struct isl_tab *tab_i, struct isl_tab *tab_j) { … } static int any(int *con, unsigned len, int status) { … } /* Return the first position of "status" in the list "con" of length "len". * Return -1 if there is no such entry. */ static int find(int *con, unsigned len, int status) { … } static int count(int *con, unsigned len, int status) { … } static int all(int *con, unsigned len, int status) { … } /* Internal information associated to a basic map in a map * that is to be coalesced by isl_map_coalesce. * * "bmap" is the basic map itself (or NULL if "removed" is set) * "tab" is the corresponding tableau (or NULL if "removed" is set) * "hull_hash" identifies the affine space in which "bmap" lives. * "modified" is set if this basic map may not be identical * to any of the basic maps in the input. * "removed" is set if this basic map has been removed from the map * "simplify" is set if this basic map may have some unknown integer * divisions that were not present in the input basic maps. The basic * map should then be simplified such that we may be able to find * a definition among the constraints. * * "eq" and "ineq" are only set if we are currently trying to coalesce * this basic map with another basic map, in which case they represent * the position of the inequalities of this basic map with respect to * the other basic map. The number of elements in the "eq" array * is twice the number of equalities in the "bmap", corresponding * to the two inequalities that make up each equality. */ struct isl_coalesce_info { … }; /* Is there any (half of an) equality constraint in the description * of the basic map represented by "info" that * has position "status" with respect to the other basic map? */ static int any_eq(struct isl_coalesce_info *info, int status) { … } /* Is there any inequality constraint in the description * of the basic map represented by "info" that * has position "status" with respect to the other basic map? */ static int any_ineq(struct isl_coalesce_info *info, int status) { … } /* Return the position of the first half on an equality constraint * in the description of the basic map represented by "info" that * has position "status" with respect to the other basic map. * The returned value is twice the position of the equality constraint * plus zero for the negative half and plus one for the positive half. * Return -1 if there is no such entry. */ static int find_eq(struct isl_coalesce_info *info, int status) { … } /* Return the position of the first inequality constraint in the description * of the basic map represented by "info" that * has position "status" with respect to the other basic map. * Return -1 if there is no such entry. */ static int find_ineq(struct isl_coalesce_info *info, int status) { … } /* Return the number of (halves of) equality constraints in the description * of the basic map represented by "info" that * have position "status" with respect to the other basic map. */ static int count_eq(struct isl_coalesce_info *info, int status) { … } /* Return the number of inequality constraints in the description * of the basic map represented by "info" that * have position "status" with respect to the other basic map. */ static int count_ineq(struct isl_coalesce_info *info, int status) { … } /* Are all non-redundant constraints of the basic map represented by "info" * either valid or cut constraints with respect to the other basic map? */ static int all_valid_or_cut(struct isl_coalesce_info *info) { … } /* Compute the hash of the (apparent) affine hull of info->bmap (with * the existentially quantified variables removed) and store it * in info->hash. */ static int coalesce_info_set_hull_hash(struct isl_coalesce_info *info) { … } /* Free all the allocated memory in an array * of "n" isl_coalesce_info elements. */ static void clear_coalesce_info(int n, struct isl_coalesce_info *info) { … } /* Clear the memory associated to "info". */ static void clear(struct isl_coalesce_info *info) { … } /* Drop the basic map represented by "info". * That is, clear the memory associated to the entry and * mark it as having been removed. */ static void drop(struct isl_coalesce_info *info) { … } /* Exchange the information in "info1" with that in "info2". */ static void exchange(struct isl_coalesce_info *info1, struct isl_coalesce_info *info2) { … } /* This type represents the kind of change that has been performed * while trying to coalesce two basic maps. * * isl_change_none: nothing was changed * isl_change_drop_first: the first basic map was removed * isl_change_drop_second: the second basic map was removed * isl_change_fuse: the two basic maps were replaced by a new basic map. */ enum isl_change { … }; /* Update "change" based on an interchange of the first and the second * basic map. That is, interchange isl_change_drop_first and * isl_change_drop_second. */ static enum isl_change invert_change(enum isl_change change) { … } /* Add the valid constraints of the basic map represented by "info" * to "bmap". "len" is the size of the constraints. * If only one of the pair of inequalities that make up an equality * is valid, then add that inequality. */ static __isl_give isl_basic_map *add_valid_constraints( __isl_take isl_basic_map *bmap, struct isl_coalesce_info *info, unsigned len) { … } /* Is "bmap" defined by a number of (non-redundant) constraints that * is greater than the number of constraints of basic maps i and j combined? * Equalities are counted as two inequalities. */ static int number_of_constraints_increases(int i, int j, struct isl_coalesce_info *info, __isl_keep isl_basic_map *bmap, struct isl_tab *tab) { … } /* Replace the pair of basic maps i and j by the basic map bounded * by the valid constraints in both basic maps and the constraints * in extra (if not NULL). * Place the fused basic map in the position that is the smallest of i and j. * * If "detect_equalities" is set, then look for equalities encoded * as pairs of inequalities. * If "check_number" is set, then the original basic maps are only * replaced if the total number of constraints does not increase. * While the number of integer divisions in the two basic maps * is assumed to be the same, the actual definitions may be different. * We only copy the definition from one of the basic maps if it is * the same as that of the other basic map. Otherwise, we mark * the integer division as unknown and simplify the basic map * in an attempt to recover the integer division definition. * If any extra constraints get introduced, then these may * involve integer divisions with a unit coefficient. * Eliminate those that do not appear with any other coefficient * in other constraints, to ensure they get eliminated completely, * improving the chances of further coalescing. */ static enum isl_change fuse(int i, int j, struct isl_coalesce_info *info, __isl_keep isl_mat *extra, int detect_equalities, int check_number) { … } /* Given a pair of basic maps i and j such that all constraints are either * "valid" or "cut", check if the facets corresponding to the "cut" * constraints of i lie entirely within basic map j. * If so, replace the pair by the basic map consisting of the valid * constraints in both basic maps. * Checking whether the facet lies entirely within basic map j * is performed by checking whether the constraints of basic map j * are valid for the facet. These tests are performed on a rational * tableau to avoid the theoretical possibility that a constraint * that was considered to be a cut constraint for the entire basic map i * happens to be considered to be a valid constraint for the facet, * even though it cuts off the same rational points. * * To see that we are not introducing any extra points, call the * two basic maps A and B and the resulting map U and let x * be an element of U \setminus ( A \cup B ). * A line connecting x with an element of A \cup B meets a facet F * of either A or B. Assume it is a facet of B and let c_1 be * the corresponding facet constraint. We have c_1(x) < 0 and * so c_1 is a cut constraint. This implies that there is some * (possibly rational) point x' satisfying the constraints of A * and the opposite of c_1 as otherwise c_1 would have been marked * valid for A. The line connecting x and x' meets a facet of A * in a (possibly rational) point that also violates c_1, but this * is impossible since all cut constraints of B are valid for all * cut facets of A. * In case F is a facet of A rather than B, then we can apply the * above reasoning to find a facet of B separating x from A \cup B first. */ static enum isl_change check_facets(int i, int j, struct isl_coalesce_info *info) { … } /* Check if info->bmap contains the basic map represented * by the tableau "tab". * For each equality, we check both the constraint itself * (as an inequality) and its negation. Make sure the * equality is returned to its original state before returning. */ static isl_bool contains(struct isl_coalesce_info *info, struct isl_tab *tab) { … } /* Basic map "i" has an inequality "k" that is adjacent * to some inequality of basic map "j". All the other inequalities * are valid for "j". * If not NULL, then "extra" contains extra wrapping constraints that are valid * for both "i" and "j". * Check if basic map "j" forms an extension of basic map "i", * taking into account the extra constraints, if any. * * Note that this function is only called if some of the equalities or * inequalities of basic map "j" do cut basic map "i". The function is * correct even if there are no such cut constraints, but in that case * the additional checks performed by this function are overkill. * * In particular, we replace constraint k, say f >= 0, by constraint * f <= -1, add the inequalities of "j" that are valid for "i", * as well as the "extra" constraints, if any, * and check if the result is a subset of basic map "j". * To improve the chances of the subset relation being detected, * any variable that only attains a single integer value * in the tableau of "i" is first fixed to that value. * If the result is a subset, then we know that this result is exactly equal * to basic map "j" since all its constraints are valid for basic map "j". * By combining the valid constraints of "i" (all equalities and all * inequalities except "k"), the valid constraints of "j" and * the "extra" constraints, if any, we therefore * obtain a basic map that is equal to their union. * In this case, there is no need to perform a rollback of the tableau * since it is going to be destroyed in fuse(). * * * |\__ |\__ * | \__ | \__ * | \_ => | \__ * |_______| _ |_________\ * * * |\ |\ * | \ | \ * | \ | \ * | | | \ * | ||\ => | \ * | || \ | \ * | || | | | * |__||_/ |_____/ * * * _______ _______ * | | __ | \__ * | ||__| => | __| * |_______| |_______/ */ static enum isl_change is_adj_ineq_extension_with_wraps(int i, int j, int k, struct isl_coalesce_info *info, __isl_keep isl_mat *extra) { … } /* Given an affine transformation matrix "T", does row "row" represent * anything other than a unit vector (possibly shifted by a constant) * that is not involved in any of the other rows? * * That is, if a constraint involves the variable corresponding to * the row, then could its preimage by "T" have any coefficients * that are different from those in the original constraint? */ static int not_unique_unit_row(__isl_keep isl_mat *T, int row) { … } /* Does inequality constraint "ineq" of "bmap" involve any of * the variables marked in "affected"? * "total" is the total number of variables, i.e., the number * of entries in "affected". */ static isl_bool is_affected(__isl_keep isl_basic_map *bmap, int ineq, int *affected, int total) { … } /* Given the compressed version of inequality constraint "ineq" * of info->bmap in "v", check if the constraint can be tightened, * where the compression is based on an equality constraint valid * for info->tab. * If so, add the tightened version of the inequality constraint * to info->tab. "v" may be modified by this function. * * That is, if the compressed constraint is of the form * * m f() + c >= 0 * * with 0 < c < m, then it is equivalent to * * f() >= 0 * * This means that c can also be subtracted from the original, * uncompressed constraint without affecting the integer points * in info->tab. Add this tightened constraint as an extra row * to info->tab to make this information explicitly available. */ static __isl_give isl_vec *try_tightening(struct isl_coalesce_info *info, int ineq, __isl_take isl_vec *v) { … } /* Tighten the (non-redundant) constraints on the facet represented * by info->tab. * In particular, on input, info->tab represents the result * of relaxing the "n" inequality constraints of info->bmap in "relaxed" * by one, i.e., replacing f_i >= 0 by f_i + 1 >= 0, and then * replacing the one at index "l" by the corresponding equality, * i.e., f_k + 1 = 0, with k = relaxed[l]. * * Compute a variable compression from the equality constraint f_k + 1 = 0 * and use it to tighten the other constraints of info->bmap * (that is, all constraints that have not been relaxed), * updating info->tab (and leaving info->bmap untouched). * The compression handles essentially two cases, one where a variable * is assigned a fixed value and can therefore be eliminated, and one * where one variable is a shifted multiple of some other variable and * can therefore be replaced by that multiple. * Gaussian elimination would also work for the first case, but for * the second case, the effectiveness would depend on the order * of the variables. * After compression, some of the constraints may have coefficients * with a common divisor. If this divisor does not divide the constant * term, then the constraint can be tightened. * The tightening is performed on the tableau info->tab by introducing * extra (temporary) constraints. * * Only constraints that are possibly affected by the compression are * considered. In particular, if the constraint only involves variables * that are directly mapped to a distinct set of other variables, then * no common divisor can be introduced and no tightening can occur. * * It is important to only consider the non-redundant constraints * since the facet constraint has been relaxed prior to the call * to this function, meaning that the constraints that were redundant * prior to the relaxation may no longer be redundant. * These constraints will be ignored in the fused result, so * the fusion detection should not exploit them. */ static isl_stat tighten_on_relaxed_facet(struct isl_coalesce_info *info, int n, int *relaxed, int l) { … } /* Replace the basic maps "i" and "j" by an extension of "i" * along the "n" inequality constraints in "relax" by one. * The tableau info[i].tab has already been extended. * Extend info[i].bmap accordingly by relaxing all constraints in "relax" * by one. * Each integer division that does not have exactly the same * definition in "i" and "j" is marked unknown and the basic map * is scheduled to be simplified in an attempt to recover * the integer division definition. * Place the extension in the position that is the smallest of i and j. */ static enum isl_change extend(int i, int j, int n, int *relax, struct isl_coalesce_info *info) { … } /* Basic map "i" has "n" inequality constraints (collected in "relax") * that are such that they include basic map "j" if they are relaxed * by one. All the other inequalities are valid for "j". * Check if basic map "j" forms an extension of basic map "i". * * In particular, relax the constraints in "relax", compute the corresponding * facets one by one and check whether each of these is included * in the other basic map. * Before testing for inclusion, the constraints on each facet * are tightened to increase the chance of an inclusion being detected. * (Adding the valid constraints of "j" to the tableau of "i", as is done * in is_adj_ineq_extension, may further increase those chances, but this * is not currently done.) * If each facet is included, we know that relaxing the constraints extends * the basic map with exactly the other basic map (we already know that this * other basic map is included in the extension, because all other * inequality constraints are valid of "j") and we can replace the * two basic maps by this extension. * * If any of the relaxed constraints turn out to be redundant, then bail out. * isl_tab_select_facet refuses to handle such constraints. It may be * possible to handle them anyway by making a distinction between * redundant constraints with a corresponding facet that still intersects * the set (allowing isl_tab_select_facet to handle them) and * those where the facet does not intersect the set (which can be ignored * because the empty facet is trivially included in the other disjunct). * However, relaxed constraints that turn out to be redundant should * be fairly rare and no such instance has been reported where * coalescing would be successful. * ____ _____ * / || / | * / || / | * \ || => \ | * \ || \ | * \___|| \____| * * * \ |\ * |\\ | \ * | \\ | \ * | | => | / * | / | / * |/ |/ */ static enum isl_change is_relaxed_extension(int i, int j, int n, int *relax, struct isl_coalesce_info *info) { … } /* Data structure that keeps track of the wrapping constraints * and of information to bound the coefficients of those constraints. * * "failed" is set if wrapping has failed. * bound is set if we want to apply a bound on the coefficients * mat contains the wrapping constraints * max is the bound on the coefficients (if bound is set) */ struct isl_wraps { … }; /* Update wraps->max to be greater than or equal to the coefficients * in the equalities and inequalities of info->bmap that can be removed * if we end up applying wrapping. */ static isl_stat wraps_update_max(struct isl_wraps *wraps, struct isl_coalesce_info *info) { … } /* Initialize the isl_wraps data structure. * If we want to bound the coefficients of the wrapping constraints, * we set wraps->max to the largest coefficient * in the equalities and inequalities that can be removed if we end up * applying wrapping. */ static isl_stat wraps_init(struct isl_wraps *wraps, __isl_take isl_mat *mat, struct isl_coalesce_info *info, int i, int j) { … } /* Free the contents of the isl_wraps data structure. */ static void wraps_free(struct isl_wraps *wraps) { … } /* Mark the wrapping as failed. */ static isl_stat wraps_mark_failed(struct isl_wraps *wraps) { … } /* Is the wrapping constraint in row "row" allowed? * * If wraps->bound is set, we check that none of the coefficients * is greater than wraps->max. */ static int allow_wrap(struct isl_wraps *wraps, int row) { … } /* Wrap "ineq" (or its opposite if "negate" is set) around "bound" * to include "set" and add the result in position "w" of "wraps". * "len" is the total number of coefficients in "bound" and "ineq". * Return 1 on success, 0 on failure and -1 on error. * Wrapping can fail if the result of wrapping is equal to "bound" * or if we want to bound the sizes of the coefficients and * the wrapped constraint does not satisfy this bound. */ static int add_wrap(struct isl_wraps *wraps, int w, isl_int *bound, isl_int *ineq, unsigned len, __isl_keep isl_set *set, int negate) { … } /* This function has two modes of operations. * * If "add_valid" is set, then all the constraints of info->bmap * (except the opposite of "bound") are valid for the other basic map. * In this case, attempts are made to wrap some of these valid constraints * to more tightly fit around "set". Only successful wrappings are recorded * and failed wrappings are ignored. * * If "add_valid" is not set, then some of the constraints of info->bmap * are not valid for the other basic map, and only those are considered * for wrapping. In this case all attempted wrappings need to succeed. * Otherwise "wraps" is marked as failed. * Note that the constraints that are valid for the other basic map * will be added to the combined basic map by default, so there is * no need to wrap them. * The caller wrap_in_facets even relies on this function not wrapping * any constraints that are already valid. * * Only consider constraints that are not redundant (as determined * by info->tab) and that are valid or invalid depending on "add_valid". * Wrap each constraint around "bound" such that it includes the whole * set "set" and append the resulting constraint to "wraps". * "wraps" is assumed to have been pre-allocated to the appropriate size. * wraps->n_row is the number of actual wrapped constraints that have * been added. * If any of the wrapping problems results in a constraint that is * identical to "bound", then this means that "set" is unbounded in such * a way that no wrapping is possible. * Similarly, if we want to bound the coefficients of the wrapping * constraints and a newly added wrapping constraint does not * satisfy the bound, then the wrapping is considered to have failed. * Note though that "wraps" is only marked failed if "add_valid" is not set. */ static isl_stat add_selected_wraps(struct isl_wraps *wraps, struct isl_coalesce_info *info, isl_int *bound, __isl_keep isl_set *set, int add_valid) { … } /* For each constraint in info->bmap that is not redundant (as determined * by info->tab) and that is not a valid constraint for the other basic map, * wrap the constraint around "bound" such that it includes the whole * set "set" and append the resulting constraint to "wraps". * Note that the constraints that are valid for the other basic map * will be added to the combined basic map by default, so there is * no need to wrap them. * The caller wrap_in_facets even relies on this function not wrapping * any constraints that are already valid. * "wraps" is assumed to have been pre-allocated to the appropriate size. * wraps->n_row is the number of actual wrapped constraints that have * been added. * If any of the wrapping problems results in a constraint that is * identical to "bound", then this means that "set" is unbounded in such * a way that no wrapping is possible. If this happens then "wraps" * is marked as failed. * Similarly, if we want to bound the coefficients of the wrapping * constraints and a newly added wrapping constraint does not * satisfy the bound, then "wraps" is also marked as failed. */ static isl_stat add_wraps(struct isl_wraps *wraps, struct isl_coalesce_info *info, isl_int *bound, __isl_keep isl_set *set) { … } /* Check if the constraints in "wraps" from "first" until the last * are all valid for the basic set represented by "tab", * dropping the invalid constraints if "keep" is set and * marking the wrapping as failed if "keep" is not set and * any constraint turns out to be invalid. */ static isl_stat check_wraps(struct isl_wraps *wraps, int first, struct isl_tab *tab, int keep) { … } /* Return a set that corresponds to the non-redundant constraints * (as recorded in tab) of bmap. * * It's important to remove the redundant constraints as some * of the other constraints may have been modified after the * constraints were marked redundant. * In particular, a constraint may have been relaxed. * Redundant constraints are ignored when a constraint is relaxed * and should therefore continue to be ignored ever after. * Otherwise, the relaxation might be thwarted by some of * these constraints. * * Update the underlying set to ensure that the dimension doesn't change. * Otherwise the integer divisions could get dropped if the tab * turns out to be empty. */ static __isl_give isl_set *set_from_updated_bmap(__isl_keep isl_basic_map *bmap, struct isl_tab *tab) { … } /* Does "info" have any cut constraints that are redundant? */ static isl_bool has_redundant_cuts(struct isl_coalesce_info *info) { … } /* Wrap some constraints of info->bmap that bound the facet defined * by inequality "k" around (the opposite of) this inequality to * include "set". "bound" may be used to store the negated inequality. * * If "add_valid" is set, then all ridges are already valid and * the purpose is to wrap "set" more tightly. In this case, * wrapping doesn't fail, although it is possible that no constraint * gets wrapped. * * If "add_valid" is not set, then some of the ridges are cut constraints * and only those are wrapped around "set". * * Since the wrapped constraints are not guaranteed to contain the whole * of info->bmap, we check them in check_wraps. * If any of the wrapped constraints turn out to be invalid, then * check_wraps will mark "wraps" as failed if "add_valid" is not set. * If "add_valid" is set, then the offending constraints are * simply removed. * * If the facet turns out to be empty, then no wrapping can be performed. * This is considered a failure, unless "add_valid" is set. * * If any of the cut constraints of info->bmap turn out * to be redundant with respect to other constraints * then these will neither be wrapped nor added directly to the result. * The result may therefore not be correct. * Skip wrapping and mark "wraps" as failed in this case. */ static isl_stat add_selected_wraps_around_facet(struct isl_wraps *wraps, struct isl_coalesce_info *info, int k, isl_int *bound, __isl_keep isl_set *set, int add_valid) { … } /* Wrap the constraints of info->bmap that bound the facet defined * by inequality "k" around (the opposite of) this inequality to * include "set". "bound" may be used to store the negated inequality. * If any of the wrapped constraints turn out to be invalid for info->bmap * itself, then mark "wraps" as failed. */ static isl_stat add_wraps_around_facet(struct isl_wraps *wraps, struct isl_coalesce_info *info, int k, isl_int *bound, __isl_keep isl_set *set) { … } /* Wrap the (valid) constraints of info->bmap that bound the facet defined * by inequality "k" around (the opposite of) this inequality to * include "set" more tightly. * "bound" may be used to store the negated inequality. * Remove any wrapping constraints that turn out to be invalid * for info->bmap itself. */ static isl_stat add_valid_wraps_around_facet(struct isl_wraps *wraps, struct isl_coalesce_info *info, int k, isl_int *bound, __isl_keep isl_set *set) { … } /* Basic map "i" has an inequality (say "k") that is adjacent * to some inequality of basic map "j". All the other inequalities * are valid for "j". * Check if basic map "j" forms an extension of basic map "i". * * Note that this function is only called if some of the equalities or * inequalities of basic map "j" do cut basic map "i". The function is * correct even if there are no such cut constraints, but in that case * the additional checks performed by this function are overkill. * * First try and wrap the ridges of "k" around "j". * Note that those ridges are already valid for "j", * but the wrapped versions may wrap "j" more tightly, * increasing the chances of "j" being detected as an extension of "i" */ static enum isl_change is_adj_ineq_extension(int i, int j, struct isl_coalesce_info *info) { … } /* Both basic maps have at least one inequality with and adjacent * (but opposite) inequality in the other basic map. * Check that there are no cut constraints and that there is only * a single pair of adjacent inequalities. * If so, we can replace the pair by a single basic map described * by all but the pair of adjacent inequalities. * Any additional points introduced lie strictly between the two * adjacent hyperplanes and can therefore be integral. * * ____ _____ * / ||\ / \ * / || \ / \ * \ || \ => \ \ * \ || / \ / * \___||_/ \_____/ * * The test for a single pair of adjacent inequalities is important * for avoiding the combination of two basic maps like the following * * /| * / | * /__| * _____ * | | * | | * |___| * * If there are some cut constraints on one side, then we may * still be able to fuse the two basic maps, but we need to perform * some additional checks in is_adj_ineq_extension. */ static enum isl_change check_adj_ineq(int i, int j, struct isl_coalesce_info *info) { … } /* Given a basic set i with a constraint k that is adjacent to * basic set j, check if we can wrap * both the facet corresponding to k (if "wrap_facet" is set) and basic map j * (always) around their ridges to include the other set. * If so, replace the pair of basic sets by their union. * * All constraints of i (except k) are assumed to be valid or * cut constraints for j. * Wrapping the cut constraints to include basic map j may result * in constraints that are no longer valid of basic map i * we have to check that the resulting wrapping constraints are valid for i. * If "wrap_facet" is not set, then all constraints of i (except k) * are assumed to be valid for j. * ____ _____ * / | / \ * / || / | * \ || => \ | * \ || \ | * \___|| \____| * */ static enum isl_change can_wrap_in_facet(int i, int j, int k, struct isl_coalesce_info *info, int wrap_facet) { … } /* Given a cut constraint t(x) >= 0 of basic map i, stored in row "w" * of wrap.mat, replace it by its relaxed version t(x) + 1 >= 0, and * add wrapping constraints to wrap.mat for all constraints * of basic map j that bound the part of basic map j that sticks out * of the cut constraint. * "set_i" is the underlying set of basic map i. * If any wrapping fails, then wraps->mat.n_row is reset to zero. * * In particular, we first intersect basic map j with t(x) + 1 = 0. * If the result is empty, then t(x) >= 0 was actually a valid constraint * (with respect to the integer points), so we add t(x) >= 0 instead. * Otherwise, we wrap the constraints of basic map j that are not * redundant in this intersection and that are not already valid * for basic map i over basic map i. * Note that it is sufficient to wrap the constraints to include * basic map i, because we will only wrap the constraints that do * not include basic map i already. The wrapped constraint will * therefore be more relaxed compared to the original constraint. * Since the original constraint is valid for basic map j, so is * the wrapped constraint. */ static isl_stat wrap_in_facet(struct isl_wraps *wraps, int w, struct isl_coalesce_info *info_j, __isl_keep isl_set *set_i, struct isl_tab_undo *snap) { … } /* Given a pair of basic maps i and j such that j sticks out * of i at n cut constraints, each time by at most one, * try to compute wrapping constraints and replace the two * basic maps by a single basic map. * The other constraints of i are assumed to be valid for j. * "set_i" is the underlying set of basic map i. * "wraps" has been initialized to be of the right size. * * For each cut constraint t(x) >= 0 of i, we add the relaxed version * t(x) + 1 >= 0, along with wrapping constraints for all constraints * of basic map j that bound the part of basic map j that sticks out * of the cut constraint. * * If any wrapping fails, i.e., if we cannot wrap to touch * the union, then we give up. * Otherwise, the pair of basic maps is replaced by their union. */ static enum isl_change try_wrap_in_facets(int i, int j, struct isl_coalesce_info *info, struct isl_wraps *wraps, __isl_keep isl_set *set_i) { … } /* Given a pair of basic maps i and j such that j sticks out * of i at n cut constraints, each time by at most one, * try to compute wrapping constraints and replace the two * basic maps by a single basic map. * The other constraints of i are assumed to be valid for j. * * The core computation is performed by try_wrap_in_facets. * This function simply extracts an underlying set representation * of basic map i and initializes the data structure for keeping * track of wrapping constraints. */ static enum isl_change wrap_in_facets(int i, int j, int n, struct isl_coalesce_info *info) { … } /* Return the effect of inequality "ineq" on the tableau "tab", * after relaxing the constant term of "ineq" by one. */ static enum isl_ineq_type type_of_relaxed(struct isl_tab *tab, isl_int *ineq) { … } /* Given two basic sets i and j, * check if relaxing all the cut constraints of i by one turns * them into valid constraint for j and check if we can wrap in * the bits that are sticking out. * If so, replace the pair by their union. * * We first check if all relaxed cut inequalities of i are valid for j * and then try to wrap in the intersections of the relaxed cut inequalities * with j. * * During this wrapping, we consider the points of j that lie at a distance * of exactly 1 from i. In particular, we ignore the points that lie in * between this lower-dimensional space and the basic map i. * We can therefore only apply this to integer maps. * ____ _____ * / ___|_ / \ * / | | / | * \ | | => \ | * \|____| \ | * \___| \____/ * * _____ ______ * | ____|_ | \ * | | | | | * | | | => | | * |_| | | | * |_____| \______| * * _______ * | | * | |\ | * | | \ | * | | \ | * | | \| * | | \ * | |_____\ * | | * |_______| * * Wrapping can fail if the result of wrapping one of the facets * around its edges does not produce any new facet constraint. * In particular, this happens when we try to wrap in unbounded sets. * * _______________________________________________________________________ * | * | ___ * | | | * |_| |_________________________________________________________________ * |___| * * The following is not an acceptable result of coalescing the above two * sets as it includes extra integer points. * _______________________________________________________________________ * | * | * | * | * \______________________________________________________________________ */ static enum isl_change can_wrap_in_set(int i, int j, struct isl_coalesce_info *info) { … } /* Check if either i or j has only cut constraints that can * be used to wrap in (a facet of) the other basic set. * if so, replace the pair by their union. */ static enum isl_change check_wrap(int i, int j, struct isl_coalesce_info *info) { … } /* Check if all inequality constraints of "i" that cut "j" cease * to be cut constraints if they are relaxed by one. * If so, collect the cut constraints in "list". * The caller is responsible for allocating "list". */ static isl_bool all_cut_by_one(int i, int j, struct isl_coalesce_info *info, int *list) { … } /* Given two basic maps such that "j" has at least one equality constraint * that is adjacent to an inequality constraint of "i" and such that "i" has * exactly one inequality constraint that is adjacent to an equality * constraint of "j", check whether "i" can be extended to include "j" or * whether "j" can be wrapped into "i". * All remaining constraints of "i" and "j" are assumed to be valid * or cut constraints of the other basic map. * However, none of the equality constraints of "i" are cut constraints. * * If "i" has any "cut" inequality constraints, then check if relaxing * each of them by one is sufficient for them to become valid. * If so, check if the inequality constraint adjacent to an equality * constraint of "j" along with all these cut constraints * can be relaxed by one to contain exactly "j". * Otherwise, or if this fails, check if "j" can be wrapped into "i". */ static enum isl_change check_single_adj_eq(int i, int j, struct isl_coalesce_info *info) { … } /* At least one of the basic maps has an equality that is adjacent * to an inequality. Make sure that only one of the basic maps has * such an equality and that the other basic map has exactly one * inequality adjacent to an equality. * If the other basic map does not have such an inequality, then * check if all its constraints are either valid or cut constraints * and, if so, try wrapping in the first map into the second. * Otherwise, try to extend one basic map with the other or * wrap one basic map in the other. */ static enum isl_change check_adj_eq(int i, int j, struct isl_coalesce_info *info) { … } /* Disjunct "j" lies on a hyperplane that is adjacent to disjunct "i". * In particular, disjunct "i" has an inequality constraint that is adjacent * to a (combination of) equality constraint(s) of disjunct "j", * but disjunct "j" has no explicit equality constraint adjacent * to an inequality constraint of disjunct "i". * * Disjunct "i" is already known not to have any equality constraints * that are adjacent to an equality or inequality constraint. * Check that, other than the inequality constraint mentioned above, * all other constraints of disjunct "i" are valid for disjunct "j". * If so, try and wrap in disjunct "j". */ static enum isl_change check_ineq_adj_eq(int i, int j, struct isl_coalesce_info *info) { … } /* The two basic maps lie on adjacent hyperplanes. In particular, * basic map "i" has an equality that lies parallel to basic map "j". * Check if we can wrap the facets around the parallel hyperplanes * to include the other set. * * We perform basically the same operations as can_wrap_in_facet, * except that we don't need to select a facet of one of the sets. * _ * \\ \\ * \\ => \\ * \ \| * * If there is more than one equality of "i" adjacent to an equality of "j", * then the result will satisfy one or more equalities that are a linear * combination of these equalities. These will be encoded as pairs * of inequalities in the wrapping constraints and need to be made * explicit. */ static enum isl_change check_eq_adj_eq(int i, int j, struct isl_coalesce_info *info) { … } /* Initialize the "eq" and "ineq" fields of "info". */ static void init_status(struct isl_coalesce_info *info) { … } /* Set info->eq to the positions of the equalities of info->bmap * with respect to the basic map represented by "tab". * If info->eq has already been computed, then do not compute it again. */ static void set_eq_status_in(struct isl_coalesce_info *info, struct isl_tab *tab) { … } /* Set info->ineq to the positions of the inequalities of info->bmap * with respect to the basic map represented by "tab". * If info->ineq has already been computed, then do not compute it again. */ static void set_ineq_status_in(struct isl_coalesce_info *info, struct isl_tab *tab) { … } /* Free the memory allocated by the "eq" and "ineq" fields of "info". * This function assumes that init_status has been called on "info" first, * after which the "eq" and "ineq" fields may or may not have been * assigned a newly allocated array. */ static void clear_status(struct isl_coalesce_info *info) { … } /* Are all inequality constraints of the basic map represented by "info" * valid for the other basic map, except for a single constraint * that is adjacent to an inequality constraint of the other basic map? */ static int all_ineq_valid_or_single_adj_ineq(struct isl_coalesce_info *info) { … } /* Basic map "i" has one or more equality constraints that separate it * from basic map "j". Check if it happens to be an extension * of basic map "j". * In particular, check that all constraints of "j" are valid for "i", * except for one inequality constraint that is adjacent * to an inequality constraints of "i". * If so, check for "i" being an extension of "j" by calling * is_adj_ineq_extension. * * Clean up the memory allocated for keeping track of the status * of the constraints before returning. */ static enum isl_change separating_equality(int i, int j, struct isl_coalesce_info *info) { … } /* Check if the union of the given pair of basic maps * can be represented by a single basic map. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * The two basic maps are assumed to live in the same local space. * The "eq" and "ineq" fields of info[i] and info[j] are assumed * to have been initialized by the caller, either to NULL or * to valid information. * * We first check the effect of each constraint of one basic map * on the other basic map. * The constraint may be * redundant the constraint is redundant in its own * basic map and should be ignore and removed * in the end * valid all (integer) points of the other basic map * satisfy the constraint * separate no (integer) point of the other basic map * satisfies the constraint * cut some but not all points of the other basic map * satisfy the constraint * adj_eq the given constraint is adjacent (on the outside) * to an equality of the other basic map * adj_ineq the given constraint is adjacent (on the outside) * to an inequality of the other basic map * * We consider seven cases in which we can replace the pair by a single * basic map. We ignore all "redundant" constraints. * * 1. all constraints of one basic map are valid * => the other basic map is a subset and can be removed * * 2. all constraints of both basic maps are either "valid" or "cut" * and the facets corresponding to the "cut" constraints * of one of the basic maps lies entirely inside the other basic map * => the pair can be replaced by a basic map consisting * of the valid constraints in both basic maps * * 3. there is a single pair of adjacent inequalities * (all other constraints are "valid") * => the pair can be replaced by a basic map consisting * of the valid constraints in both basic maps * * 4. one basic map has a single adjacent inequality, while the other * constraints are "valid". The other basic map has some * "cut" constraints, but replacing the adjacent inequality by * its opposite and adding the valid constraints of the other * basic map results in a subset of the other basic map * => the pair can be replaced by a basic map consisting * of the valid constraints in both basic maps * * 5. there is a single adjacent pair of an inequality and an equality, * the other constraints of the basic map containing the inequality are * "valid". Moreover, if the inequality the basic map is relaxed * and then turned into an equality, then resulting facet lies * entirely inside the other basic map * => the pair can be replaced by the basic map containing * the inequality, with the inequality relaxed. * * 6. there is a single inequality adjacent to an equality, * the other constraints of the basic map containing the inequality are * "valid". Moreover, the facets corresponding to both * the inequality and the equality can be wrapped around their * ridges to include the other basic map * => the pair can be replaced by a basic map consisting * of the valid constraints in both basic maps together * with all wrapping constraints * * 7. one of the basic maps extends beyond the other by at most one. * Moreover, the facets corresponding to the cut constraints and * the pieces of the other basic map at offset one from these cut * constraints can be wrapped around their ridges to include * the union of the two basic maps * => the pair can be replaced by a basic map consisting * of the valid constraints in both basic maps together * with all wrapping constraints * * 8. the two basic maps live in adjacent hyperplanes. In principle * such sets can always be combined through wrapping, but we impose * that there is only one such pair, to avoid overeager coalescing. * * Throughout the computation, we maintain a collection of tableaus * corresponding to the basic maps. When the basic maps are dropped * or combined, the tableaus are modified accordingly. */ static enum isl_change coalesce_local_pair_reuse(int i, int j, struct isl_coalesce_info *info) { … } /* Check if the union of the given pair of basic maps * can be represented by a single basic map. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * The two basic maps are assumed to live in the same local space. */ static enum isl_change coalesce_local_pair(int i, int j, struct isl_coalesce_info *info) { … } /* Shift the integer division at position "div" of the basic map * represented by "info" by "shift". * * That is, if the integer division has the form * * floor(f(x)/d) * * then replace it by * * floor((f(x) + shift * d)/d) - shift */ static isl_stat shift_div(struct isl_coalesce_info *info, int div, isl_int shift) { … } /* If the integer division at position "div" is defined by an equality, * i.e., a stride constraint, then change the integer division expression * to have a constant term equal to zero. * * Let the equality constraint be * * c + f + m a = 0 * * The integer division expression is then typically of the form * * a = floor((-f - c')/m) * * The integer division is first shifted by t = floor(c/m), * turning the equality constraint into * * c - m floor(c/m) + f + m a' = 0 * * i.e., * * (c mod m) + f + m a' = 0 * * That is, * * a' = (-f - (c mod m))/m = floor((-f)/m) * * because a' is an integer and 0 <= (c mod m) < m. * The constant term of a' can therefore be zeroed out, * but only if the integer division expression is of the expected form. */ static isl_stat normalize_stride_div(struct isl_coalesce_info *info, int div) { … } /* The basic maps represented by "info1" and "info2" are known * to have the same number of integer divisions. * Check if pairs of integer divisions are equal to each other * despite the fact that they differ by a rational constant. * * In particular, look for any pair of integer divisions that * only differ in their constant terms. * If either of these integer divisions is defined * by stride constraints, then modify it to have a zero constant term. * If both are defined by stride constraints then in the end they will have * the same (zero) constant term. */ static isl_stat harmonize_stride_divs(struct isl_coalesce_info *info1, struct isl_coalesce_info *info2) { … } /* If "shift" is an integer constant, then shift the integer division * at position "div" of the basic map represented by "info" by "shift". * If "shift" is not an integer constant, then do nothing. * If "shift" is equal to zero, then no shift needs to be performed either. * * That is, if the integer division has the form * * floor(f(x)/d) * * then replace it by * * floor((f(x) + shift * d)/d) - shift */ static isl_stat shift_if_cst_int(struct isl_coalesce_info *info, int div, __isl_keep isl_aff *shift) { … } /* Check if some of the divs in the basic map represented by "info1" * are shifts of the corresponding divs in the basic map represented * by "info2", taking into account the equality constraints "eq1" of "info1" * and "eq2" of "info2". If so, align them with those of "info2". * "info1" and "info2" are assumed to have the same number * of integer divisions. * * An integer division is considered to be a shift of another integer * division if, after simplification with respect to the equality * constraints of the other basic map, one is equal to the other * plus a constant. * * In particular, for each pair of integer divisions, if both are known, * have the same denominator and are not already equal to each other, * simplify each with respect to the equality constraints * of the other basic map. If the difference is an integer constant, * then move this difference outside. * That is, if, after simplification, one integer division is of the form * * floor((f(x) + c_1)/d) * * while the other is of the form * * floor((f(x) + c_2)/d) * * and n = (c_2 - c_1)/d is an integer, then replace the first * integer division by * * floor((f_1(x) + c_1 + n * d)/d) - n, * * where floor((f_1(x) + c_1 + n * d)/d) = floor((f2(x) + c_2)/d) * after simplification with respect to the equality constraints. */ static isl_stat harmonize_divs_with_hulls(struct isl_coalesce_info *info1, struct isl_coalesce_info *info2, __isl_keep isl_basic_set *eq1, __isl_keep isl_basic_set *eq2) { … } /* Check if some of the divs in the basic map represented by "info1" * are shifts of the corresponding divs in the basic map represented * by "info2". If so, align them with those of "info2". * Only do this if "info1" and "info2" have the same number * of integer divisions. * * An integer division is considered to be a shift of another integer * division if, after simplification with respect to the equality * constraints of the other basic map, one is equal to the other * plus a constant. * * First check if pairs of integer divisions are equal to each other * despite the fact that they differ by a rational constant. * If so, try and arrange for them to have the same constant term. * * Then, extract the equality constraints and continue with * harmonize_divs_with_hulls. * * If the equality constraints of both basic maps are the same, * then there is no need to perform any shifting since * the coefficients of the integer divisions should have been * reduced in the same way. */ static isl_stat harmonize_divs(struct isl_coalesce_info *info1, struct isl_coalesce_info *info2) { … } /* Do the two basic maps live in the same local space, i.e., * do they have the same (known) divs? * If either basic map has any unknown divs, then we can only assume * that they do not live in the same local space. */ static isl_bool same_divs(__isl_keep isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2) { … } /* Assuming that "tab" contains the equality constraints and * the initial inequality constraints of "bmap", copy the remaining * inequality constraints of "bmap" to "Tab". */ static isl_stat copy_ineq(struct isl_tab *tab, __isl_keep isl_basic_map *bmap) { … } /* Description of an integer division that is added * during an expansion. * "pos" is the position of the corresponding variable. * "cst" indicates whether this integer division has a fixed value. * "val" contains the fixed value, if the value is fixed. */ struct isl_expanded { … }; /* For each of the "n" integer division variables "expanded", * if the variable has a fixed value, then add two inequality * constraints expressing the fixed value. * Otherwise, add the corresponding div constraints. * The caller is responsible for removing the div constraints * that it added for all these "n" integer divisions. * * The div constraints and the pair of inequality constraints * forcing the fixed value cannot both be added for a given variable * as the combination may render some of the original constraints redundant. * These would then be ignored during the coalescing detection, * while they could remain in the fused result. * * The two added inequality constraints are * * -a + v >= 0 * a - v >= 0 * * with "a" the variable and "v" its fixed value. * The facet corresponding to one of these two constraints is selected * in the tableau to ensure that the pair of inequality constraints * is treated as an equality constraint. * * The information in info->ineq is thrown away because it was * computed in terms of div constraints, while some of those * have now been replaced by these pairs of inequality constraints. */ static isl_stat fix_constant_divs(struct isl_coalesce_info *info, int n, struct isl_expanded *expanded) { … } /* Insert the "n" integer division variables "expanded" * into info->tab and info->bmap and * update info->ineq with respect to the redundant constraints * in the resulting tableau. * "bmap" contains the result of this insertion in info->bmap, * while info->bmap is the original version * of "bmap", i.e., the one that corresponds to the current * state of info->tab. The number of constraints in info->bmap * is assumed to be the same as the number of constraints * in info->tab. This is required to be able to detect * the extra constraints in "bmap". * * In particular, introduce extra variables corresponding * to the extra integer divisions and add the div constraints * that were added to "bmap" after info->tab was created * from info->bmap. * Furthermore, check if these extra integer divisions happen * to attain a fixed integer value in info->tab. * If so, replace the corresponding div constraints by pairs * of inequality constraints that fix these * integer divisions to their single integer values. * Replace info->bmap by "bmap" to match the changes to info->tab. * info->ineq was computed without a tableau and therefore * does not take into account the redundant constraints * in the tableau. Mark them here. * There is no need to check the newly added div constraints * since they cannot be redundant. * The redundancy check is not performed when constants have been discovered * since info->ineq is completely thrown away in this case. */ static isl_stat tab_insert_divs(struct isl_coalesce_info *info, int n, struct isl_expanded *expanded, __isl_take isl_basic_map *bmap) { … } /* Expand info->tab and info->bmap in the same way "bmap" was expanded * in isl_basic_map_expand_divs using the expansion "exp" and * update info->ineq with respect to the redundant constraints * in the resulting tableau. info->bmap is the original version * of "bmap", i.e., the one that corresponds to the current * state of info->tab. The number of constraints in info->bmap * is assumed to be the same as the number of constraints * in info->tab. This is required to be able to detect * the extra constraints in "bmap". * * Extract the positions where extra local variables are introduced * from "exp" and call tab_insert_divs. */ static isl_stat expand_tab(struct isl_coalesce_info *info, int *exp, __isl_take isl_basic_map *bmap) { … } /* Check if the union of the basic maps represented by info[i] and info[j] * can be represented by a single basic map, * after expanding the divs of info[i] to match those of info[j]. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * * The caller has already checked for info[j] being a subset of info[i]. * If some of the divs of info[j] are unknown, then the expanded info[i] * will not have the corresponding div constraints. The other patterns * therefore cannot apply. Skip the computation in this case. * * The expansion is performed using the divs "div" and expansion "exp" * computed by the caller. * info[i].bmap has already been expanded and the result is passed in * as "bmap". * The "eq" and "ineq" fields of info[i] reflect the status of * the constraints of the expanded "bmap" with respect to info[j].tab. * However, inequality constraints that are redundant in info[i].tab * have not yet been marked as such because no tableau was available. * * Replace info[i].bmap by "bmap" and expand info[i].tab as well, * updating info[i].ineq with respect to the redundant constraints. * Then try and coalesce the expanded info[i] with info[j], * reusing the information in info[i].eq and info[i].ineq. * If this does not result in any coalescing or if it results in info[j] * getting dropped (which should not happen in practice, since the case * of info[j] being a subset of info[i] has already been checked by * the caller), then revert info[i] to its original state. */ static enum isl_change coalesce_expand_tab_divs(__isl_take isl_basic_map *bmap, int i, int j, struct isl_coalesce_info *info, __isl_keep isl_mat *div, int *exp) { … } /* Check if the union of "bmap" and the basic map represented by info[j] * can be represented by a single basic map, * after expanding the divs of "bmap" to match those of info[j]. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * * In particular, check if the expanded "bmap" contains the basic map * represented by the tableau info[j].tab. * The expansion is performed using the divs "div" and expansion "exp" * computed by the caller. * Then we check if all constraints of the expanded "bmap" are valid for * info[j].tab. * * If "i" is not equal to -1, then "bmap" is equal to info[i].bmap. * In this case, the positions of the constraints of info[i].bmap * with respect to the basic map represented by info[j] are stored * in info[i]. * * If the expanded "bmap" does not contain the basic map * represented by the tableau info[j].tab and if "i" is not -1, * i.e., if the original "bmap" is info[i].bmap, then expand info[i].tab * as well and check if that results in coalescing. */ static enum isl_change coalesce_with_expanded_divs( __isl_keep isl_basic_map *bmap, int i, int j, struct isl_coalesce_info *info, __isl_keep isl_mat *div, int *exp) { … } /* Check if the union of "bmap_i" and the basic map represented by info[j] * can be represented by a single basic map, * after aligning the divs of "bmap_i" to match those of info[j]. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * * In particular, check if "bmap_i" contains the basic map represented by * info[j] after aligning the divs of "bmap_i" to those of info[j]. * Note that this can only succeed if the number of divs of "bmap_i" * is smaller than (or equal to) the number of divs of info[j]. * * We first check if the divs of "bmap_i" are all known and form a subset * of those of info[j].bmap. If so, we pass control over to * coalesce_with_expanded_divs. * * If "i" is not equal to -1, then "bmap" is equal to info[i].bmap. */ static enum isl_change coalesce_after_aligning_divs( __isl_keep isl_basic_map *bmap_i, int i, int j, struct isl_coalesce_info *info) { … } /* Check if basic map "j" is a subset of basic map "i" after * exploiting the extra equalities of "j" to simplify the divs of "i". * If so, remove basic map "j" and return isl_change_drop_second. * * If "j" does not have any equalities or if they are the same * as those of "i", then we cannot exploit them to simplify the divs. * Similarly, if there are no divs in "i", then they cannot be simplified. * If, on the other hand, the affine hulls of "i" and "j" do not intersect, * then "j" cannot be a subset of "i". * * Otherwise, we intersect "i" with the affine hull of "j" and then * check if "j" is a subset of the result after aligning the divs. * If so, then "j" is definitely a subset of "i" and can be removed. * Note that if after intersection with the affine hull of "j". * "i" still has more divs than "j", then there is no way we can * align the divs of "i" to those of "j". */ static enum isl_change coalesce_subset_with_equalities(int i, int j, struct isl_coalesce_info *info) { … } /* Check if the union of the basic maps represented by info[i] and info[j] * can be represented by a single basic map, by aligning or equating * their integer divisions. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * * Note that we only perform any test if the number of divs is different * in the two basic maps. In case the number of divs is the same, * we have already established that the divs are different * in the two basic maps. * In particular, if the number of divs of basic map i is smaller than * the number of divs of basic map j, then we check if j is a subset of i * and vice versa. */ static enum isl_change coalesce_divs(int i, int j, struct isl_coalesce_info *info) { … } /* Does "bmap" involve any divs that themselves refer to divs? */ static isl_bool has_nested_div(__isl_keep isl_basic_map *bmap) { … } /* Return a list of affine expressions, one for each integer division * in "bmap_i". For each integer division that also appears in "bmap_j", * the affine expression is set to NaN. The number of NaNs in the list * is equal to the number of integer divisions in "bmap_j". * For the other integer divisions of "bmap_i", the corresponding * element in the list is a purely affine expression equal to the integer * division in "hull". * If no such list can be constructed, then the number of elements * in the returned list is smaller than the number of integer divisions * in "bmap_i". * The integer division of "bmap_i" and "bmap_j" are assumed to be known and * not contain any nested divs. */ static __isl_give isl_aff_list *set_up_substitutions( __isl_keep isl_basic_map *bmap_i, __isl_keep isl_basic_map *bmap_j, __isl_take isl_basic_map *hull) { … } /* Add variables to info->bmap and info->tab corresponding to the elements * in "list" that are not set to NaN. * "extra_var" is the number of these elements. * "dim" is the offset in the variables of "tab" where we should * start considering the elements in "list". * When this function returns, the total number of variables in "tab" * is equal to "dim" plus the number of elements in "list". * * The newly added existentially quantified variables are not given * an explicit representation because the corresponding div constraints * do not appear in info->bmap. These constraints are not added * to info->bmap because for internal consistency, they would need to * be added to info->tab as well, where they could combine with the equality * that is added later to result in constraints that do not hold * in the original input. */ static isl_stat add_sub_vars(struct isl_coalesce_info *info, __isl_keep isl_aff_list *list, int dim, int extra_var) { … } /* For each element in "list" that is not set to NaN, fix the corresponding * variable in "tab" to the purely affine expression defined by the element. * "dim" is the offset in the variables of "tab" where we should * start considering the elements in "list". * * This function assumes that a sufficient number of rows and * elements in the constraint array are available in the tableau. */ static isl_stat add_sub_equalities(struct isl_tab *tab, __isl_keep isl_aff_list *list, int dim) { … } /* Add variables to info->tab and info->bmap corresponding to the elements * in "list" that are not set to NaN. The value of the added variable * in info->tab is fixed to the purely affine expression defined by the element. * "dim" is the offset in the variables of info->tab where we should * start considering the elements in "list". * When this function returns, the total number of variables in info->tab * is equal to "dim" plus the number of elements in "list". */ static isl_stat add_subs(struct isl_coalesce_info *info, __isl_keep isl_aff_list *list, int dim) { … } /* Coalesce basic map "j" into basic map "i" after adding the extra integer * divisions in "i" but not in "j" to basic map "j", with values * specified by "list". The total number of elements in "list" * is equal to the number of integer divisions in "i", while the number * of NaN elements in the list is equal to the number of integer divisions * in "j". * * If no coalescing can be performed, then we need to revert basic map "j" * to its original state. We do the same if basic map "i" gets dropped * during the coalescing, even though this should not happen in practice * since we have already checked for "j" being a subset of "i" * before we reach this stage. */ static enum isl_change coalesce_with_subs(int i, int j, struct isl_coalesce_info *info, __isl_keep isl_aff_list *list) { … } /* Check if we can coalesce basic map "j" into basic map "i" after copying * those extra integer divisions in "i" that can be simplified away * using the extra equalities in "j". * All divs are assumed to be known and not contain any nested divs. * * We first check if there are any extra equalities in "j" that we * can exploit. Then we check if every integer division in "i" * either already appears in "j" or can be simplified using the * extra equalities to a purely affine expression. * If these tests succeed, then we try to coalesce the two basic maps * by introducing extra dimensions in "j" corresponding to * the extra integer divisions "i" fixed to the corresponding * purely affine expression. */ static enum isl_change check_coalesce_into_eq(int i, int j, struct isl_coalesce_info *info) { … } /* Check if we can coalesce basic maps "i" and "j" after copying * those extra integer divisions in one of the basic maps that can * be simplified away using the extra equalities in the other basic map. * We require all divs to be known in both basic maps. * Furthermore, to simplify the comparison of div expressions, * we do not allow any nested integer divisions. */ static enum isl_change check_coalesce_eq(int i, int j, struct isl_coalesce_info *info) { … } /* Check if the union of the given pair of basic maps * can be represented by a single basic map. * If so, replace the pair by the single basic map and return * isl_change_drop_first, isl_change_drop_second or isl_change_fuse. * Otherwise, return isl_change_none. * * We first check if the two basic maps live in the same local space, * after aligning the divs that differ by only an integer constant. * If so, we do the complete check. Otherwise, we check if they have * the same number of integer divisions and can be coalesced, if one is * an obvious subset of the other or if the extra integer divisions * of one basic map can be simplified away using the extra equalities * of the other basic map. * * Note that trying to coalesce pairs of disjuncts with the same * number, but different local variables may drop the explicit * representation of some of these local variables. * This operation is therefore not performed when * the "coalesce_preserve_locals" option is set. */ static enum isl_change coalesce_pair(int i, int j, struct isl_coalesce_info *info) { … } /* Return the maximum of "a" and "b". */ static int isl_max(int a, int b) { … } /* Pairwise coalesce the basic maps in the range [start1, end1[ of "info" * with those in the range [start2, end2[, skipping basic maps * that have been removed (either before or within this function). * * For each basic map i in the first range, we check if it can be coalesced * with respect to any previously considered basic map j in the second range. * If i gets dropped (because it was a subset of some j), then * we can move on to the next basic map. * If j gets dropped, we need to continue checking against the other * previously considered basic maps. * If the two basic maps got fused, then we recheck the fused basic map * against the previously considered basic maps, starting at i + 1 * (even if start2 is greater than i + 1). */ static int coalesce_range(isl_ctx *ctx, struct isl_coalesce_info *info, int start1, int end1, int start2, int end2) { … } /* Pairwise coalesce the basic maps described by the "n" elements of "info". * * We consider groups of basic maps that live in the same apparent * affine hull and we first coalesce within such a group before we * coalesce the elements in the group with elements of previously * considered groups. If a fuse happens during the second phase, * then we also reconsider the elements within the group. */ static int coalesce(isl_ctx *ctx, int n, struct isl_coalesce_info *info) { … } /* Update the basic maps in "map" based on the information in "info". * In particular, remove the basic maps that have been marked removed and * update the others based on the information in the corresponding tableau. * Since we detected implicit equalities without calling * isl_basic_map_gauss, we need to do it now. * Also call isl_basic_map_simplify if we may have lost the definition * of one or more integer divisions. * If a basic map is still equal to the one from which the corresponding "info" * entry was created, then redundant constraint and * implicit equality constraint detection have been performed * on the corresponding tableau and the basic map can be marked as such. */ static __isl_give isl_map *update_basic_maps(__isl_take isl_map *map, int n, struct isl_coalesce_info *info) { … } /* For each pair of basic maps in the map, check if the union of the two * can be represented by a single basic map. * If so, replace the pair by the single basic map and start over. * * We factor out any (hidden) common factor from the constraint * coefficients to improve the detection of adjacent constraints. * Note that this function does not call isl_basic_map_gauss, * but it does make sure that only a single copy of the basic map * is affected. This means that isl_basic_map_gauss may have * to be called at the end of the computation (in update_basic_maps) * on this single copy to ensure that * the basic maps are not left in an unexpected state. * * Since we are constructing the tableaus of the basic maps anyway, * we exploit them to detect implicit equalities and redundant constraints. * This also helps the coalescing as it can ignore the redundant constraints. * In order to avoid confusion, we make all implicit equalities explicit * in the basic maps. If the basic map only has a single reference * (this happens in particular if it was modified by * isl_basic_map_reduce_coefficients), then isl_basic_map_gauss * does not get called on the result. The call to * isl_basic_map_gauss in update_basic_maps resolves this as well. * For each basic map, we also compute the hash of the apparent affine hull * for use in coalesce. */ __isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map) { … } /* For each pair of basic sets in the set, check if the union of the two * can be represented by a single basic set. * If so, replace the pair by the single basic set and start over. */ __isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set) { … }
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