From 3fe110a4d75e491805b34f25bb333dc62b2aef4f Mon Sep 17 00:00:00 2001 From: Sven Verdoolaege Date: Wed, 4 Mar 2009 18:12:30 +0100 Subject: [PATCH] add internal representation of LP tableaus --- Makefile.am | 2 + isl_tab.c | 1760 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ isl_tab.h | 145 +++++ 3 files changed, 1907 insertions(+) create mode 100644 isl_tab.c create mode 100644 isl_tab.h diff --git a/Makefile.am b/Makefile.am index 3dc4fd4..df72ac2 100644 --- a/Makefile.am +++ b/Makefile.am @@ -70,6 +70,8 @@ libisl_la_SOURCES = \ isl_stream.c \ isl_stream.h \ isl_seq.c \ + isl_tab.c \ + isl_tab.h \ isl_vec.c \ isl_vec.h EXTRA_libisl_la_SOURCES = \ diff --git a/isl_tab.c b/isl_tab.c new file mode 100644 index 0000000..3fe08eb --- /dev/null +++ b/isl_tab.c @@ -0,0 +1,1760 @@ +#include "isl_map_private.h" +#include "isl_tab.h" + +/* + * The implementation of tableaus in this file was inspired by Section 8 + * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem + * prover for program checking". + */ + +struct isl_tab *isl_tab_alloc(struct isl_ctx *ctx, + unsigned n_row, unsigned n_var) +{ + int i; + struct isl_tab *tab; + + tab = isl_calloc_type(ctx, struct isl_tab); + if (!tab) + return NULL; + tab->mat = isl_mat_alloc(ctx, n_row, 2 + n_var); + if (!tab->mat) + goto error; + tab->var = isl_alloc_array(ctx, struct isl_tab_var, n_var); + if (!tab->var) + goto error; + tab->con = isl_alloc_array(ctx, struct isl_tab_var, n_row); + if (!tab->con) + goto error; + tab->col_var = isl_alloc_array(ctx, int, n_var); + if (!tab->col_var) + goto error; + tab->row_var = isl_alloc_array(ctx, int, n_row); + if (!tab->row_var) + goto error; + for (i = 0; i < n_var; ++i) { + tab->var[i].index = i; + tab->var[i].is_row = 0; + tab->var[i].is_nonneg = 0; + tab->var[i].is_zero = 0; + tab->var[i].is_redundant = 0; + tab->var[i].frozen = 0; + tab->col_var[i] = i; + } + tab->n_row = 0; + tab->n_con = 0; + tab->max_con = n_row; + tab->n_col = n_var; + tab->n_var = n_var; + tab->n_dead = 0; + tab->n_redundant = 0; + tab->need_undo = 0; + tab->rational = 0; + tab->empty = 0; + tab->killed_col = 0; + tab->bottom.type = isl_tab_undo_bottom; + tab->bottom.next = NULL; + tab->top = &tab->bottom; + return tab; +error: + isl_tab_free(ctx, tab); + return NULL; +} + +static int extend_cons(struct isl_ctx *ctx, struct isl_tab *tab, unsigned n_new) +{ + if (tab->max_con < tab->n_con + n_new) { + struct isl_tab_var *con; + + con = isl_realloc_array(ctx, tab->con, + struct isl_tab_var, tab->max_con + n_new); + if (!con) + return -1; + tab->con = con; + tab->max_con += n_new; + } + if (tab->mat->n_row < tab->n_row + n_new) { + int *row_var; + + tab->mat = isl_mat_extend(ctx, tab->mat, + tab->n_row + n_new, tab->n_col); + if (!tab->mat) + return -1; + row_var = isl_realloc_array(ctx, tab->row_var, + int, tab->mat->n_row); + if (!row_var) + return -1; + tab->row_var = row_var; + } + return 0; +} + +struct isl_tab *isl_tab_extend(struct isl_ctx *ctx, struct isl_tab *tab, + unsigned n_new) +{ + if (extend_cons(ctx, tab, n_new) >= 0) + return tab; + + isl_tab_free(ctx, tab); + return NULL; +} + +static void free_undo(struct isl_ctx *ctx, struct isl_tab *tab) +{ + struct isl_tab_undo *undo, *next; + + for (undo = tab->top; undo && undo != &tab->bottom; undo = next) { + next = undo->next; + free(undo); + } + tab->top = undo; +} + +void isl_tab_free(struct isl_ctx *ctx, struct isl_tab *tab) +{ + if (!tab) + return; + free_undo(ctx, tab); + isl_mat_free(ctx, tab->mat); + free(tab->var); + free(tab->con); + free(tab->row_var); + free(tab->col_var); + free(tab); +} + +static struct isl_tab_var *var_from_index(struct isl_ctx *ctx, + struct isl_tab *tab, int i) +{ + if (i >= 0) + return &tab->var[i]; + else + return &tab->con[~i]; +} + +static struct isl_tab_var *var_from_row(struct isl_ctx *ctx, + struct isl_tab *tab, int i) +{ + return var_from_index(ctx, tab, tab->row_var[i]); +} + +static struct isl_tab_var *var_from_col(struct isl_ctx *ctx, + struct isl_tab *tab, int i) +{ + return var_from_index(ctx, tab, tab->col_var[i]); +} + +/* Check if there are any upper bounds on column variable "var", + * i.e., non-negative rows where var appears with a negative coefficient. + * Return 1 if there are no such bounds. + */ +static int max_is_manifestly_unbounded(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int i; + + if (var->is_row) + return 0; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + if (!isl_int_is_neg(tab->mat->row[i][2 + var->index])) + continue; + if (var_from_row(ctx, tab, i)->is_nonneg) + return 0; + } + return 1; +} + +/* Check if there are any lower bounds on column variable "var", + * i.e., non-negative rows where var appears with a positive coefficient. + * Return 1 if there are no such bounds. + */ +static int min_is_manifestly_unbounded(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int i; + + if (var->is_row) + return 0; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + if (!isl_int_is_pos(tab->mat->row[i][2 + var->index])) + continue; + if (var_from_row(ctx, tab, i)->is_nonneg) + return 0; + } + return 1; +} + +/* Given the index of a column "c", return the index of a row + * that can be used to pivot the column in, with either an increase + * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable. + * If "var" is not NULL, then the row returned will be different from + * the one associated with "var". + * + * Each row in the tableau is of the form + * + * x_r = a_r0 + \sum_i a_ri x_i + * + * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn" + * impose any limit on the increase or decrease in the value of x_c + * and this bound is equal to a_r0 / |a_rc|. We are therefore looking + * for the row with the smallest (most stringent) such bound. + * Note that the common denominator of each row drops out of the fraction. + * To check if row j has a smaller bound than row r, i.e., + * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|, + * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0, + * where -sign(a_jc) is equal to "sgn". + */ +static int pivot_row(struct isl_ctx *ctx, struct isl_tab *tab, + struct isl_tab_var *var, int sgn, int c) +{ + int j, r, tsgn; + isl_int t; + + isl_int_init(t); + r = -1; + for (j = tab->n_redundant; j < tab->n_row; ++j) { + if (var && j == var->index) + continue; + if (!var_from_row(ctx, tab, j)->is_nonneg) + continue; + if (sgn * isl_int_sgn(tab->mat->row[j][2 + c]) >= 0) + continue; + if (r < 0) { + r = j; + continue; + } + isl_int_mul(t, tab->mat->row[r][1], tab->mat->row[j][2 + c]); + isl_int_submul(t, tab->mat->row[j][1], tab->mat->row[r][2 + c]); + tsgn = sgn * isl_int_sgn(t); + if (tsgn < 0 || (tsgn == 0 && + tab->row_var[j] < tab->row_var[r])) + r = j; + } + isl_int_clear(t); + return r; +} + +/* Find a pivot (row and col) that will increase (sgn > 0) or decrease + * (sgn < 0) the value of row variable var. + * As the given row in the tableau is of the form + * + * x_r = a_r0 + \sum_i a_ri x_i + * + * we need to find a column such that the sign of a_ri is equal to "sgn" + * (such that an increase in x_i will have the desired effect) or a + * column with a variable that may attain negative values. + * If a_ri is positive, then we need to move x_i in the same direction + * to obtain the desired effect. Otherwise, x_i has to move in the + * opposite direction. + */ +static void find_pivot(struct isl_ctx *ctx, struct isl_tab *tab, + struct isl_tab_var *var, int sgn, int *row, int *col) +{ + int j, r, c; + isl_int *tr; + + *row = *col = -1; + + isl_assert(ctx, var->is_row, return); + tr = tab->mat->row[var->index]; + + c = -1; + for (j = tab->n_dead; j < tab->n_col; ++j) { + if (isl_int_is_zero(tr[2 + j])) + continue; + if (isl_int_sgn(tr[2 + j]) != sgn && + var_from_col(ctx, tab, j)->is_nonneg) + continue; + if (c < 0 || tab->col_var[j] < tab->col_var[c]) + c = j; + } + if (c < 0) + return; + + sgn *= isl_int_sgn(tr[2 + c]); + r = pivot_row(ctx, tab, var, sgn, c); + *row = r < 0 ? var->index : r; + *col = c; +} + +/* Return 1 if row "row" represents an obviously redundant inequality. + * This means + * - it represents an inequality or a variable + * - that is the sum of a non-negative sample value and a positive + * combination of zero or more non-negative variables. + */ +static int is_redundant(struct isl_ctx *ctx, struct isl_tab *tab, int row) +{ + int i; + + if (tab->row_var[row] < 0 && !var_from_row(ctx, tab, row)->is_nonneg) + return 0; + + if (isl_int_is_neg(tab->mat->row[row][1])) + return 0; + + for (i = tab->n_dead; i < tab->n_col; ++i) { + if (isl_int_is_zero(tab->mat->row[row][2 + i])) + continue; + if (isl_int_is_neg(tab->mat->row[row][2 + i])) + return 0; + if (!var_from_col(ctx, tab, i)->is_nonneg) + return 0; + } + return 1; +} + +static void swap_rows(struct isl_ctx *ctx, + struct isl_tab *tab, int row1, int row2) +{ + int t; + t = tab->row_var[row1]; + tab->row_var[row1] = tab->row_var[row2]; + tab->row_var[row2] = t; + var_from_row(ctx, tab, row1)->index = row1; + var_from_row(ctx, tab, row2)->index = row2; + tab->mat = isl_mat_swap_rows(ctx, tab->mat, row1, row2); +} + +static void push(struct isl_ctx *ctx, struct isl_tab *tab, + enum isl_tab_undo_type type, struct isl_tab_var *var) +{ + struct isl_tab_undo *undo; + + if (!tab->need_undo) + return; + + undo = isl_alloc_type(ctx, struct isl_tab_undo); + if (!undo) { + free_undo(ctx, tab); + tab->top = NULL; + return; + } + undo->type = type; + undo->var = var; + undo->next = tab->top; + tab->top = undo; +} + +/* Mark row with index "row" as being redundant. + * If we may need to undo the operation or if the row represents + * a variable of the original problem, the row is kept, + * but no longer considered when looking for a pivot row. + * Otherwise, the row is simply removed. + * + * The row may be interchanged with some other row. If it + * is interchanged with a later row, return 1. Otherwise return 0. + * If the rows are checked in order in the calling function, + * then a return value of 1 means that the row with the given + * row number may now contain a different row that hasn't been checked yet. + */ +static int mark_redundant(struct isl_ctx *ctx, + struct isl_tab *tab, int row) +{ + struct isl_tab_var *var = var_from_row(ctx, tab, row); + var->is_redundant = 1; + isl_assert(ctx, row >= tab->n_redundant, return); + if (tab->need_undo || tab->row_var[row] >= 0) { + if (tab->row_var[row] >= 0) { + var->is_nonneg = 1; + push(ctx, tab, isl_tab_undo_nonneg, var); + } + if (row != tab->n_redundant) + swap_rows(ctx, tab, row, tab->n_redundant); + push(ctx, tab, isl_tab_undo_redundant, var); + tab->n_redundant++; + return 0; + } else { + if (row != tab->n_row - 1) + swap_rows(ctx, tab, row, tab->n_row - 1); + var_from_row(ctx, tab, tab->n_row - 1)->index = -1; + tab->n_row--; + return 1; + } +} + +static void mark_empty(struct isl_ctx *ctx, struct isl_tab *tab) +{ + if (!tab->empty && tab->need_undo) + push(ctx, tab, isl_tab_undo_empty, NULL); + tab->empty = 1; +} + +/* Given a row number "row" and a column number "col", pivot the tableau + * such that the associated variable are interchanged. + * The given row in the tableau expresses + * + * x_r = a_r0 + \sum_i a_ri x_i + * + * or + * + * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc + * + * Substituting this equality into the other rows + * + * x_j = a_j0 + \sum_i a_ji x_i + * + * with a_jc \ne 0, we obtain + * + * x_j = a_jc/a_rc x_r + a_j0 - a_jc a_r0/a_rc + sum a_ji - a_jc a_ri/a_rc + * + * The tableau + * + * n_rc/d_r n_ri/d_r + * n_jc/d_j n_ji/d_j + * + * where i is any other column and j is any other row, + * is therefore transformed into + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j) + * + * The transformation is performed along the following steps + * + * d_r/n_rc n_ri/n_rc + * n_jc/d_j n_ji/d_j + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * n_jc/d_j n_ji/d_j + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j) + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j) + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j) + * + * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc| + * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j) + * + */ +static void pivot(struct isl_ctx *ctx, + struct isl_tab *tab, int row, int col) +{ + int i, j; + int sgn; + int t; + struct isl_mat *mat = tab->mat; + struct isl_tab_var *var; + + isl_int_swap(mat->row[row][0], mat->row[row][2 + col]); + sgn = isl_int_sgn(mat->row[row][0]); + if (sgn < 0) { + isl_int_neg(mat->row[row][0], mat->row[row][0]); + isl_int_neg(mat->row[row][2 + col], mat->row[row][2 + col]); + } else + for (j = 0; j < 1 + tab->n_col; ++j) { + if (j == 1 + col) + continue; + isl_int_neg(mat->row[row][1 + j], mat->row[row][1 + j]); + } + if (!isl_int_is_one(mat->row[row][0])) + isl_seq_normalize(mat->row[row], 2 + tab->n_col); + for (i = 0; i < tab->n_row; ++i) { + if (i == row) + continue; + if (isl_int_is_zero(mat->row[i][2 + col])) + continue; + isl_int_mul(mat->row[i][0], mat->row[i][0], mat->row[row][0]); + for (j = 0; j < 1 + tab->n_col; ++j) { + if (j == 1 + col) + continue; + isl_int_mul(mat->row[i][1 + j], + mat->row[i][1 + j], mat->row[row][0]); + isl_int_addmul(mat->row[i][1 + j], + mat->row[i][2 + col], mat->row[row][1 + j]); + } + isl_int_mul(mat->row[i][2 + col], + mat->row[i][2 + col], mat->row[row][2 + col]); + if (!isl_int_is_one(mat->row[row][0])) + isl_seq_normalize(mat->row[i], 2 + tab->n_col); + } + t = tab->row_var[row]; + tab->row_var[row] = tab->col_var[col]; + tab->col_var[col] = t; + var = var_from_row(ctx, tab, row); + var->is_row = 1; + var->index = row; + var = var_from_col(ctx, tab, col); + var->is_row = 0; + var->index = col; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + if (isl_int_is_zero(mat->row[i][2 + col])) + continue; + if (!var_from_row(ctx, tab, i)->frozen && + is_redundant(ctx, tab, i)) + if (mark_redundant(ctx, tab, i)) + --i; + } +} + +/* If "var" represents a column variable, then pivot is up (sgn > 0) + * or down (sgn < 0) to a row. The variable is assumed not to be + * unbounded in the specified direction. + */ +static void to_row(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var, int sign) +{ + int r; + + if (var->is_row) + return; + + r = pivot_row(ctx, tab, NULL, sign, var->index); + isl_assert(ctx, r >= 0, return); + pivot(ctx, tab, r, var->index); +} + +static void check_table(struct isl_ctx *ctx, struct isl_tab *tab) +{ + int i; + + if (tab->empty) + return; + for (i = 0; i < tab->n_row; ++i) { + if (!var_from_row(ctx, tab, i)->is_nonneg) + continue; + assert(!isl_int_is_neg(tab->mat->row[i][1])); + } +} + +/* Return the sign of the maximal value of "var". + * If the sign is not negative, then on return from this function, + * the sample value will also be non-negative. + * + * If "var" is manifestly unbounded wrt positive values, we are done. + * Otherwise, we pivot the variable up to a row if needed + * Then we continue pivoting down until either + * - no more down pivots can be performed + * - the sample value is positive + * - the variable is pivoted into a manifestly unbounded column + */ +static int sign_of_max(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + + if (max_is_manifestly_unbounded(ctx, tab, var)) + return 1; + to_row(ctx, tab, var, 1); + while (!isl_int_is_pos(tab->mat->row[var->index][1])) { + find_pivot(ctx, tab, var, 1, &row, &col); + if (row == -1) + return isl_int_sgn(tab->mat->row[var->index][1]); + pivot(ctx, tab, row, col); + if (!var->is_row) /* manifestly unbounded */ + return 1; + } + return 1; +} + +/* Perform pivots until the row variable "var" has a non-negative + * sample value or until no more upward pivots can be performed. + * Return the sign of the sample value after the pivots have been + * performed. + */ +static int restore_row(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + + while (isl_int_is_neg(tab->mat->row[var->index][1])) { + find_pivot(ctx, tab, var, 1, &row, &col); + if (row == -1) + return; + pivot(ctx, tab, row, col); + if (!var->is_row) /* manifestly unbounded */ + return; + } +} + +/* Perform pivots until we are sure that the row variable "var" + * can attain non-negative values. After return from this + * function, "var" is still a row variable, but its sample + * value may not be non-negative, even if the function returns 1. + */ +static int at_least_zero(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + + while (isl_int_is_neg(tab->mat->row[var->index][1])) { + find_pivot(ctx, tab, var, 1, &row, &col); + if (row == -1) + break; + if (row == var->index) /* manifestly unbounded */ + return 1; + pivot(ctx, tab, row, col); + } + return !isl_int_is_neg(tab->mat->row[var->index][1]); +} + +/* Return a negative value if "var" can attain negative values. + * Return a non-negative value otherwise. + * + * If "var" is manifestly unbounded wrt negative values, we are done. + * Otherwise, if var is in a column, we can pivot it down to a row. + * Then we continue pivoting down until either + * - the pivot would result in a manifestly unbounded column + * => we don't perform the pivot, but simply return -1 + * - no more down pivots can be performed + * - the sample value is negative + * If the sample value becomes negative and the variable is supposed + * to be nonnegative, then we undo the last pivot. + * However, if the last pivot has made the pivoting variable + * obviously redundant, then it may have moved to another row. + * In that case we look for upward pivots until we reach a non-negative + * value again. + */ +static int sign_of_min(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + struct isl_tab_var *pivot_var; + + if (min_is_manifestly_unbounded(ctx, tab, var)) + return -1; + if (!var->is_row) { + col = var->index; + row = pivot_row(ctx, tab, NULL, -1, col); + pivot_var = var_from_col(ctx, tab, col); + pivot(ctx, tab, row, col); + if (var->is_redundant) + return 0; + if (isl_int_is_neg(tab->mat->row[var->index][1])) { + if (var->is_nonneg) { + if (!pivot_var->is_redundant && + pivot_var->index == row) + pivot(ctx, tab, row, col); + else + restore_row(ctx, tab, var); + } + return -1; + } + } + if (var->is_redundant) + return 0; + while (!isl_int_is_neg(tab->mat->row[var->index][1])) { + find_pivot(ctx, tab, var, -1, &row, &col); + if (row == var->index) + return -1; + if (row == -1) + return isl_int_sgn(tab->mat->row[var->index][1]); + pivot_var = var_from_col(ctx, tab, col); + pivot(ctx, tab, row, col); + if (var->is_redundant) + return 0; + } + if (var->is_nonneg) { + /* pivot back to non-negative value */ + if (!pivot_var->is_redundant && pivot_var->index == row) + pivot(ctx, tab, row, col); + else + restore_row(ctx, tab, var); + } + return -1; +} + +/* Return 1 if "var" can attain values <= -1. + * Return 0 otherwise. + * + * The sample value of "var" is assumed to be non-negative when the + * the function is called and will be made non-negative again before + * the function returns. + */ +static int min_at_most_neg_one(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + struct isl_tab_var *pivot_var; + + if (min_is_manifestly_unbounded(ctx, tab, var)) + return 1; + if (!var->is_row) { + col = var->index; + row = pivot_row(ctx, tab, NULL, -1, col); + pivot_var = var_from_col(ctx, tab, col); + pivot(ctx, tab, row, col); + if (var->is_redundant) + return 0; + if (isl_int_is_neg(tab->mat->row[var->index][1]) && + isl_int_abs_ge(tab->mat->row[var->index][1], + tab->mat->row[var->index][0])) { + if (var->is_nonneg) { + if (!pivot_var->is_redundant && + pivot_var->index == row) + pivot(ctx, tab, row, col); + else + restore_row(ctx, tab, var); + } + return 1; + } + } + if (var->is_redundant) + return 0; + do { + find_pivot(ctx, tab, var, -1, &row, &col); + if (row == var->index) + return 1; + if (row == -1) + return 0; + pivot_var = var_from_col(ctx, tab, col); + pivot(ctx, tab, row, col); + if (var->is_redundant) + return 0; + } while (!isl_int_is_neg(tab->mat->row[var->index][1]) || + isl_int_abs_lt(tab->mat->row[var->index][1], + tab->mat->row[var->index][0])); + if (var->is_nonneg) { + /* pivot back to non-negative value */ + if (!pivot_var->is_redundant && pivot_var->index == row) + pivot(ctx, tab, row, col); + restore_row(ctx, tab, var); + } + return 1; +} + +/* Return 1 if "var" can attain values >= 1. + * Return 0 otherwise. + */ +static int at_least_one(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int row, col; + isl_int *r; + + if (max_is_manifestly_unbounded(ctx, tab, var)) + return 1; + to_row(ctx, tab, var, 1); + r = tab->mat->row[var->index]; + while (isl_int_lt(r[1], r[0])) { + find_pivot(ctx, tab, var, 1, &row, &col); + if (row == -1) + return isl_int_ge(r[1], r[0]); + if (row == var->index) /* manifestly unbounded */ + return 1; + pivot(ctx, tab, row, col); + } + return 1; +} + +static void swap_cols(struct isl_ctx *ctx, + struct isl_tab *tab, int col1, int col2) +{ + int t; + t = tab->col_var[col1]; + tab->col_var[col1] = tab->col_var[col2]; + tab->col_var[col2] = t; + var_from_col(ctx, tab, col1)->index = col1; + var_from_col(ctx, tab, col2)->index = col2; + tab->mat = isl_mat_swap_cols(ctx, tab->mat, 2 + col1, 2 + col2); +} + +/* Mark column with index "col" as representing a zero variable. + * If we may need to undo the operation the column is kept, + * but no longer considered. + * Otherwise, the column is simply removed. + * + * The column may be interchanged with some other column. If it + * is interchanged with a later column, return 1. Otherwise return 0. + * If the columns are checked in order in the calling function, + * then a return value of 1 means that the column with the given + * column number may now contain a different column that + * hasn't been checked yet. + */ +static int kill_col(struct isl_ctx *ctx, + struct isl_tab *tab, int col) +{ + tab->killed_col = 1; + var_from_col(ctx, tab, col)->is_zero = 1; + if (tab->need_undo) { + push(ctx, tab, isl_tab_undo_zero, var_from_col(ctx, tab, col)); + if (col != tab->n_dead) + swap_cols(ctx, tab, col, tab->n_dead); + tab->n_dead++; + return 0; + } else { + if (col != tab->n_col - 1) + swap_cols(ctx, tab, col, tab->n_col - 1); + var_from_col(ctx, tab, tab->n_col - 1)->index = -1; + tab->n_col--; + return 1; + } +} + +/* Row variable "var" is non-negative and cannot attain any values + * larger than zero. This means that the coefficients of the unrestricted + * column variables are zero and that the coefficients of the non-negative + * column variables are zero or negative. + * Each of the non-negative variables with a negative coefficient can + * then also be written as the negative sum of non-negative variables + * and must therefore also be zero. + */ +static void close_row(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + int j; + struct isl_mat *mat = tab->mat; + + isl_assert(ctx, var->is_nonneg, return); + var->is_zero = 1; + for (j = tab->n_dead; j < tab->n_col; ++j) { + if (isl_int_is_zero(mat->row[var->index][2 + j])) + continue; + isl_assert(ctx, isl_int_is_neg(mat->row[var->index][2 + j]), + return); + if (kill_col(ctx, tab, j)) + --j; + } + mark_redundant(ctx, tab, var->index); +} + +/* Add a row to the tableau. The row is given as an affine combination + * of the original variables and needs to be expressed in terms of the + * column variables. + * + * We add each term in turn. + * If r = n/d_r is the current sum and we need to add k x, then + * if x is a column variable, we increase the numerator of + * this column by k d_r + * if x = f/d_x is a row variable, then the new representation of r is + * + * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f + * --- + --- = ------------------- = ------------------- + * d_r d_r d_r d_x/g m + * + * with g the gcd of d_r and d_x and m the lcm of d_r and d_x. + */ +static int add_row(struct isl_ctx *ctx, struct isl_tab *tab, isl_int *line) +{ + int i; + unsigned r; + isl_int *row; + isl_int a, b; + + isl_assert(ctx, tab->n_row < tab->mat->n_row, return -1); + + isl_int_init(a); + isl_int_init(b); + r = tab->n_con; + tab->con[r].index = tab->n_row; + tab->con[r].is_row = 1; + tab->con[r].is_nonneg = 0; + tab->con[r].is_zero = 0; + tab->con[r].is_redundant = 0; + tab->con[r].frozen = 0; + tab->row_var[tab->n_row] = ~r; + row = tab->mat->row[tab->n_row]; + isl_int_set_si(row[0], 1); + isl_int_set(row[1], line[0]); + isl_seq_clr(row + 2, tab->n_col); + for (i = 0; i < tab->n_var; ++i) { + if (tab->var[i].is_zero) + continue; + if (tab->var[i].is_row) { + isl_int_lcm(a, + row[0], tab->mat->row[tab->var[i].index][0]); + isl_int_swap(a, row[0]); + isl_int_divexact(a, row[0], a); + isl_int_divexact(b, + row[0], tab->mat->row[tab->var[i].index][0]); + isl_int_mul(b, b, line[1 + i]); + isl_seq_combine(row + 1, a, row + 1, + b, tab->mat->row[tab->var[i].index] + 1, + 1 + tab->n_col); + } else + isl_int_addmul(row[2 + tab->var[i].index], + line[1 + i], row[0]); + } + isl_seq_normalize(row, 2 + tab->n_col); + tab->n_row++; + tab->n_con++; + push(ctx, tab, isl_tab_undo_allocate, &tab->con[r]); + isl_int_clear(a); + isl_int_clear(b); + + return r; +} + +static int drop_row(struct isl_ctx *ctx, struct isl_tab *tab, int row) +{ + isl_assert(ctx, ~tab->row_var[row] == tab->n_con - 1, return -1); + if (row != tab->n_row - 1) + swap_rows(ctx, tab, row, tab->n_row - 1); + tab->n_row--; + tab->n_con--; + return 0; +} + +/* Add inequality "ineq" and check if it conflicts with the + * previously added constraints or if it is obviously redundant. + */ +struct isl_tab *isl_tab_add_ineq(struct isl_ctx *ctx, + struct isl_tab *tab, isl_int *ineq) +{ + int r; + int sgn; + + if (!tab) + return NULL; + r = add_row(ctx, tab, ineq); + if (r < 0) + goto error; + tab->con[r].is_nonneg = 1; + push(ctx, tab, isl_tab_undo_nonneg, &tab->con[r]); + if (is_redundant(ctx, tab, tab->con[r].index)) { + mark_redundant(ctx, tab, tab->con[r].index); + return tab; + } + + sgn = sign_of_max(ctx, tab, &tab->con[r]); + if (sgn < 0) + mark_empty(ctx, tab); + else { + if (sgn == 0) + close_row(ctx, tab, &tab->con[r]); + else if (tab->con[r].is_row && + is_redundant(ctx, tab, tab->con[r].index)) + mark_redundant(ctx, tab, tab->con[r].index); + } + return tab; +error: + isl_tab_free(ctx, tab); + return NULL; +} + +/* We assume Gaussian elimination has been performed on the equalities. + * The equalities can therefore never conflict. + * Adding the equalities is currently only really useful for a later call + * to isl_tab_ineq_type. + */ +static struct isl_tab *add_eq(struct isl_ctx *ctx, + struct isl_tab *tab, isl_int *eq) +{ + int i; + int r; + + if (!tab) + return NULL; + r = add_row(ctx, tab, eq); + if (r < 0) + goto error; + + r = tab->con[r].index; + for (i = tab->n_dead; i < tab->n_col; ++i) { + if (isl_int_is_zero(tab->mat->row[r][2 + i])) + continue; + pivot(ctx, tab, r, i); + kill_col(ctx, tab, i); + break; + } + + return tab; +error: + isl_tab_free(ctx, tab); + return NULL; +} + +struct isl_tab *isl_tab_from_basic_map(struct isl_basic_map *bmap) +{ + int i; + struct isl_tab *tab; + + if (!bmap) + return NULL; + tab = isl_tab_alloc(bmap->ctx, + isl_basic_map_total_dim(bmap) + bmap->n_ineq + 1, + isl_basic_map_total_dim(bmap)); + if (!tab) + return NULL; + tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); + if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) { + mark_empty(bmap->ctx, tab); + return tab; + } + for (i = 0; i < bmap->n_eq; ++i) { + tab = add_eq(bmap->ctx, tab, bmap->eq[i]); + if (!tab) + return tab; + } + tab->killed_col = 0; + for (i = 0; i < bmap->n_ineq; ++i) { + tab = isl_tab_add_ineq(bmap->ctx, tab, bmap->ineq[i]); + if (!tab || tab->empty) + return tab; + } + return tab; +} + +struct isl_tab *isl_tab_from_basic_set(struct isl_basic_set *bset) +{ + return isl_tab_from_basic_map((struct isl_basic_map *)bset); +} + +/* Construct a tableau corresponding to the recession cone of "bmap". + */ +struct isl_tab *isl_tab_from_recession_cone(struct isl_basic_map *bmap) +{ + isl_int cst; + int i; + struct isl_tab *tab; + + if (!bmap) + return NULL; + tab = isl_tab_alloc(bmap->ctx, bmap->n_eq + bmap->n_ineq, + isl_basic_map_total_dim(bmap)); + if (!tab) + return NULL; + tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); + + isl_int_init(cst); + for (i = 0; i < bmap->n_eq; ++i) { + isl_int_swap(bmap->eq[i][0], cst); + tab = add_eq(bmap->ctx, tab, bmap->eq[i]); + isl_int_swap(bmap->eq[i][0], cst); + if (!tab) + goto done; + } + tab->killed_col = 0; + for (i = 0; i < bmap->n_ineq; ++i) { + int r; + isl_int_swap(bmap->ineq[i][0], cst); + r = add_row(bmap->ctx, tab, bmap->ineq[i]); + isl_int_swap(bmap->ineq[i][0], cst); + if (r < 0) + goto error; + tab->con[r].is_nonneg = 1; + push(bmap->ctx, tab, isl_tab_undo_nonneg, &tab->con[r]); + } +done: + isl_int_clear(cst); + return tab; +error: + isl_int_clear(cst); + isl_tab_free(bmap->ctx, tab); + return NULL; +} + +/* Assuming "tab" is the tableau of a cone, check if the cone is + * bounded, i.e., if it is empty or only contains the origin. + */ +int isl_tab_cone_is_bounded(struct isl_ctx *ctx, struct isl_tab *tab) +{ + int i; + + if (!tab) + return -1; + if (tab->empty) + return 1; + if (tab->n_dead == tab->n_col) + return 1; + + for (i = tab->n_redundant; i < tab->n_row; ++i) { + struct isl_tab_var *var; + var = var_from_row(ctx, tab, i); + if (!var->is_nonneg) + continue; + if (sign_of_max(ctx, tab, var) == 0) + close_row(ctx, tab, var); + else + return 0; + if (tab->n_dead == tab->n_col) + return 1; + } + return 0; +} + +static int sample_is_integer(struct isl_ctx *ctx, struct isl_tab *tab) +{ + int i; + + for (i = 0; i < tab->n_var; ++i) { + int row; + if (!tab->var[i].is_row) + continue; + row = tab->var[i].index; + if (!isl_int_is_divisible_by(tab->mat->row[row][1], + tab->mat->row[row][0])) + return 0; + } + return 1; +} + +static struct isl_vec *extract_integer_sample(struct isl_ctx *ctx, + struct isl_tab *tab) +{ + int i; + struct isl_vec *vec; + + vec = isl_vec_alloc(ctx, 1 + tab->n_var); + if (!vec) + return NULL; + + isl_int_set_si(vec->block.data[0], 1); + for (i = 0; i < tab->n_var; ++i) { + if (!tab->var[i].is_row) + isl_int_set_si(vec->block.data[1 + i], 0); + else { + int row = tab->var[i].index; + isl_int_divexact(vec->block.data[1 + i], + tab->mat->row[row][1], tab->mat->row[row][0]); + } + } + + return vec; +} + +/* Update "bmap" based on the results of the tableau "tab". + * In particular, implicit equalities are made explicit, redundant constraints + * are removed and if the sample value happens to be integer, it is stored + * in "bmap" (unless "bmap" already had an integer sample). + * + * The tableau is assumed to have been created from "bmap" using + * isl_tab_from_basic_map. + */ +struct isl_basic_map *isl_basic_map_update_from_tab(struct isl_basic_map *bmap, + struct isl_tab *tab) +{ + int i; + unsigned n_eq; + + if (!bmap) + return NULL; + if (!tab) + return bmap; + + n_eq = bmap->n_eq; + if (tab->empty) + bmap = isl_basic_map_set_to_empty(bmap); + else + for (i = bmap->n_ineq - 1; i >= 0; --i) { + if (isl_tab_is_equality(bmap->ctx, tab, n_eq + i)) + isl_basic_map_inequality_to_equality(bmap, i); + else if (isl_tab_is_redundant(bmap->ctx, tab, n_eq + i)) + isl_basic_map_drop_inequality(bmap, i); + } + if (!tab->rational && + !bmap->sample && sample_is_integer(bmap->ctx, tab)) + bmap->sample = extract_integer_sample(bmap->ctx, tab); + return bmap; +} + +struct isl_basic_set *isl_basic_set_update_from_tab(struct isl_basic_set *bset, + struct isl_tab *tab) +{ + return (struct isl_basic_set *)isl_basic_map_update_from_tab( + (struct isl_basic_map *)bset, tab); +} + +/* Given a non-negative variable "var", add a new non-negative variable + * that is the opposite of "var", ensuring that var can only attain the + * value zero. + * If var = n/d is a row variable, then the new variable = -n/d. + * If var is a column variables, then the new variable = -var. + * If the new variable cannot attain non-negative values, then + * the resulting tableau is empty. + * Otherwise, we know the value will be zero and we close the row. + */ +static struct isl_tab *cut_to_hyperplane(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + unsigned r; + isl_int *row; + int sgn; + + if (extend_cons(ctx, tab, 1) < 0) + goto error; + + r = tab->n_con; + tab->con[r].index = tab->n_row; + tab->con[r].is_row = 1; + tab->con[r].is_nonneg = 0; + tab->con[r].is_zero = 0; + tab->con[r].is_redundant = 0; + tab->con[r].frozen = 0; + tab->row_var[tab->n_row] = ~r; + row = tab->mat->row[tab->n_row]; + + if (var->is_row) { + isl_int_set(row[0], tab->mat->row[var->index][0]); + isl_seq_neg(row + 1, + tab->mat->row[var->index] + 1, 1 + tab->n_col); + } else { + isl_int_set_si(row[0], 1); + isl_seq_clr(row + 1, 1 + tab->n_col); + isl_int_set_si(row[2 + var->index], -1); + } + + tab->n_row++; + tab->n_con++; + push(ctx, tab, isl_tab_undo_allocate, &tab->con[r]); + + sgn = sign_of_max(ctx, tab, &tab->con[r]); + if (sgn < 0) + mark_empty(ctx, tab); + else { + tab->con[r].is_nonneg = 1; + push(ctx, tab, isl_tab_undo_nonneg, &tab->con[r]); + /* sgn == 0 */ + close_row(ctx, tab, &tab->con[r]); + } + + return tab; +error: + isl_tab_free(ctx, tab); + return NULL; +} + +/* Given a tableau "tab" and an inequality constraint "con" of the tableau, + * relax the inequality by one. That is, the inequality r >= 0 is replaced + * by r' = r + 1 >= 0. + * If r is a row variable, we simply increase the constant term by one + * (taking into account the denominator). + * If r is a column variable, then we need to modify each row that + * refers to r = r' - 1 by substituting this equality, effectively + * subtracting the coefficient of the column from the constant. + */ +struct isl_tab *isl_tab_relax(struct isl_ctx *ctx, + struct isl_tab *tab, int con) +{ + struct isl_tab_var *var; + if (!tab) + return NULL; + + var = &tab->con[con]; + + if (!var->is_row && !max_is_manifestly_unbounded(ctx, tab, var)) + to_row(ctx, tab, var, 1); + + if (var->is_row) + isl_int_add(tab->mat->row[var->index][1], + tab->mat->row[var->index][1], tab->mat->row[var->index][0]); + else { + int i; + + for (i = 0; i < tab->n_row; ++i) { + if (isl_int_is_zero(tab->mat->row[i][2 + var->index])) + continue; + isl_int_sub(tab->mat->row[i][1], tab->mat->row[i][1], + tab->mat->row[i][2 + var->index]); + } + + } + + push(ctx, tab, isl_tab_undo_relax, var); + + return tab; +} + +struct isl_tab *isl_tab_select_facet(struct isl_ctx *ctx, + struct isl_tab *tab, int con) +{ + if (!tab) + return NULL; + + return cut_to_hyperplane(ctx, tab, &tab->con[con]); +} + +static int may_be_equality(struct isl_tab *tab, int row) +{ + return (tab->rational ? isl_int_is_zero(tab->mat->row[row][1]) + : isl_int_lt(tab->mat->row[row][1], + tab->mat->row[row][0])) && + isl_seq_first_non_zero(tab->mat->row[row] + 2 + tab->n_dead, + tab->n_col - tab->n_dead) != -1; +} + +/* Check for (near) equalities among the constraints. + * A constraint is an equality if it is non-negative and if + * its maximal value is either + * - zero (in case of rational tableaus), or + * - strictly less than 1 (in case of integer tableaus) + * + * When the rows are added to the tableau, they are already + * checked for being equal to zero. If none of the rows + * have been determined to be zero (killed_col is not set) + * and we are dealing with a rational tableau, then we wouldn't + * be able to find any zero row, so we can return immediately. + * + * We first mark all non-redundant and non-dead variables that + * are not frozen and not obviously not an equality. + * Then we iterate over all marked variables if they can attain + * any values larger than zero or at least one. + * If the maximal value is zero, we mark any column variables + * that appear in the row as being zero and mark the row as being redundant. + * Otherwise, if the maximal value is strictly less than one (and the + * tableau is integer), then we restrict the value to being zero + * by adding an opposite non-negative variable. + */ +struct isl_tab *isl_tab_detect_equalities(struct isl_ctx *ctx, + struct isl_tab *tab) +{ + int i; + unsigned n_marked; + + if (!tab) + return NULL; + if (tab->empty) + return tab; + if (tab->rational && !tab->killed_col) + return tab; + if (tab->n_dead == tab->n_col) + return tab; + + n_marked = 0; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + struct isl_tab_var *var = var_from_row(ctx, tab, i); + var->marked = !var->frozen && var->is_nonneg && + may_be_equality(tab, i); + if (var->marked) + n_marked++; + } + for (i = tab->n_dead; i < tab->n_col; ++i) { + struct isl_tab_var *var = var_from_col(ctx, tab, i); + var->marked = !var->frozen && var->is_nonneg; + if (var->marked) + n_marked++; + } + while (n_marked) { + struct isl_tab_var *var; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + var = var_from_row(ctx, tab, i); + if (var->marked) + break; + } + if (i == tab->n_row) { + for (i = tab->n_dead; i < tab->n_col; ++i) { + var = var_from_col(ctx, tab, i); + if (var->marked) + break; + } + if (i == tab->n_col) + break; + } + var->marked = 0; + n_marked--; + if (sign_of_max(ctx, tab, var) == 0) + close_row(ctx, tab, var); + else if (!tab->rational && !at_least_one(ctx, tab, var)) { + tab = cut_to_hyperplane(ctx, tab, var); + return isl_tab_detect_equalities(ctx, tab); + } + for (i = tab->n_redundant; i < tab->n_row; ++i) { + var = var_from_row(ctx, tab, i); + if (!var->marked) + continue; + if (may_be_equality(tab, i)) + continue; + var->marked = 0; + n_marked--; + } + } + + tab->killed_col = 0; + return tab; +} + +/* Check for (near) redundant constraints. + * A constraint is redundant if it is non-negative and if + * its minimal value (temporarily ignoring the non-negativity) is either + * - zero (in case of rational tableaus), or + * - strictly larger than -1 (in case of integer tableaus) + * + * We first mark all non-redundant and non-dead variables that + * are not frozen and not obviously negatively unbounded. + * Then we iterate over all marked variables if they can attain + * any values smaller than zero or at most negative one. + * If not, we mark the row as being redundant (assuming it hasn't + * been detected as being obviously redundant in the mean time). + */ +struct isl_tab *isl_tab_detect_redundant(struct isl_ctx *ctx, + struct isl_tab *tab) +{ + int i; + unsigned n_marked; + + if (!tab) + return NULL; + if (tab->empty) + return tab; + if (tab->n_redundant == tab->n_row) + return tab; + + n_marked = 0; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + struct isl_tab_var *var = var_from_row(ctx, tab, i); + var->marked = !var->frozen && var->is_nonneg; + if (var->marked) + n_marked++; + } + for (i = tab->n_dead; i < tab->n_col; ++i) { + struct isl_tab_var *var = var_from_col(ctx, tab, i); + var->marked = !var->frozen && var->is_nonneg && + !min_is_manifestly_unbounded(ctx, tab, var); + if (var->marked) + n_marked++; + } + while (n_marked) { + struct isl_tab_var *var; + for (i = tab->n_redundant; i < tab->n_row; ++i) { + var = var_from_row(ctx, tab, i); + if (var->marked) + break; + } + if (i == tab->n_row) { + for (i = tab->n_dead; i < tab->n_col; ++i) { + var = var_from_col(ctx, tab, i); + if (var->marked) + break; + } + if (i == tab->n_col) + break; + } + var->marked = 0; + n_marked--; + if ((tab->rational ? (sign_of_min(ctx, tab, var) >= 0) + : !min_at_most_neg_one(ctx, tab, var)) && + !var->is_redundant) + mark_redundant(ctx, tab, var->index); + for (i = tab->n_dead; i < tab->n_col; ++i) { + var = var_from_col(ctx, tab, i); + if (!var->marked) + continue; + if (!min_is_manifestly_unbounded(ctx, tab, var)) + continue; + var->marked = 0; + n_marked--; + } + } + + return tab; +} + +int isl_tab_is_equality(struct isl_ctx *ctx, struct isl_tab *tab, int con) +{ + int row; + + if (!tab) + return -1; + if (tab->con[con].is_zero) + return 1; + if (tab->con[con].is_redundant) + return 0; + if (!tab->con[con].is_row) + return tab->con[con].index < tab->n_dead; + + row = tab->con[con].index; + + return isl_int_is_zero(tab->mat->row[row][1]) && + isl_seq_first_non_zero(tab->mat->row[row] + 2 + tab->n_dead, + tab->n_col - tab->n_dead) == -1; +} + +/* Return the minimial value of the affine expression "f" with denominator + * "denom" in *opt, *opt_denom, assuming the tableau is not empty and + * the expression cannot attain arbitrarily small values. + * If opt_denom is NULL, then *opt is rounded up to the nearest integer. + * The return value reflects the nature of the result (empty, unbounded, + * minmimal value returned in *opt). + */ +enum isl_lp_result isl_tab_min(struct isl_ctx *ctx, struct isl_tab *tab, + isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom) +{ + int r; + enum isl_lp_result res = isl_lp_ok; + struct isl_tab_var *var; + + if (tab->empty) + return isl_lp_empty; + + r = add_row(ctx, tab, f); + if (r < 0) + return isl_lp_error; + var = &tab->con[r]; + isl_int_mul(tab->mat->row[var->index][0], + tab->mat->row[var->index][0], denom); + for (;;) { + int row, col; + find_pivot(ctx, tab, var, -1, &row, &col); + if (row == var->index) { + res = isl_lp_unbounded; + break; + } + if (row == -1) + break; + pivot(ctx, tab, row, col); + } + if (drop_row(ctx, tab, var->index) < 0) + return isl_lp_error; + if (res == isl_lp_ok) { + if (opt_denom) { + isl_int_set(*opt, tab->mat->row[var->index][1]); + isl_int_set(*opt_denom, tab->mat->row[var->index][0]); + } else + isl_int_cdiv_q(*opt, tab->mat->row[var->index][1], + tab->mat->row[var->index][0]); + } + return res; +} + +int isl_tab_is_redundant(struct isl_ctx *ctx, struct isl_tab *tab, int con) +{ + int row; + unsigned n_col; + + if (!tab) + return -1; + if (tab->con[con].is_zero) + return 0; + if (tab->con[con].is_redundant) + return 1; + return tab->con[con].is_row && tab->con[con].index < tab->n_redundant; +} + +/* Take a snapshot of the tableau that can be restored by s call to + * isl_tab_rollback. + */ +struct isl_tab_undo *isl_tab_snap(struct isl_ctx *ctx, struct isl_tab *tab) +{ + if (!tab) + return NULL; + tab->need_undo = 1; + return tab->top; +} + +/* Undo the operation performed by isl_tab_relax. + */ +static void unrelax(struct isl_ctx *ctx, + struct isl_tab *tab, struct isl_tab_var *var) +{ + if (!var->is_row && !max_is_manifestly_unbounded(ctx, tab, var)) + to_row(ctx, tab, var, 1); + + if (var->is_row) + isl_int_sub(tab->mat->row[var->index][1], + tab->mat->row[var->index][1], tab->mat->row[var->index][0]); + else { + int i; + + for (i = 0; i < tab->n_row; ++i) { + if (isl_int_is_zero(tab->mat->row[i][2 + var->index])) + continue; + isl_int_add(tab->mat->row[i][1], tab->mat->row[i][1], + tab->mat->row[i][2 + var->index]); + } + + } +} + +static void perform_undo(struct isl_ctx *ctx, struct isl_tab *tab, + struct isl_tab_undo *undo) +{ + switch(undo->type) { + case isl_tab_undo_empty: + tab->empty = 0; + break; + case isl_tab_undo_nonneg: + undo->var->is_nonneg = 0; + break; + case isl_tab_undo_redundant: + undo->var->is_redundant = 0; + tab->n_redundant--; + break; + case isl_tab_undo_zero: + undo->var->is_zero = 0; + tab->n_dead--; + break; + case isl_tab_undo_allocate: + if (!undo->var->is_row) { + if (max_is_manifestly_unbounded(ctx, tab, undo->var)) + to_row(ctx, tab, undo->var, -1); + else + to_row(ctx, tab, undo->var, 1); + } + drop_row(ctx, tab, undo->var->index); + break; + case isl_tab_undo_relax: + unrelax(ctx, tab, undo->var); + break; + } +} + +/* Return the tableau to the state it was in when the snapshot "snap" + * was taken. + */ +int isl_tab_rollback(struct isl_ctx *ctx, struct isl_tab *tab, + struct isl_tab_undo *snap) +{ + struct isl_tab_undo *undo, *next; + + if (!tab) + return -1; + + for (undo = tab->top; undo && undo != &tab->bottom; undo = next) { + next = undo->next; + if (undo == snap) + break; + perform_undo(ctx, tab, undo); + free(undo); + } + tab->top = undo; + if (!undo) + return -1; + return 0; +} + +/* The given row "row" represents an inequality violated by all + * points in the tableau. Check for some special cases of such + * separating constraints. + * In particular, if the row has been reduced to the constant -1, + * then we know the inequality is adjacent (but opposite) to + * an equality in the tableau. + * If the row has been reduced to r = -1 -r', with r' an inequality + * of the tableau, then the inequality is adjacent (but opposite) + * to the inequality r'. + */ +static enum isl_ineq_type separation_type(struct isl_ctx *ctx, + struct isl_tab *tab, unsigned row) +{ + int pos; + + if (tab->rational) + return isl_ineq_separate; + + if (!isl_int_is_one(tab->mat->row[row][0])) + return isl_ineq_separate; + if (!isl_int_is_negone(tab->mat->row[row][1])) + return isl_ineq_separate; + + pos = isl_seq_first_non_zero(tab->mat->row[row] + 2 + tab->n_dead, + tab->n_col - tab->n_dead); + if (pos == -1) + return isl_ineq_adj_eq; + + if (!isl_int_is_negone(tab->mat->row[row][2 + tab->n_dead + pos])) + return isl_ineq_separate; + + pos = isl_seq_first_non_zero( + tab->mat->row[row] + 2 + tab->n_dead + pos + 1, + tab->n_col - tab->n_dead - pos - 1); + + return pos == -1 ? isl_ineq_adj_ineq : isl_ineq_separate; +} + +/* Check the effect of inequality "ineq" on the tableau "tab". + * The result may be + * isl_ineq_redundant: satisfied by all points in the tableau + * isl_ineq_separate: satisfied by no point in tha tableau + * isl_ineq_cut: satisfied by some by not all points + * isl_ineq_adj_eq: adjacent to an equality + * isl_ineq_adj_ineq: adjacent to an inequality. + */ +enum isl_ineq_type isl_tab_ineq_type(struct isl_ctx *ctx, struct isl_tab *tab, + isl_int *ineq) +{ + enum isl_ineq_type type = isl_ineq_error; + struct isl_tab_undo *snap = NULL; + int con; + int row; + + if (!tab) + return isl_ineq_error; + + if (extend_cons(ctx, tab, 1) < 0) + return isl_ineq_error; + + snap = isl_tab_snap(ctx, tab); + + con = add_row(ctx, tab, ineq); + if (con < 0) + goto error; + + row = tab->con[con].index; + if (is_redundant(ctx, tab, row)) + type = isl_ineq_redundant; + else if (isl_int_is_neg(tab->mat->row[row][1]) && + (tab->rational || + isl_int_abs_ge(tab->mat->row[row][1], + tab->mat->row[row][0]))) { + if (at_least_zero(ctx, tab, &tab->con[con])) + type = isl_ineq_cut; + else + type = separation_type(ctx, tab, row); + } else if (tab->rational ? (sign_of_min(ctx, tab, &tab->con[con]) < 0) + : min_at_most_neg_one(ctx, tab, &tab->con[con])) + type = isl_ineq_cut; + else + type = isl_ineq_redundant; + + if (isl_tab_rollback(ctx, tab, snap)) + return isl_ineq_error; + return type; +error: + isl_tab_rollback(ctx, tab, snap); + return isl_ineq_error; +} + +void isl_tab_dump(struct isl_ctx *ctx, struct isl_tab *tab, + FILE *out, int indent) +{ + unsigned r, c; + int i; + + if (!tab) { + fprintf(out, "%*snull tab\n", indent, ""); + return; + } + fprintf(out, "%*sn_redundant: %d, n_dead: %d", indent, "", + tab->n_redundant, tab->n_dead); + if (tab->rational) + fprintf(out, ", rational"); + if (tab->empty) + fprintf(out, ", empty"); + if (tab->killed_col) + fprintf(out, ", killed_col"); + fprintf(out, "\n"); + fprintf(out, "%*s[", indent, ""); + for (i = 0; i < tab->n_var; ++i) { + if (i) + fprintf(out, ", "); + fprintf(out, "%c%d%s", tab->var[i].is_row ? 'r' : 'c', + tab->var[i].index, + tab->var[i].is_zero ? " [=0]" : + tab->var[i].is_redundant ? " [R]" : ""); + } + fprintf(out, "]\n"); + fprintf(out, "%*s[", indent, ""); + for (i = 0; i < tab->n_con; ++i) { + if (i) + fprintf(out, ", "); + fprintf(out, "%c%d%s", tab->con[i].is_row ? 'r' : 'c', + tab->con[i].index, + tab->con[i].is_zero ? " [=0]" : + tab->con[i].is_redundant ? " [R]" : ""); + } + fprintf(out, "]\n"); + fprintf(out, "%*s[", indent, ""); + for (i = 0; i < tab->n_row; ++i) { + if (i) + fprintf(out, ", "); + fprintf(out, "r%d: %d%s", i, tab->row_var[i], + var_from_row(ctx, tab, i)->is_nonneg ? " [>=0]" : ""); + } + fprintf(out, "]\n"); + fprintf(out, "%*s[", indent, ""); + for (i = 0; i < tab->n_col; ++i) { + if (i) + fprintf(out, ", "); + fprintf(out, "c%d: %d%s", i, tab->col_var[i], + var_from_col(ctx, tab, i)->is_nonneg ? " [>=0]" : ""); + } + fprintf(out, "]\n"); + r = tab->mat->n_row; + tab->mat->n_row = tab->n_row; + c = tab->mat->n_col; + tab->mat->n_col = 2 + tab->n_col; + isl_mat_dump(ctx, tab->mat, out, indent); + tab->mat->n_row = r; + tab->mat->n_col = c; +} diff --git a/isl_tab.h b/isl_tab.h new file mode 100644 index 0000000..179457a --- /dev/null +++ b/isl_tab.h @@ -0,0 +1,145 @@ +#ifndef ISL_TAB_H +#define ISL_TAB_H + +#include "isl_lp.h" +#include "isl_map.h" +#include "isl_mat.h" + +struct isl_tab_var { + int index; + unsigned is_row : 1; + unsigned is_nonneg : 1; + unsigned is_zero : 1; + unsigned is_redundant : 1; + unsigned marked : 1; + unsigned frozen : 1; +}; + +enum isl_tab_undo_type { + isl_tab_undo_bottom, + isl_tab_undo_empty, + isl_tab_undo_nonneg, + isl_tab_undo_redundant, + isl_tab_undo_zero, + isl_tab_undo_allocate, + isl_tab_undo_relax, +}; + +struct isl_tab_undo { + enum isl_tab_undo_type type; + struct isl_tab_var *var; + struct isl_tab_undo *next; +}; + +/* The tableau maintains equality relations. + * Each column and each row is associated to a variable or a constraint. + * The "value" of an inequality constraint is the value of the corresponding + * slack variable. + * The "row_var" and "col_var" arrays map column and row indices + * to indices in the "var" and "con" arrays. The elements of these + * arrays maintain extra information about the variables and the constraints. + * Each row expresses the corresponding row variable as an affine expression + * of the column variables. + * The first two columns in the matrix contain the common denominator of + * the row and the numerator of the constant term. The third column + * in the matrix is called column 0 with respect to the col_var array. + * The sample value of the tableau is the value that assigns zero + * to all the column variables and the constant term of each affine + * expression to the corresponding row variable. + * The operations on the tableau maintain the property that the sample + * value satisfies the non-negativity constraints (usually on the slack + * variables). + * + * The first n_dead column variables have their values fixed to zero. + * The corresponding tab_vars are flagged "is_zero". + * Some of the rows that have have zero coefficients in all but + * the dead columns are also flagged "is_zero". + * + * The first n_redundant rows correspond to inequality constraints + * that are always satisfied for any value satisfying the non-redundant + * rows. The corresponding tab_vars are flagged "is_redundant". + * A row variable that is flagged "is_zero" is also flagged "is_redundant" + * since the constraint has been reduced to 0 = 0 and is therefore always + * satisfied. + * + * Dead columns and redundant rows are detected on the fly. + * However, the basic operations do not ensure that all dead columns + * or all redundant rows are detected. + * isl_tab_detect_equalities and isl_tab_detect_redundant can be used + * to peform and exhaustive search for dead columns and redundant rows. + */ +struct isl_tab { + struct isl_mat *mat; + + unsigned n_row; + unsigned n_col; + unsigned n_dead; + unsigned n_redundant; + + unsigned n_var; + unsigned n_con; + unsigned max_con; + struct isl_tab_var *var; + struct isl_tab_var *con; + int *row_var; /* v >= 0 -> var v; v < 0 -> con ~v */ + int *col_var; /* v >= 0 -> var v; v < 0 -> con ~v */ + + struct isl_tab_undo bottom; + struct isl_tab_undo *top; + + unsigned need_undo : 1; + unsigned rational : 1; + unsigned empty : 1; + unsigned killed_col : 1; +}; + +void isl_tab_free(struct isl_ctx *ctx, struct isl_tab *tab); + +struct isl_tab *isl_tab_from_basic_map(struct isl_basic_map *bmap); +struct isl_tab *isl_tab_from_basic_set(struct isl_basic_set *bset); +struct isl_tab *isl_tab_from_recession_cone(struct isl_basic_map *bmap); +int isl_tab_cone_is_bounded(struct isl_ctx *ctx, struct isl_tab *tab); +struct isl_basic_map *isl_basic_map_update_from_tab(struct isl_basic_map *bmap, + struct isl_tab *tab); +struct isl_basic_set *isl_basic_set_update_from_tab(struct isl_basic_set *bset, + struct isl_tab *tab); +struct isl_tab *isl_tab_detect_equalities(struct isl_ctx *ctx, + struct isl_tab *tab); +struct isl_tab *isl_tab_detect_redundant(struct isl_ctx *ctx, + struct isl_tab *tab); +enum isl_lp_result isl_tab_min(struct isl_ctx *ctx, struct isl_tab *tab, + isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom); + +struct isl_tab *isl_tab_extend(struct isl_ctx *ctx, struct isl_tab *tab, + unsigned n_new); +struct isl_tab *isl_tab_add_ineq(struct isl_ctx *ctx, + struct isl_tab *tab, isl_int *ineq); + +int isl_tab_is_equality(struct isl_ctx *ctx, struct isl_tab *tab, int con); +int isl_tab_is_redundant(struct isl_ctx *ctx, struct isl_tab *tab, int con); + +enum isl_ineq_type { + isl_ineq_error = -1, + isl_ineq_redundant, + isl_ineq_separate, + isl_ineq_cut, + isl_ineq_adj_eq, + isl_ineq_adj_ineq, +}; + +enum isl_ineq_type isl_tab_ineq_type(struct isl_ctx *ctx, struct isl_tab *tab, + isl_int *ineq); + +struct isl_tab_undo *isl_tab_snap(struct isl_ctx *ctx, struct isl_tab *tab); +int isl_tab_rollback(struct isl_ctx *ctx, struct isl_tab *tab, + struct isl_tab_undo *snap); + +struct isl_tab *isl_tab_relax(struct isl_ctx *ctx, + struct isl_tab *tab, int con); +struct isl_tab *isl_tab_select_facet(struct isl_ctx *ctx, + struct isl_tab *tab, int con); + +void isl_tab_dump(struct isl_ctx *ctx, struct isl_tab *tab, + FILE *out, int indent); + +#endif -- 2.7.4