For an array A[10][20] with two subscript locations s0 and s1, the
linear memory access is 20 * s0 + s1: a stride of 1 in subscript s0
- corresponds to a memory stride of 20. */
+ corresponds to a memory stride of 20.
+
+ OFFSET is a number of dimensions to prepend before the
+ subscript dimensions: s_0, s_1, ..., s_n.
+
+ Thus, the final linear expression has the following format:
+ 0 .. 0_{offset} | 0 .. 0_{nit} | 0 .. 0_{gd} | 0 | c_0 c_1 ... c_n
+ where the expression itself is:
+ c_0 * s_0 + c_1 * s_1 + ... c_n * s_n. */
static ppl_Linear_Expression_t
-build_linearized_memory_access (poly_dr_p pdr)
+build_linearized_memory_access (ppl_dimension_type offset, poly_dr_p pdr)
{
ppl_Linear_Expression_t res;
ppl_Linear_Expression_t le;
ppl_dimension_type first = pdr_subscript_dim (pdr, 0);
ppl_dimension_type last = pdr_subscript_dim (pdr, PDR_NB_SUBSCRIPTS (pdr));
Value size, sub_size;
- graphite_dim_t dim = pdr_dim (pdr);
+ graphite_dim_t dim = offset + pdr_dim (pdr);
ppl_new_Linear_Expression_with_dimension (&res, dim);
for (i = last - 1; i >= first; i--)
{
- ppl_set_coef_gmp (res, i, size);
+ ppl_set_coef_gmp (res, i + offset, size);
- ppl_new_Linear_Expression_with_dimension (&le, dim);
+ ppl_new_Linear_Expression_with_dimension (&le, dim - offset);
ppl_set_coef (le, i, 1);
ppl_max_for_le_pointset (PDR_ACCESSES (pdr), le, sub_size);
value_multiply (size, size, sub_size);
}
/* Set STRIDE to the stride of PDR in memory by advancing by one in
- loop DEPTH. */
+ time dimension DEPTH. */
static void
memory_stride_in_loop (Value stride, graphite_dim_t depth, poly_dr_p pdr)
{
ppl_Linear_Expression_t le, lma;
ppl_Constraint_t new_cstr;
- ppl_Pointset_Powerset_C_Polyhedron_t p1, p2;
- graphite_dim_t nb_subscripts = PDR_NB_SUBSCRIPTS (pdr);
ppl_dimension_type i, *map;
- ppl_dimension_type dim = pdr_dim (pdr);
- ppl_dimension_type dim_i = pdr_iterator_dim (pdr, depth);
- ppl_dimension_type dim_k = dim;
- ppl_dimension_type dim_L1 = dim + nb_subscripts + 1;
- ppl_dimension_type dim_L2 = dim + nb_subscripts + 2;
- ppl_dimension_type new_dim = dim + nb_subscripts + 3;
-
- /* Add new dimensions to the polyhedron corresponding to
- k, s0', s1',..., L1, and L2. These new variables are at
- dimensions dim, dim + 1,... of the polyhedron P1 respectively. */
- ppl_new_Pointset_Powerset_C_Polyhedron_from_Pointset_Powerset_C_Polyhedron
- (&p1, PDR_ACCESSES (pdr));
- ppl_Pointset_Powerset_C_Polyhedron_add_space_dimensions_and_embed
- (p1, nb_subscripts + 3);
-
- lma = build_linearized_memory_access (pdr);
- ppl_set_coef (lma, dim_L1, -1);
- ppl_new_Constraint (&new_cstr, lma, PPL_CONSTRAINT_TYPE_EQUAL);
- ppl_Pointset_Powerset_C_Polyhedron_add_constraint (p1, new_cstr);
-
- /* Build P2. */
- ppl_new_Pointset_Powerset_C_Polyhedron_from_Pointset_Powerset_C_Polyhedron
- (&p2, p1);
- map = ppl_new_id_map (new_dim);
- ppl_interchange (map, dim_L1, dim_L2);
- ppl_interchange (map, dim_i, dim_k);
- for (i = 0; i < PDR_NB_SUBSCRIPTS (pdr); i++)
- ppl_interchange (map, pdr_subscript_dim (pdr, i), dim + i + 1);
- ppl_Pointset_Powerset_C_Polyhedron_map_space_dimensions (p2, map, new_dim);
- free (map);
-
- /* Add constraint k = i + 1. */
- ppl_new_Linear_Expression_with_dimension (&le, new_dim);
- ppl_set_coef (le, dim_i, 1);
- ppl_set_coef (le, dim_k, -1);
- ppl_set_inhomogeneous (le, 1);
- ppl_new_Constraint (&new_cstr, le, PPL_CONSTRAINT_TYPE_EQUAL);
- ppl_Pointset_Powerset_C_Polyhedron_add_constraint (p2, new_cstr);
- ppl_delete_Linear_Expression (le);
- ppl_delete_Constraint (new_cstr);
+ ppl_Pointset_Powerset_C_Polyhedron_t p1, p2, sctr;
+ graphite_dim_t nb_subscripts = PDR_NB_SUBSCRIPTS (pdr) + 1;
+ poly_bb_p pbb = PDR_PBB (pdr);
+ ppl_dimension_type offset = pbb_nb_scattering_transform (pbb)
+ + pbb_nb_local_vars (pbb)
+ + pbb_dim_iter_domain (pbb);
+ ppl_dimension_type offsetg = offset + pbb_nb_params (pbb);
+ ppl_dimension_type dim_sctr = pbb_nb_scattering_transform (pbb)
+ + pbb_nb_local_vars (pbb);
+ ppl_dimension_type dim_L1 = offset + offsetg + 2 * nb_subscripts;
+ ppl_dimension_type dim_L2 = offset + offsetg + 2 * nb_subscripts + 1;
+ ppl_dimension_type new_dim = offset + offsetg + 2 * nb_subscripts + 2;
+
+ /* The resulting polyhedron should have the following format:
+ T|I|T'|I'|G|S|S'|l1|l2
+ where:
+ | T = t_1..t_{dim_sctr}
+ | I = i_1..i_{dim_iter_domain}
+ | T'= t'_1..t'_{dim_sctr}
+ | I'= i'_1..i'_{dim_iter_domain}
+ | G = g_1..g_{nb_params}
+ | S = s_1..s_{nb_subscripts}
+ | S'= s'_1..s'_{nb_subscripts}
+ | l1 and l2 are scalars.
+
+ Some invariants:
+ offset = dim_sctr + dim_iter_domain + nb_local_vars
+ offsetg = dim_sctr + dim_iter_domain + nb_local_vars + nb_params. */
+
+ /* Construct the T|I|0|0|G|0|0|0|0 part. */
+ {
+ ppl_new_Pointset_Powerset_C_Polyhedron_from_C_Polyhedron
+ (&sctr, PBB_TRANSFORMED_SCATTERING (pbb));
+ ppl_Pointset_Powerset_C_Polyhedron_add_space_dimensions_and_embed
+ (sctr, 2 * nb_subscripts + 2);
+ ppl_insert_dimensions_pointset (sctr, offset, offset);
+ }
+
+ /* Construct the 0|I|0|0|G|S|0|0|0 part. */
+ {
+ ppl_new_Pointset_Powerset_C_Polyhedron_from_Pointset_Powerset_C_Polyhedron
+ (&p1, PDR_ACCESSES (pdr));
+ ppl_Pointset_Powerset_C_Polyhedron_add_space_dimensions_and_embed
+ (p1, nb_subscripts + 2);
+ ppl_insert_dimensions_pointset (p1, 0, dim_sctr);
+ ppl_insert_dimensions_pointset (p1, offset, offset);
+ }
+
+ /* Construct the 0|0|0|0|0|S|0|l1|0 part. */
+ {
+ lma = build_linearized_memory_access (offset + dim_sctr, pdr);
+ ppl_set_coef (lma, dim_L1, -1);
+ ppl_new_Constraint (&new_cstr, lma, PPL_CONSTRAINT_TYPE_EQUAL);
+ ppl_Pointset_Powerset_C_Polyhedron_add_constraint (p1, new_cstr);
+ }
+
+ /* Now intersect all the parts to get:
+ T|I|0|0|G|0|0|0 |0
+ 0|I|0|0|G|S|0|0 |0
+ 0|0|0|0|0|S|0|l1|0
+ ------------------
+ T|I|0|0|G|S|0|l1|0. */
+
+ ppl_Pointset_Powerset_C_Polyhedron_intersection_assign (p1, sctr);
+ ppl_delete_Pointset_Powerset_C_Polyhedron (sctr);
+
+ /* Build P2, which would have the following form:
+ 0|0|T'|I'|G|0|S'|0|l2
+
+ P2 is built, by remapping the P1 polyhedron:
+ T|I|0|0|G|S|0|l1|0
+
+ using the following mapping:
+ T->T'
+ I->I'
+ S->S'
+ l1->l2. */
+ {
+ ppl_new_Pointset_Powerset_C_Polyhedron_from_Pointset_Powerset_C_Polyhedron
+ (&p2, p1);
+
+ map = ppl_new_id_map (new_dim);
+
+ /* T->T' and I->I'. */
+ for (i = 0; i < offset; i++)
+ ppl_interchange (map, i, i + offset);
+
+ /* l1->l2. */
+ ppl_interchange (map, dim_L1, dim_L2);
+
+ /* S->S'. */
+ for (i = 0; i < nb_subscripts; i++)
+ ppl_interchange (map, offset + offsetg + i,
+ offset + offsetg + nb_subscripts + i);
+
+ ppl_Pointset_Powerset_C_Polyhedron_map_space_dimensions (p2, map, new_dim);
+ free (map);
+ }
+
+ /* Add equalities:
+ | t1 = t1'
+ | ...
+ | t_{depth-1} = t'_{depth-1}
+ | t_{depth+1} = t'_{depth+1}
+ | ...
+ | t_{dim_sctr} = t'_{dim_sctr}
+
+ This means that all the time dimensions are equal except for
+ depth, where we will add t_{depth} = t'_{depth} + 1 in the next
+ step. */
+ for (i = 0; i < dim_sctr; i++)
+ if (i != depth)
+ {
+ ppl_new_Linear_Expression_with_dimension (&le, new_dim);
+ ppl_set_coef (le, i, 1);
+ ppl_set_coef (le, i + offset, -1);
+ ppl_new_Constraint (&new_cstr, le, PPL_CONSTRAINT_TYPE_EQUAL);
+ ppl_Pointset_Powerset_C_Polyhedron_add_constraint (p2, new_cstr);
+ ppl_delete_Linear_Expression (le);
+ ppl_delete_Constraint (new_cstr);
+ }
+
+ /* Add equality : t_{depth} = t'_{depth} + 1.
+ This is the core part of this alogrithm, since this
+ constraint asks for the memory access stride (difference)
+ between two consecutive points in time dimensions. */
+ {
+ ppl_new_Linear_Expression_with_dimension (&le, new_dim);
+ ppl_set_coef (le, depth, 1);
+ ppl_set_coef (le, depth + offset, -1);
+ ppl_set_inhomogeneous (le, 1);
+ ppl_new_Constraint (&new_cstr, le, PPL_CONSTRAINT_TYPE_EQUAL);
+ ppl_Pointset_Powerset_C_Polyhedron_add_constraint (p2, new_cstr);
+ ppl_delete_Linear_Expression (le);
+ ppl_delete_Constraint (new_cstr);
+ }
/* P1 = P1 inter P2. */
- ppl_Pointset_Powerset_C_Polyhedron_intersection_assign (p1, p2);
- ppl_delete_Pointset_Powerset_C_Polyhedron (p2);
+ {
+ ppl_Pointset_Powerset_C_Polyhedron_intersection_assign (p1, p2);
+ ppl_delete_Pointset_Powerset_C_Polyhedron (p2);
+ }
/* Maximise the expression L2 - L1. */
- ppl_new_Linear_Expression_with_dimension (&le, new_dim);
- ppl_set_coef (le, dim_L2, 1);
- ppl_set_coef (le, dim_L1, -1);
- ppl_max_for_le_pointset (p1, le, stride);
- ppl_delete_Linear_Expression (le);
+ {
+ ppl_new_Linear_Expression_with_dimension (&le, new_dim);
+ ppl_set_coef (le, dim_L2, 1);
+ ppl_set_coef (le, dim_L1, -1);
+ ppl_max_for_le_pointset (p1, le, stride);
+ ppl_delete_Linear_Expression (le);
+ }
}
-
-/* Returns true when it is profitable to interchange loop at DEPTH1
- and loop at DEPTH2 with DEPTH1 < DEPTH2 for PBB.
+/* Returns true when it is profitable to interchange time dimensions DEPTH1
+ and DEPTH2 with DEPTH1 < DEPTH2 for PBB.
Example:
| i j N a s0 s1 1
| 0 0 0 0 100 1 0
+ TODO: the shown format is not valid as it does not show the fact
+ that the iteration domain "i j" is transformed using the scattering.
+
Next, to measure the impact of iterating once in loop "i", we build
a maximization problem: first, we add to DR accesses the dimensions
k, s2, s3, L1 = 100 * s0 + s1, L2, and D1: polyhedron P1.
| 0 0 0 0 100 1 0 0 0 -1 0 0 0 = 0 L1 = 100 * s0 + s1
Then, we generate the polyhedron P2 by interchanging the dimensions
- (s0, s2), (s1, s3), (L1, L2), (i0, i)
+ (s0, s2), (s1, s3), (L1, L2), (k, i)
| i j N a s0 s1 k s2 s3 L1 L2 D1 1
| 0 0 0 1 0 0 0 0 0 0 0 0 -5 = 0 alias = 5
and finally we maximize the expression "D1 = max (P1 inter P2, L2 - L1)".
- For determining the impact of one iteration on loop "j", we
+ Similarly, to determine the impact of one iteration on loop "j", we
interchange (k, j), we add "k = j + 1", and we compute D2 the
maximal value of the difference.
scattering. */
static void
-pbb_interchange_loop_depths (graphite_dim_t depth1, graphite_dim_t depth2, poly_bb_p pbb)
+pbb_interchange_loop_depths (graphite_dim_t depth1, graphite_dim_t depth2,
+ poly_bb_p pbb)
{
ppl_dimension_type i, dim;
ppl_dimension_type *map;
ppl_Polyhedron_t poly = PBB_TRANSFORMED_SCATTERING (pbb);
- ppl_dimension_type dim1 = psct_iterator_dim (pbb, depth1);
- ppl_dimension_type dim2 = psct_iterator_dim (pbb, depth2);
+ ppl_dimension_type dim1 = psct_dynamic_dim (pbb, depth1);
+ ppl_dimension_type dim2 = psct_dynamic_dim (pbb, depth2);
ppl_Polyhedron_space_dimension (poly, &dim);
map = (ppl_dimension_type *) XNEWVEC (ppl_dimension_type, dim);
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