def _build_dag(self, cgroups, prefix):
"""
A DAG representing the data dependences across the ClusterGroups within
a given scope.
"""
prefix = {i.dim for i in as_tuple(prefix)}
dag = DAG(nodes=cgroups)
for n, cg0 in enumerate(cgroups):
for cg1 in cgroups[n + 1:]:
# A Scope to compute all cross-ClusterGroup anti-dependences
rule = lambda i: i.is_cross
scope = Scope(exprs=cg0.exprs + cg1.exprs, rules=rule)
# Optimization: we exploit the following property:
# no prefix => (edge <=> at least one (any) dependence)
# to jump out of this potentially expensive loop as quickly as possible
if not prefix and any(scope.d_all_gen()):
dag.add_edge(cg0, cg1)
# Anti-dependences along `prefix` break the execution flow
# (intuitively, "the loop nests are to be kept separated")
# * All ClusterGroups between `cg0` and `cg1` must precede `cg1`
# * All ClusterGroups after `cg1` cannot precede `cg1`
elif any(i.cause & prefix for i in scope.d_anti_gen()):
for cg2 in cgroups[n:cgroups.index(cg1)]:
dag.add_edge(cg2, cg1)
for cg2 in cgroups[cgroups.index(cg1) + 1:]:
dag.add_edge(cg1, cg2)
break
# Any anti- and iaw-dependences impose that `cg1` follows `cg0`
# while not being its immediate successor (unless it already is),
# to avoid they are fused together (thus breaking the dependence)
# TODO: the "not being its immediate successor" part *seems* to be
# a work around to the fact that any two Clusters characterized
# by anti-dependence should have been given a different stamp,
# and same for guarded Clusters, but that is not the case (yet)
elif any(scope.d_anti_gen()) or\
any(i.is_iaw for i in scope.d_output_gen()):
dag.add_edge(cg0, cg1)
index = cgroups.index(cg1) - 1
if index > n and self._key(cg0) == self._key(cg1):
dag.add_edge(cg0, cgroups[index])
dag.add_edge(cgroups[index], cg1)
# Any flow-dependences along an inner Dimension (i.e., a Dimension
# that doesn't appear in `prefix`) impose that `cg1` follows `cg0`
elif any(not (i.cause and i.cause & prefix)
for i in scope.d_flow_gen()):
dag.add_edge(cg0, cg1)
# Clearly, output dependences must be honored
elif any(scope.d_output_gen()):
dag.add_edge(cg0, cg1)
return dag
def _build_dag(self, cgroups, prefix):
"""
A DAG captures data dependences between ClusterGroups up to the iteration
space depth dictated by ``prefix``.
Examples
--------
Consider two ClusterGroups `c0` and `c1`, and ``prefix=[i]``.
1) cg0 := b[i, j] = ...
cg1 := ... = ... b[i, j] ...
Non-carried flow-dependence, so `cg1` must go after `cg0`.
2) cg0 := b[i, j] = ...
cg1 := ... = ... b[i, j-1] ...
Carried flow-dependence in `j`, so `cg1` must go after `cg0`.
3) cg0 := b[i, j] = ...
cg1 := ... = ... b[i, j+1] ...
Carried anti-dependence in `j`, so `cg1` must go after `cg0`.
4) cg0 := b[i, j] = ...
cg1 := ... = ... b[i-1, j+1] ...
Carried flow-dependence in `i`, so `cg1` can safely go before or after
`cg0`. Note: the `j+1` in `cg1` has no impact -- the actual dependence
betweeb `b[i, j]` and `b[i-1, j+1]` is along `i`.
"""
prefix = {i.dim for i in as_tuple(prefix)}
dag = DAG(nodes=cgroups)
for n, cg0 in enumerate(cgroups):
for cg1 in cgroups[n + 1:]:
rule = lambda i: i.is_cross # Only retain dep if cross-ClusterGroup
scope = Scope(exprs=cg0.exprs + cg1.exprs, rules=rule)
# Optimization: we exploit the following property:
# no prefix => (edge <=> at least one (any) dependence)
# To jump out of this potentially expensive loop as quickly as possible
if not prefix and any(scope.d_all_gen()):
dag.add_edge(cg0, cg1)
continue
# Handle anti-dependences
if any(i.cause & prefix for i in scope.d_anti_gen()):
# Anti-dependences break the execution flow
# i) ClusterGroups between `cg0` and `cg1` must precede `cg1`
for cg2 in cgroups[n:cgroups.index(cg1)]:
dag.add_edge(cg2, cg1)
# ii) ClusterGroups after `cg1` cannot precede `cg1`
for cg2 in cgroups[cgroups.index(cg1) + 1:]:
dag.add_edge(cg1, cg2)
break
elif any(scope.d_anti_gen()):
dag.add_edge(cg0, cg1)
continue
# Flow-dependences along one of the `prefix` Dimensions can
# be ignored; all others require sequentialization
if any(not (i.cause and i.cause & prefix)
for i in scope.d_flow_gen()):
dag.add_edge(cg0, cg1)
continue
# Handle increment-after-write dependences
if any(i.is_iaw for i in scope.d_output_gen()):
dag.add_edge(cg0, cg1)
continue
return dag